UNIVERSIDADE FEDERAL FLUMINENSE INSTITUTO DE …um dos modelos mais usados na Din^amica de Fluidos Computacionais (CFD) para simular caracter sticas de uxo m edio para condi˘c~oes

UNIVERSIDADE FEDERAL FLUMINENSE

INSTITUTO DE MATEMATICA E ESTATISTICA

POS-GRADUACAO EM MATEMATICA

PITAGORAS PINHEIRO DE CARVALHO

THEORETICAL AND NUMERICAL ANALYSIS

AND CONTROL OF SOME PDEs

NITEROI

December 2017

PITAGORAS PINHEIRO DE CARVALHO

THEORETICAL AND NUMERICAL ANALYSIS

AND CONTROL OF SOME PDEs

Thesis presented by Pitagoras Pinheiro de

Carvalho to the Postgraduate Program in

Ph.D. in Mathematics – Universidade Federal

Fluminense, as partial fulfillment of the re-

quirements for the degree of Doctor. Re-

search Line: Partial Differential Equations.

Advisor: Juan Bautista Lımaco Ferrel

Co-advisor: Enrique Fernandez-Cara

NITEROI

December 2017

Acknowledgements

To thank is not an easy task, not by the act itself, but to do so without fear of

forgetting those who inspired me. For all the people that supported me, even if they

are not explicitly mentioned, I am very grateful.

First of all, I would like to thank my parents and my sister who started the

mathematical stimulus by choosing (unintentionally) my name. Furthermore, by

their love, support and encouragement of my personal and academic choices.

I thank Professor Juan Bautista Lımaco Ferrel, my teacher advisor in this thesis,

for including me in his active and motivating research group. Also, for your own

patience, disposition, enthusiasm and friendship, which were so important for my

academic education.

To teacher and advisor Enrique Fernandez-Cara I owe a special thanks for his

attention and hospitality throughout my stay in Seville. His knowledge, patience

and diversity of insights made every conversation very valuable. In addition, thanks

for providing me work opportunities in close collaboration with other teachers. In

particular, the possibility to work with Anna Doubova and Jairo Rocha added a lot

to my formation. I really appreciate all the availability, kindness and interest in

discussing math of these teachers.

To the professors of graduation and post-graduation, who supported and trust

in my dedication to this goal, especially for the following: Newton Luis, Marcondes

Clark, Nelson Neri Castro and Haroldo Clark.

To best friends: Jose Arimatea, Diego Araujo, Felipe Chaves, Jose Francisco,

Lecio Gustavo, Maurıcio Santos, Olımpio Sa and Serjao, who are always present in

my life with support, counseling and motivation. I think of them as an extension of

my family.

To my friends of UFF: Andre, Laurent, Dany Nina, Miguel Nunez, Bruno, Jacque-

line, Raquel and Renato.

A special thanks to my wife, Vanessa Carvalho, who has always supported me

and made my dreams come true. Certainly, it made a considerable difference to my

daily motivation.

Finally, thanks to Foundation Euclides da Cunha, CAPES, IMUS and UESPI for

financial support.

Abstract

This dissertation presents theoretical and numerical results for some Partial Dif-

ferential Equations (PDEs). For the distinction of the topics addressed during the

development of this work, we have divided its goals into four parts. Precisely, in the

first two chapters, we focus on the development of some results associated with the

k–ε model, and in the last two chapters, some numerical results related to the heat

and wave equations. The k–ε turbulence model is one of the most used models in

Computational Fluid Dynamics (CFD) to simulate medium flow characteristics for

turbulent flow conditions. It is a model formed by equations of the type Navier-

Stokes (called Reynolds equations) coupled to two equations that in general describe

the evolution in time, the transport, the diffusion and the generation of turbulence.

In Chapter 1, we prove a renormalized weak solution result for a simplified model

from the k–ε model. In the following chapters, we present some controllability results

for some models. More specifically, we proof the existence of controls which drive

the average field of velocity from a prescribed initial data to a desired final state in a

positive time given. Furthermore, we present in Chapter 2 a result of the local null

partial control for a simplified model from the k–ε model. On the other hand, aiming

the numerical development of hierarchical control problems, Chapters 3 and 4 are

presented. These two chapters show the theoretical and numerical developments that

were performed, as well as Freefem++ which is a software that was used to present

some graphical results of experiments associated with the problems. Particularly

in Chapter 3, we develop numerical results in optimal controls for heat equations

(linear and semi-linear cases) in combination with the notions of cooperative and

non-cooperative games, according to Pareto and Nash strategies, respectively. Moti-

vated by the results found in Chapter 3, we developed Chapter 4, where we obtained

similar numerical results of optimal control in bi-objective problems associated with

the notions of Nash and Pareto equilibrium for wave equations.

Keywords: Partial Differential Equations, Optimal Multi-Objective Control, Nash

and Pareto Equilibrium, Renomalized Solution, Turbulence Models, Navier-Stokes,

Partial Null Control, Nonlinear Parabolic PDE, Carleman Estimates.

Resumo

Nesta tese apresentamos resultados teoricos e numericos para algumas Equacoes

Diferenciais Parciais (EDPs). Pela distincao dos temas abordados durante o desen-

volvimento do presente trabalho, fracionamos os objetivos em quatro partes. Mais

precisamente, nos dois primeiros capıtulos, focamos no desenvolvimento de alguns

resultados associados ao modelo k-ε, e nos dois ultimos capıtulos, alguns resultados

numericos associados as equacoes de calor e onda. O modelo de turbulencia k-ε e

um dos modelos mais usados na Dinamica de Fluidos Computacionais (CFD) para

simular caracterısticas de fluxo medio para condicoes de fluxo turbulento. E um mod-

elo formado por equacoes do tipo Navier-Stokes (chamadas equacoes de Reynolds)

acopladas a duas equacoes que descreve de uma maneira geral a evolucao no tempo,

o transporte, a difusao e a geracao de turbulencia. No capıtulo 1, provamos um

resultado de solucao fraca renormalizada para um modelo simplificado a partir do

modelo k-ε. Nos capıtulos seguintes, apresentamos alguns resultados de controlabili-

dade para alguns modelos. Mais precisamente, prova-se a existencia de controles que

conduzem o campo medio de velocidades de um dado inicial prescrito, a um estado

final desejado, em um tempo positivo dado. Nessa linha, apresentamos no capıtulo 2

um resultado de controle parcial nulo local para um modelo simplificado a partir do

modelo k-ε. Por outro lado, ambicionando o desenvolvimento numerico de problemas

de controle hierarquico, apresentamos os capıtulos 3 e 4. Nesses capıtulos sao realiza-

dos desenvolvimentos teoricos e numericos, e utilizamos o software Freefem++ para

apresentar alguns resultados graficos de experimentos associados aos problemas. Em

particular, no capıtulo 3, desenvolvemos resultados numericos em controle optimo

para equacoes do calor (casos lineares e semilineares) em combinacao com as nocoes

de jogos cooperativos e nao-cooperativos, segundo as estrategias de Pareto e Nash

(respectivamente). Motivados pelos resultados obtidos no capıtulo 3, desenvolve-

mos o capıtulo 4, onde obtivemos similares resultados numericos de controle optimo

em problemas bi-objetivos associados as nocoes de equılibrio de Nash e Pareto para

equacoes de ondas.

Palavras-chave: Equacoes Diferenciais Parciais, Controle Otimo Multi-objetivo,

Equilibrio de Nash e Pareto, Solucao Renormalizada, Modelos de Turbulencia, Navier-

Stokes, Controle Nulo Parcial, EDP Parabolica nao-linear, Estimativas de Carleman.

Contents

Introduction 7

1 Weak-Renormalized Solutions for a Simplified k-ε Model of Turbu-

lence 26

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.2.1 Notation and spaces . . . . . . . . . . . . . . . . . . . . . . . 30

1.2.2 Hypotheses and main result . . . . . . . . . . . . . . . . . . . 32

1.3 Some auxiliary problems . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.4 Proof of Theorem 1.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.4.1 An auxiliary regularized problem . . . . . . . . . . . . . . . . 34

1.4.2 Some a priori estimates . . . . . . . . . . . . . . . . . . . . . 37

1.4.3 Passage to the limit and conclusions . . . . . . . . . . . . . . 39

2 On the Control of a Simplified k-ε Model of Turbulence 43

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.2 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.2.1 Some Carleman estimates . . . . . . . . . . . . . . . . . . . . 48

2.2.2 The null controllability of the linear system (2.7) . . . . . . . 52

2.2.3 Some additional estimates . . . . . . . . . . . . . . . . . . . 53

2.3 Proof of Theorem 2.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.4 Some additional comments and questions . . . . . . . . . . . . . . . . 64

3 On the Computation of Nash and Pareto Equilibria for some Bi-

Objective Control Problems 66

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5

3.2 Formulation of the Problems . . . . . . . . . . . . . . . . . . . . . . . 68

3.3 Computation of Nash Equilibria for (3.1) . . . . . . . . . . . . . . . . 71

3.3.1 A Formulation Equivalent to (3.5) . . . . . . . . . . . . . . . . 71

3.3.2 A Fixed-Point Algorithm for Solving (3.10)–(3.12) . . . . . . . 74

3.3.3 Reduction to Finite Dimension . . . . . . . . . . . . . . . . . 75

3.3.4 Fixed-Point Solution of the Discretized Linear Problem . . . . 76

3.4 Computation of Pareto Equilibria for (3.1) . . . . . . . . . . . . . . . 77

3.4.1 A Formulation Equivalent to (3.7) and a Fixed-Point Algorithm 77

3.4.2 Finite Dimensional Approximation and Solution . . . . . . . . 79



3.7 Some Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . 85

3.7.1 Description of the Problems . . . . . . . . . . . . . . . . . . . 86

3.7.2 Computation of Nash and Pareto Equilibria in the Linear Case 89

3.7.3 Computation of Nash and Pareto Equilibria in the Semilinear

Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.8.1 Existence and uniqueness of a solution to (3.5) . . . . . . . . . 91

3.8.2 The convergence of ALG 1 . . . . . . . . . . . . . . . . . . . 97

3.8.3 Existence and uniqueness of a solution to (3.22)–(3.24) . . . . 98

4 On the Computation of Nash and Pareto Equilibria for some Bi-

Objective Optimal Control Problems for the Wave Equation 100

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.2 The Problems and Their Motivations . . . . . . . . . . . . . . . . . . 103


4.3.1 Equivalent Formulation of the Optimality System (4.5) . . . . 106

4.3.2 A Fixed-Point Algorithm for Solving (4.11)–(4.13) . . . . . . . 108

4.3.3 Reduction to Finite Dimension . . . . . . . . . . . . . . . . . 110

4.3.4 Fixed-Point Solution of the Discretized Linear Problem . . . . 111


4.4.1 A Formulation Equivalent to (4.7) and a Fixed-Point Algorithm112

4.4.2 Finite Dimensional Approximation and Solution . . . . . . . . 114



4.7 Some Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.7.1 Description of the Problems . . . . . . . . . . . . . . . . . . . 121

4.7.2 Computation of Nash and Pareto Equilibria in the Linear Case 124

4.7.3 Computation of Nash and Pareto Equilibria in the Semilinear

Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.8.1 Existence and uniqueness of a solution to (4.5) . . . . . . . . . 129

4.8.2 The convergence of ALG 1 . . . . . . . . . . . . . . . . . . . 132

4.8.3 Existence and uniqueness of a solution to the semilinear sys-

tem (4.23)–(4.25) . . . . . . . . . . . . . . . . . . . . . . . . . 133

References 133

Introduction

In this PhD dissertation, we present some existence, controllability and numerical

analysis results (in terms that will be explained hereafter) of several problems of

partial differential equations. In this introduction, we briefly describe the questions

that will be addressed in the following chapters.

First, we will start with a brief historical introduction to the mathematical theory

of fluid dynamics. This began in the 17th century, with the advances of Sir Isaac

Newton, who applied his recent developed laws of mechanics to the movement of flu-

ids. Leonhard Paul Euler (1755) then wrote partial differential equations describing

an ideal fluid, i.e. a fluid where no dissipation is produced due to the interaction

between molecules (inviscid fluid; such equations known as Euler equations). Later,

Claude-Louis Henri Navier (1822) and, independently, George Gabiel Stokes (1845)

considered a viscous fluid and obtained the equations we know today as Navier-Stokes

Equations.

A considerable challenge for researchers working with fluids is to understand well

the phenomenon of turbulence. To summarize, we can say that turbulent fluids

are characterized by flows where the particles are mixed in a non-linear form over

time and space, i.e. chaotically. The main issues related to turbulence have been

raised since the beginning of the 20th century, and a major number of empirical and

heuristic results have been obtained, mainly motivated by engineering applications.

In mathematics, Jean Leray’s pioneering works on the equations of Navier Stokes

between 1933-1934 appear. Leray speculated that the turbulence appears due to

the formation of point or lines of vortices in which some component of the velocity

becomes infinite. To deal with this situation, Leray introduces the concept of weak

solution (non-classic solution) to the Navier-Stokes equations.

The Newtonian, homogeneous and incompressible flow of a fluid is governed by

the Navier-Stokes equations which, in Eulerian form and with vector notation can

7

8

be written as follows: vt − ν∆v + (v · ∇)v +∇p = f in Ω× (0, T ),

∇ · v = 0 in Ω× (0, T ).(1)

In [25], turbulent flows are analyzed using the RANS (Reynolds Averaged Navier-

Stokes) method and the k–ε model is deduced. This will be the focus of the results

in the first two chapters of this dissertation. For the deduction of the k–ε model,

Reynolds decomposition is used to separate the mean and the floating part of each

quantity, i.e. v := v+ v′ and p := p+ p′ where v and p are the mean parts, v′ and p′

are the turbulent zero-mean pertubations. For a no-slip fluid in Ω × (0, T ) starting

from initial conditions at t = 0, the resulting equations are the following, where for

simplicity we have put v and p instead of v and p :

vt + (v · ∇)v −∇ · ((ν + cνk2

ε)Dv) +∇(p− 2

3k) = f in Q,

∇ · v = 0 in Q,

kt + v∇k −∇ · (κ+ cκk2

ε∇k) = cν

k2

ε|Dv|2 − ε in Q,

εt + v∇ε−∇ · (β + cεk2

ε∇ε) = cηk|Dv|2 − a

ε2

kin Q,

v = 0 on Σ,

∂k

∂n= 0 and

∂ε

∂n= 0 on Σ,

v(x, 0) = v0(x), k(x, 0) = k0(x) and ε(x, 0) = ε0(x) in Ω.

(2)

Here, v = v(x, t), k = k(x, t), ε = ε(x, t) and p = p (x, t), represent the mean velocity

field, the turbulent kinetic energy, the rate of turbulent energy dissipation, and the

average pressure of a fluid in turbulent regime, respectively. The turbulent viscosity

is given by νt = cνk2

ε.

We also have that Dv := ∇v+∇Tv is the symmetric part of the spatial gradient

of v (also called deformation tensor). In addition, νt appears because of Boussinesq

hypothesis, which establishes that the Reynolds tensor, responsible for causing the

turbulence, R := −v′ ⊗ v′ can be replaced by

R := νtDv +2

3kI.

9

In Chapters 1 and 2, we will present existence and controllability results for a

simplified version of (2) that will be determined later.

On the other hand, as important as theoretical analysis, are numerical aspects

associated with PDEs. The field of numerical analysis precedes the invention of the

computer in centuries. Linear interpolation has been used for over 2000 years. Great

mathematicians have worked in the past with numerical analysis, which is obviously

perceived by the name of important algorithms such as Newton Method, Lagrange

Polynomial, Gaussian Elimination or Euler Method.

With the appearance and evolution of modern computers and the subsequent

development of programming languages, several scientific and industrial problems

have boosted the field of computational simulations. Since the 1980s, these have

been increasing in relevance due to several factors; mainly those related to progress

in science, informatics, and engineering.

Thanks to computational technologies and numerical methods advances (Finite

Differences, Elements and Volumes, among others), it is possible to simulate the

behavior of a vast class of applications of interest, allowing us to treat increasingly

more sophisticated problems and optimize costs.

Simulations of numerical problems associated with natural phenomena are of

great importance and usefulness for describing practical and real situations. Thus,

Chapters 3 and 4 are presented and have the goal to expose numerical results for

some multi-objective control problems governed by partial differential equations (in

a sense that will be explained later). In all cases, the state will be the solution of

the heat system

yt −∆y + F (y) = f + v11O1 + v21O2 in Q,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x) in Ω,

(3)

10

our the wave system

ytt −∆y + F (y) = f1O + v11O1 + v21O2 in QT ,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x), yt(x, 0) = y1(x) in Ω.

(4)

For both systems, we will separately the case F (y) ≡ 0 (linear problems) and the

more general case of a non-vanishing globally Lipschitz-continuous F : R 7→ R .

The variables v1 and v2 that appear in systems (3) and (4) are controls associated

to equilibrium problems introduced (in economics) by Nash [62] and Pareto [64]. We

will fix two different objectives or criteria. Then, our main interest will be to compute

numerically equilibria of one of the following kinds:

• A Nash Equilibrium (Competitive): It consists of a combination of strategies

(one for each player) such that no player improves their gains by altering their

strategy while the other players keep their strategies unchanged.

• A Pareto Equilibrium (Cooperative): Is a strategy for multiple player’s such

that it is impossible to improve one player’s situation without worsening that

of another; in other words, there is no way to make all players involved in a

situation improve their earnings.

We will be more specific below on the problems addressed in this dissertation.

Thus, we provide now a summary.

Chapter 1

Weak-Renormalized Solutions for a Simplified k–ε Model of Turbulence

Let Ω ⊂ RN (where N = 2 or N = 3) be a connected bounded open set with a C2

boundary (∂Ω) and let ω ⊂ Ω be a (small) nonempty open set. Let T > 0 be given

and let us consider the cylindrical domain Q = Ω × (0, T ), with lateral boundary

Σ = ∂Ω× (0, T ).

In this chapter, our goal is to get a solution result to the simplified k − ε model.

For this, firstly we present the concept of renormalized solutions (since the concept

11

of weak solution is not satisfied) and after we get a solution result (in that sense

renormalized) for the simplified k − ε model.

To deduce a simplified system for the previous model k − ε in (2), we set

ϕ :=k2

ε

and assume that ϕ depends only on t.

After replacing of ϕ in the system, for technical reasons, we simplify a little more

the system and suppose that cη = 2cν and a > 2. Accordingly, we find that ϕ, k, v

and q satisfy:

ϕt +

(2cκ|Ω|

∫Ω

|∇k|2

|α + k|2dx

)ϕ2 =

(a− 2)

|Ω|

(∫Ω

k dx

)in (0, T ), (5)

kt + v∇k −∇ · ((κ+ cκϕ)∇k) +k2

ϕ= cνϕ|D(v)|2 in Q, (6)

vt + (v·∇)v −∇ · ((ν + cνϕ)D(v)) +∇q = f , div v = 0 in Q, (7)

∂k

∂n= 0 and v = 0 on Σ, (8)

ϕ(0) = ϕ0, k(x, 0) = k0(x), v(x, 0) = v0(x) in Ω, (9)

where α > 0, ϕ0, k0 and v0 are again given, ϕ0 > 0 and k0 ≥ 0 a.e.

In the equation (6), needs a special treatment due to the nonlinear right-hand side,

that only belongs to L1(Q) since, in general, D(v) only belongs to L2(Q)N×N . Thus,

we did not achieve a weak solution result in the distributions sense for the system

(5)-(9). For this reason, the good concepts of solution involves renormalization.

Renormalized solutions to PDEs were first introduced by DiPerna and P.-L. Li-

ons [36, 35] in the context of Boltzmann-like equations. Later, they have also been

considered in other situations; let us mention in particular the contributions by

Blanchard, Boccardo, Murat and their co-workers in the framework of second-order

elliptic and parabolic PDEs.

For a better understanding, we introduce V = v ∈ C∞0 (Ω)N : ∇ · v = 0; we will

denote the closures of V in L2(Ω)N and H10 (Ω)N respectively by H and V . Then, H

and V are Hilbert spaces for the corresponding norms and one has

H = v ∈ L2(Ω)N : ∇ · v = 0 in Ω, v · n = 0 on ∂Ω

and

V = v ∈ H10 (Ω)N : ∇ · v = 0 in Ω .

12

We will also have to consider the following set:

L(0, T ; Ω) := u ∈ L∞(0, T ;L1(Ω)) : TR(u) ∈ L2(0, T ;H10 (Ω)) ∀R > 0,

limn→+∞

1

n

∫An(u)

|∇u|2 dx dt = 0.

Here and in the sequel, An(u) stands for the set

An(u) := (x, t) ∈ Q : n ≤ |u(x, t)| ≤ 2n.

And assume that the following hypotheses hold:

(Θ) f ∈ L2(Q)N , ϕ0 ∈ R+, v0 ∈ H, k0 ∈ L1(Ω) with k0 ≥ 0 a.e.

Thus, we introduce the definition of a weak-renormalized solution to our

problem:

Definition 0.0.1 It will be said that (ϕ, k, v) is a (weak-renormalized) solution to

(5)–(9) if the following conditions are satisfied:

1. ϕ ∈ H1(0, T ), k ∈ L(0, T ; Ω) and v ∈ L2(0, T ;V ) ∩ L∞(0, T ;H).

2. ϕ solves the ODE (5) and ϕ(0) = ϕ0.

3. k solves (6) in the renormalized sense, that is: for any β ∈ W 2,∞(R) such

that Supp β′ is compact and for any η ∈ C1([0, T ];H10 (Ω)) ∩ L∞(Q) such that

η(x, T ) ≡ 0, we have

−∫∫

Q

β(k) ηt dx dt+

∫∫Q

(κ+ ckϕ)∇β(k) · ∇η dx dt

+

∫∫Q

(v · ∇)β(k) · η dx dt−∫

Ω

β(k0) η(x, 0) dx (10)

=

∫∫Q

β′(k)

[(cνϕD(u) : D(u) − k2

ϕ

)−(κ+ ckϕ

)∇β′(k) · ∇k

]· η dx dt.

4. v solves (7) in the usual weak sense (together with some q ∈ D′(Q)) and

v|t=0 = v0.

The main result in this chapter is:

13

Theorem 0.0.1 Assume that N = 2 and the hypothesis (Θ) holds. Then, there

exists at least one solution (ϕ, k, v) to (5)–(9).

The proof of this result (for two dimensional flows) is split into three main steps:

• First: Some auxiliary problems are considered;

• Second: An auxiliary regularized problem is introduced;

• Third: Some a priori estimates and passage to the limit are obtained.

It is readily seen that the previous proof of theorem (0.0.1) does not work in the

case N = 3.

The results of this work are found in [20].

Chapter 2

On the control of a simplified k-ε model of turbulence

As in Chapter 1, let Ω ⊂ RN be a bounded connected open set with regular

boundary ∂Ω (N = 2 or N = 3); let ω ⊂ Ω be a (small) nonempty open set, let

T > 0 be given and let us consider the cylindrical domain Q = Ω × (0, T ), with

lateral boundary Σ = ∂Ω× (0, T ).

In this chapter, we will investigate the partial local null controllability properties

of the simplified k − ε model of turbulence.

For this, firstly consider in the system (2) f = u1ω(with u ∈ L2(ω × (0, T ))N

)and replace

φ0 :=k2

ε

where φ0 is a function only in t.

As a consequence, we have the simplified system:

14

vt + (v · ∇)v −∇ · ((ν + cνφ0)Dv) +∇q = u1ω in Q,

∇ · v = 0 in Q,

kt + (v · ∇)k −∇ · ((κ+ c0φ0)∇k) +k2

φ0

= cνφ0|Dv|2 in Q,

v = 0 on Σ,

∂k

∂n= 0 on Σ,

v(x, 0) = v0(x) and k(x, 0) = k0(x) in Ω,

(11)

and φ0,t = −2c0

|Ω|

(∫Ω

|∇k|2

k2dx

)φ2

0 +a− 2

|Ω|

(∫Ω

k dx

)in (0, T ),

φ0(0) = φ00.

(12)

Analogous to Chapter 1, let us set V = v ∈ C∞0 (Ω)N : ∇· v = 0; we will denote

the closures of V in L2(Ω)N and H10 (Ω)N respectively by H and V , that are Hilbert

spaces for the corresponding norms, where

H = v ∈ L2(Ω)N : ∇ · v = 0 in Ω, v · n = 0 on ∂Ω

and

V = v ∈ H10 (Ω)N : ∇ · v = 0 in Ω .

In (11)–(12), u is the control and (v, q, k, φ0) is the state.

The main goal of this chapter is to prove a partial local null controllability result

for this simplified system.

Obtain a partial local null controllability result for the (11)–(12) basically boils

down to finding a control u acting only in the first equation in (11), such that for

very small initial data v0, the solution v (in time T > 0) is taken to the desired goal

v(x, T ) = 0.

Formally, we have:

15

Definition 0.0.2 Let k0 and φ00 be given, with

k0 ∈ H1(Ω), k0 ≥ 0 a.e., φ00 ∈ R+. (13)

It will be said that (11)–(12) is partially locally null-controllable at time T if there

exists ε > 0 such that, for any v0 ∈ V with

‖v0‖H10≤ ε,

there exist controls u ∈ L2(ω × (0, T ))N and associated states (v, q) satisfying (11)

and

v(x, T ) = 0 in Ω. (14)

The main result is the following:

Theorem 0.0.2 For any k0 and φ0 satisfying (13) and any T > 0, the nonlinear

system (11)–(12) is partially locally null-controllable at T .

For the proof, we have to employ several different techniques, in the following

order:

• First: We rewrite the partial null controllability problem as a fixed-point equa-

tion;

• Second: We apply an Inverse Mapping Theorem in Hilbert spaces, to proof of

the locally null-controllability condition;

• Third: We apply the Schauder’s Fixed-Point Theorem, to proof the partially

locally null-controllability condition.

The results of this work are found in [21].

16

Chapter 3

On the Computation of Nash and Pareto Equilibria for some

Bi-Objective Control Problems

This chapter is concerned with the numerical solution of some multi-objective

optimal control problems that use Nash or Pareto strategies. For an extremal prob-

lem with p objectives or functionals Ji to minimize, a Nash strategy reduces to the

search of a set of p players or controls vi, each of them optimizing Ji with respect to

the i-th variable. If the Ji are regular enough and no constraint is imposed, the vican be characterized in terms of the derivatives of Ji:

∂Ji∂vi

(v1, . . . , vp) = 0, i = 1, . . . , p.

This way, each player has to optimize his/her assigned criterion and accepts that

the other criteria are fixed by the other players. When no player can further improve

his/her criterion, we say that the system has reached a Nash equilibrium state.

On the other hand, in theoretical economics, a Pareto equilibrium is a state of

allocation of resources such that it is impossible to get an individual improvement

of the functional values without making at least one individual worse off. Such

equilibria are obviously cooperative and, in general, not unique. In the framework of

a multi-objective control problem with p regular cost functionals Ji depending on p

controls vi, in the absence of constraints, Pareto equilibria must satisfy

p∑i=1

λiJ′i(v1, . . . , vp) = 0,

for some λi ≥ 0 with

p∑i=1

λi = 1.

Initially, we introduced Ω ⊂ RN and assume that Γ1, Γ2 ⊂ ∂Ω, with Γ1 ∩ Γ2 = ∅and ∂Ω = Γ1 ∪ Γ2. We will use the notation Q := Ω × (0, T ), Σ1 := Γ1 × (0, T )

and Σ2 := Γ2 × (0, T ). To understand better the ideas, we will consider systems

with only two controls, although the arguments and results that follow can be easily

extended to cover a larger amount.

The state equation will be given by a linear or semilinear heat PDE completed

17

with appropriate boundary and initial conditions:

yt −∆y = f + v11O1 + v21O2 in Q,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x) in Ω,

(15)

or

yt −∆y + F (y) = f + v11O1 + v21O2 in Q,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x) in Ω.

(16)

In (16), we assume that the function F : R 7→ R is (globally) Lipschitz-continuous

and the right hand side f and the initial data y0 are prescribed; O1,O2 ⊂ Ω are the

control domains, with O1 ∩ O2 = ∅ (both are supposed to be small); 1O1 and 1O2

are the corresponding characteristic functions; the controls are v1 and v2. Let the

Oi,d ⊂ Ω be open sets (i = 1, 2), representing prescribed observations domains.

We will consider the following cost functionals for (15) and (16):

Ji(v1, v2) :=1

2

∫∫Oi,d×(0,T )

|y − yi,d|2 dx dt +µi2

∫∫Oi×(0,T )

|vi|2 dx dt, i = 1, 2,

where the µi are positive constants and the yi,d = yi,d(x, t) are given functions.

The process for obtaining of the numerical results of Nash and Pareto equilibria

for some Bi-Objective optimal control problems associated to the Ji, is described in

the following steps:

• First: Show that the Nash (and Pareto) equilibrium property is equivalent to

solving a given system for the state and adjoint state, and find an expression

involving control and adjoint state(s);

• Second: For the discretization, use finite differences in time and finite elements

in space;

18

• Third: The results of numerical experiments are obtained by implementing of

the informations using freefem++ package.

(a) Initial Mesh (b) Final Mesh

Figure 1: Adaptation of the mesh after the iteration process.

(a) The final state for Nash equilibria in (Linear

Case) with T=2 and µ = 0.15.

(b) The final state for Nash equilibria in (Linear

Case) with T=2 and µ = 9.5.

Figure 2: States for Nash equilibrium in linear case.

19

As example of graphical representations for our experiments, we have the Figure 1

the adaptation of the mesh in the interactive process, and the Figure 2 the final state

that depends of the controls (v1, v2) and µ variations. More details of the results,

about the used algorithms and experiments see [22].

Chapter 4

On the Computation of Nash and Pareto Equilibria for some

Bi-Objective Optimal Control Problems for the Wave Equation

This chapter presents numerical methods for solving some multi-objective opti-

mal control problems for systems governed by linear and semilinear wave equations.

More precisely, for such problems, we use Nash and Pareto equilibria, which respec-

tively correspond to appropriate noncooperative and cooperative strategies, related

to controls. As in Chapter 3, for the multi-objective control problems, various strate-

gies for the choice of good controls can appear, depending of the characteristics of

the problem. These strategies lead to what we call equilibria, that can be coopera-

tive (when the controls collaborate to achieve the goals) and noncooperative (in the

opposite case).

The main novelty of the present chapter is to extend the analysis and results of

the numerical development for obtaining optimal controls results in multi-objective

wave equations using Nash and Pareto strategies.

Of course, there are several strategies for multi-objective optimization, an exam-

ple is the Stackelberg hierarchical-cooperative strategy in [67], where we can intro-

duce goal variations, such as Stackelberg-Nash, Stackelberg-Pareto, etc.

Some previous results on strategies for the control of differential equations are de-

veloped by:

• A. Ramos and R. Glowinski in [65], in this paper the authors develop numer-

ically optimal control results, using Nash strategy for multi-objective linear

problems.

• J. L. Lions in [59], in this work the author gives some results about the Pareto

strategies.

• M. O. Bristeau and R. Glowinski in [14], in this work the authors compare

Pareto and Nash strategies by using genetic algorithms to compute numerically

the solutions corresponding to these strategies.

20

In our problems, the state equation will be given by a linear or semilinear wave

PDE completed with appropriate boundary and initial conditions:

ytt −∆y = f + v11O1 + v21O2 in QT ,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x), yt(x, 0) = y1(x) in Ω

(17)

or

ytt −∆y + F (y) = f + v11O1 + v21O2 in QT ,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x), yt(x, 0) = y1(x) in Ω,

(18)

where F : R 7→ R is (globally) Lipschitz-continuous, f and y0 are prescribed; the

sets Oi and Oi,d are also as above and we set again

Ji(v1, v2) :=1

2

∫∫Oi,d×(0,T )

|y−yi,d|2 dx dt +µi2

∫∫Oi×(0,T )

|vi|2 dx dt, i = 1, 2. (19)

Following the ideas in Chapter 3, again we will analyze numerically the Nash and

Pareto equilibria for linear and semilinear wave problems. Thus, we will follow the

steps indicated in the previous chapter (but a change in convergence algorithms will

be introduced), adapted to (17)–(18).

In resume, for our numerical results, we combine finite difference methods for the

time discretization, finite element methods for the space discretization and gradient

methods for the iterative solution of the discrete control problems.

Some representative examples of our experiments are given below. In Figure 3.(a)

we have the initial Mesh, in 3.(b) the function f is fixed in the Nash equilibrium for

the linear case. In Figures 4.(a) and 4.(b), we indicate the amount of effort that the

controls v1 and v2 (in the Nash equilibrium) perform to control the problem in the

desired goal.

21

(a) The Initial Mesh. (b) The function f .

Figure 3: Initial Mesh and the function f -fixed.

(a) The computed state at t = T for µ = 1.5. (b) The computed state at t = T for µ = 10.5.

Figure 4: Final States in the time T = 1.5.

More details of the results, about the used algorithms and experiments see [23].

22

Works in progress

In this work, we have been able to extend the numerical study in chapter 3 to the

framework of the so called Stackelberg-Nash null controllability of the heat equation

for the linear and semilinear cases.

• Stackelberg-Nash strategy: In a multi-objective control problem, for each leader

control which impose the null controllability for the state variable, we find

a Nash equilibrium associated to followers controls, considering some costs

functions. The leader control is chosen to be the one of minimal cost.

In this sense, the problem is the following (for brevity, let’s just comment the

linear case):

With N a positive integer, Ω ⊂ RN and T > 0, consider in Q = Ω × (0, T ) a

distributed system governed by a parabolic equation with a control v of the support

ω. We divide v into three parts, say f , v1 and v2; where f wants to control the

system solution to zero, v1 and v2 aim to control optimally. We will divide ω in three

regions O, O1 and O2.

Consider the heat problemyt −∆y = f1O + v11O1 + v21O2 in Q,

y = 0 on Σ,

y(x, 0) = y0(x) in Ω

(20)

the functional costs (secondary)

Ji(f ; v1, v2) :=αi2

∫∫Oi,d×(0,T )

|y − yi,d|2 dx dt+µi2

∫∫Oi×(0,T )

|vi|2 dx dt.

and the functional costs (principal)

Ji(f) :=1

2

∫∫O×(0,T )

|f |2 dx dt,

where αi > 0, µi > 0 are constants, yi,d = yi,d(x, t) (i=1,2) are given functions and

suppose that O1 = O2.

To use the Stackelber-Nash strategies idea, let’s fixe the control f (leader) and

let’s solve an optimal control problem for the controls v1 and v2 (followers), associated

23

with the Nash equilibrium. In this way, we write the pair (v1(f), v2(f)) (in function

of f) obtaining an associated optimality system, depending only on f , where we study

a controllability problem with control f .

We will summarize the numerical resolution in four steps:

• Step 1 (The optimal System):

Consider the problem (already in the conditions of the Nash equilibrium):

yt −∆y = f1O + v11O1 + v21O2 in Q,

−φi,t −∆φi = αi(y − yi,d)1Oi,d , i = 1, 2 in Q,

y(x, 0) = y0(x) on Σ,

φi(x, T ) = 0 in Ω

(21)

with,

vi =1

µiφi1Oi , for i = 1, 2 .

• Step 2 (The goal):

Find f , such that the solution for (21) satisfies y(x, T ) = 0.

In order to obtain null control, we must minimize a functional with weightsMinimize

∫∫Q

ρ2|y|2 +

∫∫O×(0,T )

ρ20|f |2

Subject to (21)

(22)

ρ and ρ0 are appropriate, blow up as t→ T .

The advantage: y necessarily vanishes exactly at t = T (and so does f).

• Step 3 (Numerical solutions):

As a consequence of the problem (22), we must solve the 4th-order Lax-Milgram

problem a((ψ, γ1, γ2) (ψ′, γ′1, γ′2)) =

⟨l, (ψ′, γ′1, γ

′2)⟩

∀(ψ′, γ′1, γ′2) ∈ W, (ψ, γ1, γ2) ∈ W(23)

24

where W is the space of adjoint states for (y, φ1, φ2).

We have introduced a reformulation for the problem above which require high

regularity.

Additional reformulation : a 2nd-order mixed problem after integration by parts

is given by α((z, m), (z′, m′)) + β((z′, m′), λ) =

⟨l, (z′, m′)

⟩β((z, m), λ′) = 0

∀(z′, m′, λ′) ∈ Z ×M × Λ, (z, m, λ) ∈ Z ×M × Λ.

(24)

(a) The Domain Mesh (b) The leader f

Figure 5: Ω = (0, 1); O = (0.2, 0.8), O1 = (0, 0.2), O2 = (0.8, 1); T = 0.5; Number

of vertices (xi, ti) : 710 ; Number of triangles: 1310

• Step 4 (Techniques used):

Mixed P1 − P2- Lagrange FEM’s (Finite Element Methods). In resume, we use:

- Finite differences in time;

- FEM divide the domain of the problem into a collection of subdomains;

- Then reconnects elements at nodes;

- This process results in a set of simultaneous algebraic equations;

25

- We implemented numerically the results using freefem ++.

Remark 0.0.1 The figures presented in this work are in the process of improvement.

• As an illustration, in Figure 5.(a) we have the discretization of the domain,

and the Figure 5.(b) the leader control f .

• The figures 6(A)-6(D) show the process of evolution of the state y(x, t).

Figure 6: Evolution of the state y for the objective y(x, T ) = 0

More information and details about this work, see [24].

Chapter 1

Weak-Renormalized Solutions for a

Simplified k-ε Model of Turbulence

27

Weak-Renormalized Solutions for a Simplified k-ε

Model of Turbulence

Pitagoras P. de Carvalho and Enrique Fernandez-Cara

Abstract. The aim of this paper is to prove the existence of a weak-

renormalized solution to a simplified model of turbulence of the k − ε

kind in spatial dimension N = 2. The unknowns are the average velocity

field and pressure, the mean turbulent kinetic energy and an appropriate

time dependent variable. The motion equation and the additional PDE

are respectively solved in the weak and renormalized senses.

1.1 Introduction

Let Ω ⊂ RN (where N = 2 or N = 3) be a connected bounded open set with a C2

boundary (∂Ω) and let ω ⊂ Ω be a (small) nonempty open set. Let T > 0 be given

and let us consider the cylindrical domain Q = Ω × (0, T ), with lateral boundary

Σ = ∂Ω× (0, T ).

In the sequel, n = n(x) will stand for the outward unit vector normal to Ω at

points x ∈ ∂Ω. We will investigate the existence of a weak-renormalized solution to

a simplified system of turbulence. This model can be justified as follows.

28

Let us start from the standard k − ε model of turbulence:

vt + (v · ∇)v −∇ · ((ν + cνk2

ε)Dv) +∇(p− 2

3k) = f in Q,

∇ · v = 0 in Q,

kt + v∇k −∇ · (κ+ cκk2

ε∇k) = cν

k2


εt + v∇ε−∇ · (β + cεk2

ε∇ε) = cηk|Dv|2 − a

ε2

kin Q,

v = 0 on Σ,

∂k

∂n= 0 and

∂ε

∂n= 0 on Σ,


Here, v = v(x, t), k = k(x, t), ε = ε(x, t) and p = p (x, t) respectively represent

the averaged velocity field, turbulent kinetic energy, turbulent energy dissipation

rate and average pressure of a fluid in turbulent regime whose particles are in Ω

during the time interval (0, T ); v0, k0 and ε0 are the initial conditions at time t = 0;

f = f(x, t) is a prescribed field of external forces; finally, ν, cν , κ, cκ, β, cε, cη and

a are positive constants.

The quantity νt = cνk2

εis the so called turbulent viscosity and Dv stands for the

symmetrized gradient of v : Dv = ∇v+∇Tv. In fact, νt appears as a consequence of

the following hypothesis of the Boussinesq kind: the Reynolds tensor R := −v′ ⊗ v′is given by

R = νtDv +2

3kI.

For a detailed motivation of the problem, see for instance [25, 55].

With the previous model in mind, let us introduce

ϕ(t) =k2

ε

and let us assume that ϕ depends only on t. This may be an acceptable hypothesis

in many cases. Note that it leads to a model that is, in some sense, intermediate

between 1-equation and 2-equation models (see [25]).

29

Then, it is not difficult to check that v, p, k and ϕ satisfy the system (1.1)-(1.2)

below, where we have introduced q := p− 2

3k :

vt + (v · ∇)v −∇ · ((ν + cνϕ)Dv) +∇q = f in Q,

∇ · v = 0 in Q,

kt + v∇k −∇ · ((κ+ cκϕ)∇k) +k2

ϕ= cνϕ|Dv|2 in Q,

∂k

∂n= 0 and v = 0 on Σ,

v(x, 0) = v0(x) and k(x, 0) = k0(x) in Ω,

(1.1)

together with the ODE problemϕt=

[(2cν−cη)|Ω|

∫Ω

|Dv|2

kdx− 2cκ

|Ω|

(∫Ω

|∇k|2

|k|2dx

)]ϕ2+

(a− 2)

|Ω|

∫Ω

k dx

ϕ(0)=ϕ0.

(1.2)

For technical reasons, we will simplify a little more the system and suppose that

cη = 2cν and a > 2. Thus, instead of (1.1)-(1.2), we will assume that ϕ, k, v and q

satisfy:

ϕt +

(2cκ|Ω|

∫Ω

|∇k|2

|α + k|2dx

)ϕ2 =

(a− 2)

|Ω|

(∫Ω

k dx

)in (0, T ), (1.3)

kt + v∇k −∇ · ((κ+ cκϕ)∇k) +k2

ϕ= cνϕ|D(v)|2 in Q, (1.4)

vt + (v·∇)v −∇ · ((ν + cνϕ)D(v)) +∇q = f , div v = 0 in Q, (1.5)

∂k

∂n= 0 and v = 0 on Σ, (1.6)

ϕ(0) = ϕ0, k(x, 0) = k0(x), v(x, 0) = v0(x) in Ω, (1.7)

where α > 0, ϕ0, k0 and v0 are again given, ϕ0 > 0 and k0 ≥ 0 a.e.

The structure of this system not only appears in turbulence modelling, but also

in non-isothermal solidification problems with melt convection [2, 6, 17]; in this

particular context, (1.4) can be viewed as a phase-field equation and is essentially

the same found in [17] with an advection term. The other equations are standard

and straightforward consequences of the usual physical balance laws (energy, linear

momentum and mass).

30

Throughout this paper, we will denote by C or M generic constants depending

only on known quantities, which will be indicated frequently.

A great deal of attention has been paid to turbulence and phase-field models

for solidification processes during the last two decades; see for example [17, 18,

34, 61, 6, 2]. In these works, many situations and many different hypotheses have

been considered. Notice that (1.4) needs a special treatment due to the nonlinear

right-hand side, that only belongs to L1(Q) since, in general, D(v) only belongs

to L2(Q)N×N . For this reason, we will consider the notion of renormalized solutions

adapted to our setting.

Renormalized solutions to PDEs were first introduced by DiPerna and P.-L. Li-

ons [36, 35] in the context of Boltzmann-like equations. Later, they have also been

considered in other situations; let us mention in particular the contributions by

Blanchard, Boccardo, Murat and their co-workers in the framework of second-order

elliptic and parabolic PDEs; see [8, 10, 9, 7, 11, 12] and the references therein; see

also [60] and [15] for more related results.

In order to solve (1.3)–(1.7), we will use regularization techniques, truncations,

appropriate estimates and the compactness of approximate solutions.

This paper is organized as follows.

In Section 1.2, we fix the notation and we introduce some functional spaces. We

also recall several technical results. We enumerate the hypotheses, we introduce the

concept of weak-renormalized solution and we state the main result of the paper.

In Section 1.3, we investigate the solvability of some auxiliary problems.

Section 1.4 is devoted to present the proof of the existence result for two-dimensional

flows; it is split into three steps, namely, the formulation and resolution of regularized

problems, the obtention of estimates, and the passage to the limit.

1.2 Preliminaries

1.2.1 Notation and spaces

For any q ≥ 1, we denote by Lq(Ω) the standard Lebesgue space with the usual

norm, denoted by ‖ · ‖q,Ω. For any nonnegative integer m, Wm,q(Ω) is the standard

Sobolev space with the usual norm denoted by ‖ · ‖m,q,Ω. The space Wm,q0 (Ω) is the

31

closure with respect to the norm ‖ · ‖m,q,Ω of the space C∞0 (Ω) of C∞ functions with

compact support in Ω. We refer for instance to [37] for more details on these spaces.

The following result from [63] will be used below:∫∫Q

|u|τ dx dt ≤ C‖u‖pq/NL∞(0,T ;Lp(Ω))

∫∫Q

|∇u|q dx dt, (1.8)

for every u ∈ Lq(0, T ;W 1,q0 (Ω))∩L∞(0, T ;Lp(Ω)) with p, q ≥ 1 and τ = q(N +p)/N .

For the analysis of the motion equation (1.5), we will need other function spaces.

Thus, let us set V = v ∈ C∞0 (Ω)N : ∇ · v = 0; we will denote the closures of Vin L2(Ω)N and H1

0 (Ω)N respectively by H and V . Then, H and V are Hilbert spaces

for the corresponding norms and one has

H = v ∈ L2(Ω)N : ∇ · v = 0 in Ω, v · n = 0 on ∂Ω

and

V = v ∈ H10 (Ω)N : ∇ · v = 0 in Ω .

The general properties of these spaces can be found for instance in [69].

In the sequel, we will use the following truncation function: for any positive real

number R, we set

TR(s) = s if |s| ≤ R and TR(s) = R sign (s) if |s| > R,

where sign (s) = 0 if s = 0 and sign (s) = s/|s| if s 6= 0.

Since TR is Lipschitz-continuous, for any function u ∈ W 1,q0 (Ω) one has TR(u) ∈

W 1,q0 (Ω) and the chain rule for the differentiation of TR(u) holds true, that is,

∇TR(u) = T ′R(u)∇v a.e. in Ω.

We will also have to consider the following set:

L(0, T ; Ω) := u ∈ L∞(0, T ;L1(Ω)) : TR(u) ∈ L2(0, T ;H10 (Ω)) ∀R > 0,

limn→+∞

1

n

∫An(u)

|∇u|2 dx dt = 0.

Here and in the sequel, An(u) stands for the set

An(u) := (x, t) ∈ Q : n ≤ |u(x, t)| ≤ 2n.

We will make use of the following lemma, due to Boccardo and Gallouet (see [11];

see also [57]):

32

Lemma 1.2.1 Assume that u ∈ L∞(0, T ;L1(Ω)), TR(u) ∈ L2(0, T ;H10 (Ω)) for all

R > 0 and there exists M > 0 such that

‖u‖L∞(0,T ;L1(Ω)) ≤M and

∫∫Q

|∇TR(u)|2 dx dt ≤MR ∀R > 0.

Then, for all 1 < q < (N + 2)/(N + 1), one has

u ∈ Lq(0, T ;W 1,q0 (Ω)) and ‖u‖Lq(0,T ;W 1,q

0 (Ω)) ≤ C(q)M.

1.2.2 Hypotheses and main result

Along this work, we will assume that the following hypotheses hold:

(Θ) f ∈ L2(Q)N , ϕ0 ∈ R+, v0 ∈ H, k0 ∈ L1(Ω) with k0 ≥ 0 a.e.

We introduce now the definition of a weak-renormalized solution:

Definition 1.2.1 It will be said that (ϕ, k, v) is a (weak-renormalized) solution to (1.3)–

(1.7) if the following conditions are satisfied:

1. ϕ ∈ H1(0, T ), k ∈ L(0, T ; Ω) and v ∈ L2(0, T ;V ) ∩ L∞(0, T ;H).

2. ϕ solves the ODE (1.3) and ϕ(0) = ϕ0.

3. k solves (1.4) in the renormalized sense, that is: for any β ∈ W 2,∞(R) such

that Supp β′ is compact and for any η ∈ C1([0, T ];H10 (Ω)) ∩ L∞(Q) such that

η(x, T ) ≡ 0, we have

−∫∫

Q

β(k) ηt dx dt+

∫∫Q

(κ+ ckϕ)∇β(k) · ∇η dx dt

+

∫∫Q

(v · ∇)β(k) · η dx dt−∫

Ω

β(k0) η(x, 0) dx (1.9)

=

∫∫Q

β′(k)

[(cνϕD(u) : D(u) − k2

ϕ

)−(κ+ ckϕ

)∇β′(k) · ∇k

]· η dx dt.

4. v solves (1.5) in the usual weak sense (together with some q ∈ D′(Q)) and

v|t=0 = v0.

We can now state our main result in this paper:

Theorem 1.2.1 Assume that N = 2 and (Θ) holds. Then, there exists at least one

solution (ϕ, k, v) to (1.3)–(1.7).

33

1.3 Some auxiliary problems

In order to prove Theorem 1.2.1, it is convenient to first consider and solve a family

of auxiliary problems.

Let ρε be a regularizing sequence in RN . For any ε > 0 and any v ∈ V , we will

denote by Rεv the following function:

Rεv := ρε ∗ v.

Here, v is the extension by zero of v to the whole RN .

Recall that Rεv ∈ C∞(RN)N , ∇ · (Rεv) = 0 in Ω and

‖Rεv‖m,q,Ω ≤ C(m, q, ε)‖v‖2,Ω

for all m and q.

The first auxiliary problem is the following:

vt + ((Rεv)·∇)v −∇ · (m(t)D(v)) +∇q = f, div v = 0 in Q, (1.10)

v = 0 on Σ, (1.11)

v(x, 0) = v0(x) in Ω. (1.12)

Here, we assume thatf ∈ L2(Q)N , v0 ∈ H,m ∈ L∞(0, T ), 0 < ν1 ≤ m(t) ≤ ν2 a.e.

(1.13)

The existence and uniqueness of a weak solution to (1.10)–(1.12) can be proved

via a Galerkin method. It suffices to argue for instance like in [58] or [69] for the

classical Navier-Stokes equations. In this way, the following is obtained:

Proposition 1.3.1 Let the assumptions (1.13) be satisfied. Then, there exists ex-

actly one solution (v, q) to (1.10)–(1.12), with

v ∈ L2(0, T ;V ) ∩ C0([0, T ];H), vt ∈ L2(0, T ;V ′).

Furthermore, one has ‖v‖L2(0,T ;V ) + ‖v‖L∞(0,T ;H) ≤ C,

‖vt‖Lσ(0,T ;V ′) ≤ C(ν2),

where σ = 2 if N = 2 and σ = 4/3 if N = 3 and C (resp. C(ν2)) depends on Ω, T ,

‖f‖L2(Q), ‖v0‖H and ν1 (resp. these data and ν2).

34

Next, we consider a second auxiliary problem, related to the ODE in our original

system: ϕt + A(t) · ϕ2 = B(t) in (0, T ),

ϕ(0) = ϕ0,

(1.14)

where

A ∈ L1(0, T ), B ∈ L∞(0, T ), A,B ≥ 0 a.e. and ϕ0 ∈ R+. (1.15)

The following result can also be easily proved:

Proposition 1.3.2 Let the assumptions (1.15) be satisfied. Then, there exists a

unique solution to (1.14), with

ϕ ∈ H1(0, T ) (1.16)

and the norm in this space bounded by a constant only depending on ‖A‖L1(0,T ) +

‖B‖L∞(0,T ). Furthermore, one has

1

‖A‖L1(0,T ) +1

ϕ0

≤ ϕ(t) ≤ ϕ0 + T‖B‖L∞(0,T ) ∀t ∈ [0, T ]. (1.17)

1.4 Proof of Theorem 1.2.1

1.4.1 An auxiliary regularized problem

We begin by introducing some notation. Thus, for any ε > 0, we set:

(i) k0,ε = T1/ε(k0) .

(ii) gε = T1/ε(cνϕε|D(vε)|2) .

We will also use a second truncation function: for any ε > 0, we set

Lε(s) = T1/ε(s) if s ≥ ε and Lε = ε otherwise.

35

We will begin the proof with no restriction on the spatial dimension (N = 2 or

N = 3). We consider the following regularized version of (1.3)–(1.7):

ϕε,t +

(2ck|Ω|

∫Ω

|∇kε|2

|α + kε|2dx

)ϕ2ε =

a− 2

|Ω|

(∫Ω

kε dx)

in (0, T ), (1.18)

ϕε(0) = ϕ0, (1.19)

kε,t + vε∇kε −∇ · ((κ+ cκLε(ϕε))∇kε) +k2ε

Lε(ϕε)= gε in Q, (1.20)

∂kε∂n

= 0 on Σ, kε(x, 0) = k0ε(x) in Ω, (1.21)

vε,t + ((Rεvε) · ∇)vε −∇ · ((ν + cνLε(ϕε))D(vε)) +∇qε = f, div vε = 0 in Q, (1.22)

vε = 0 on Σ, vε(x, 0) = v0(x) in Ω. (1.23)

We then have the following existence result:

Proposition 1.4.1 Let the assumptions (Θ) be fulfilled. Then, for each ε > 0, there

exists at least one solution (ϕε, kε, vε) to (1.18)–(1.23), withϕε ∈ H1(0, T ),

kε ∈ L2(0, T ;H10 (Ω)) ∩ C0([0, T ];L2(Ω)),

vε ∈ L2(0, T ;V ) ∩ C0([0, T ];H).

Proof: The proof can be obtained from a standard application of Schauder’s or

Leray-Schauder’s fixed-point Theorem.

Let us consider the mapping Λε that associates to each ϕ ∈ C0([0, T ]), first,

the unique solution vε to (1.22)–(1.23) with ϕε replaced by ϕ; then, the unique

solution kε to (1.20)–(1.21) with ϕε replaced by ϕ and the right hand side gε replaced

by T1/ε(cνϕ|D(uε)|2); finally, the unique solution ϕε to (1.18)–(1.19). Note that kεbelongs to the space L2(0, T ;H1(Ω)) ∩ C0([0, T ];L2(Ω)) and kε ≥ 0 a.e.

Consequently, in view of the results in Sections 1.2 and 1.3, Λε : L2(0, T ) 7→L2(0, T ) is well-defined. Furthermore, it is continuous. Indeed, let ϕ and ϕn be given

in C0([0, T ]), let us set

ϕ = Λε(ϕ), ϕn = Λε(ϕn)

and let us assume that ϕn → ϕ strongly in C0([0, T ]). Then, arguing as in [43], we

see that

36

1. The associated vn converge strongly in L2(0, T ;V ) to the velocity field v cor-

responding to ϕ.

2. The associated kn converge strongly in L2(0, T ;H1(Ω)) to the energy k corre-

sponding to ϕ and

3. Finally, ϕn → ϕ strongly in H1(0, T ) .

The previous second and third assertions are consequences of the usual energy

estimates for parabolic equations. The first one can be justified as follows.

First, from the estimates in Proposition 1.3.1, it is clear that vn converges weakly

in L2(0, T ;V ) to v. Secondly, taking into account that vt and vnt belong to L2(0, T ;V ′),

we find the energy identities

1

2‖v(T )‖2

2,Ω +

∫∫Q

(ν + cνLε(ϕ)) |D(v)|2 dx dt =

∫∫Q

f · v dx dt+1

2‖v0‖2

2,Ω

and

1

2‖vn(T )‖2

2,Ω +

∫∫Q

(ν + cνLε(ϕn)) |D(vn)|2 dx dt =

∫∫Q

f · vn dx dt+1

2‖v0‖2

2,Ω

for all n ≥ 1. Consequently,

limn→+∞

[1

2‖vn(T )‖2

2,Ω +

∫∫Q

(ν + cνLε(ϕn)) |D(vn)|2 dx dt

]=

1

2‖v(T )‖2

2,Ω +

∫∫Q

(ν + cνLε(ϕ)) |D(v)|2 dx dt.

But this yields the strong convergence of (vn(T ), (ν + cνLε(ϕn))1/2D(vn)) in the

product spaceH×L2(Q)N×N . Since (ϕn) converges uniformly and the sequence (Lε(ϕn))

is bounded from above and from below, we deduce that (D(vn)) also converges

strongly in L2(Q)N×N . In view of Korn’s inequality, this is equivalent to the strong

convergence of (∇vn) in the same space, that is, the strong convergence of (vn)

in L2(0, T ;V ).

Notice that Λε maps C0([0, T ]) into a compact set.

37

Indeed, from Proposition 1.3.2, we know that the Λε(ϕ) belong to a ball B(0;R)

in the space H1(0, T ), with R only depending on∫∫Q

|∇kε|2

|α + kε|2dx dt and sup

(0,T )

∫Ω

kε dx.

But these quantities are bounded by a constant only depending on ε, again thanks

to standard energy estimates.

Consequently, we can apply Schauder’s Theorem to Λε and deduce that this

mapping possesses at least one fixed point.

This provides a solution to (1.18)–(1.23) and ends the proof.

1.4.2 Some a priori estimates

In this section, we will deduce some a priori estimates for the solutions to (1.18)–

(1.23), uniformly with respect to ε.

To this end, we start by applying Proposition 1.3.1 to (1.22)–(1.23) and we obtain:

‖vε‖L2(0,T ;V ) + ‖vε‖L∞(0,T ;H) ≤ C. (1.24)

A first consequence is that gε = T1/ε(cνϕε|D(vε)|2) is uniformly bounded in L1(Q).

In view of the results in [5], the following estimates hold for kε:kε ∈ bounded set inL∞(0, T ;L1(Ω)) ∩ L1(0, T ;Lp(Ω))

for all 1 < p < +∞ if N = 2 and for all 1 < p < 3 if N = 3.(1.25)

Furthermore, arguing as in [8], we see that there exists M such that∫∫Q

|∇TR(kε)|2 dx dt ≤MR and1

n

∫∫n≤|kε|≤2n

|∇kε|2 dx dt ≤M (1.26)

for all R > 0 and n ≥ 1.

From Lemma 1.2.1, we also get:

kε ∈ bounded set inLq(0, T ;W 1,q0 (Ω)) ∀ 1 < q <

N + 2

N + 1. (1.27)

Combining (1.25), (1.27) and the embedding result (1.8), we deduce that

kε ∈ bounded set inLτ (Q) ∀ 1 < τ <N + 2

N. (1.28)

38

Taking into account (1.25) and Proposition 1.3.2, we deduce that (ϕε) is uniformly

bounded from above.

Let us see that they are also bounded from below by a positive constant. Thus,

let us multiply (1.20) by −(α + kε)−1 and let us integrate in Ω. This gives:

− d

dt

∫Ω

log(α + kε) +

∫Ω

(κ+ cκϕε)|∇kε|2

|α + kε|2dx−

∫Ω

k2ε

ϕε(α + kε)dx

= −∫

Ω

gεα + kε

dx.

After integration in time, we see at once that

ϕε

∫Ω

|∇kε|2

|α + kε|2dx ≤ d

dt

∫Ω

log(α + kε) dx+

∫Ω

|kε|2

α + kεdx

and, in particular, ∫ T

0

(∫Ω

|∇kε|2

|α + kε|2dx

)ϕε dt ≤ C. (1.29)

In view of (1.29) introducing

Mε(t) :=2cκ|Ω|

(∫Ω

|∇kε|2

|α + kε|2dx

)ϕε ,

we note that (Mε) is uniformly bounded in L1(0, T ) and (ϕε) can be viewed as the

unique solution to the linear problemϕε,t +Mε(t)ϕε =

a− 2

|Ω|

∫Ω

kε in (0, T ) ,

ϕε(0) = ϕ0.

Obviously, this implies ϕε ≥ C > 0, as desired. Summarizing, we have

0 < ϕ ≤ ϕε ≤ ϕ (1.30)

for some positive ϕ and ϕ. Using again Proposition 1.3.2, one has

‖ϕε‖H1(0,T ) ≤ C. (1.31)

From (1.30) and Proposition 1.3.1, we deduce that

39

‖vε,t‖Lσ(0,T ;V ′) ≤ C. (1.32)

Also, from the PDE satisfied by kε, the fact that gε is uniformly bounded in L1(Q),

(1.24) and (1.25), one has:

kε,t ∈ bounded set inL1(0, T ;W−1,a(Ω)) ∀ 1 < a < a, (1.33)

where a = 4/3 if N = 2 and a = 6/5 if N = 3.

Some consequences of these estimates are the following:

• ϕε ∈ compact set in C0([0, T ])

(in view of the compactness of the embedding H1(0, T ) → L2(0, T )).

• kε ∈ compact set in Lq(0, T ;Lb(Ω)) ∀ 1 < q <N + 2

N + 1, 1 < b <

Nq

N − q(in view of (1.27), (1.33) and the compactness of the embedding W 1,q

0 (Ω) →Lb(Ω)).

• vε ∈ compact set in L2(0, T ;H)

(in view of (1.24) and the compactness of the embedding V → H).

Therefore, at least for a subsequence, we have:

ϕε → ϕ strongly in C0([0, T ]) and a.e., (1.34)

kε→kweakly inLq(0,T ;W 1,q0 (Ω)), strongly inLq(0,T ;Lb(Ω)) and a.e., (1.35)

vε → v weakly in L2(0, T ;V ), strongly in L2(Q)N and a.e., (1.36)

ϕε,t → ϕt weakly in Lσ(0, T ), (1.37)

vε,t → vt weakly in Lσ(0, T ;V ′). (1.38)

1.4.3 Passage to the limit and conclusions

The convergence properties (1.34)–(1.38) are enough to prove that we can pass

to the limit in the equations and initial conditions satisfied by ϕε and vε. This is well

known.

We will show now that k solves (1.4) in the renormalized sense. In fact, it is just

here where we have to assume that N = 2.

40

Since N = 2, we have v ∈ L2(0, T ;V ′) and, therefore,

1

2‖v(t2)‖2

2,Ω −1

2‖v(t1)‖2

2,Ω +

∫ t2

t1

∫Ω

(ν + cνϕ)|D(v)|2 dx dt=∫ t2

t1

∫Ω

f · v dx dt

for all t1, t2 ∈ [0, T ].

One of the delicate points of the argument is to prove that D(vε)→ D(v) strongly

in L2(Q)2×2. To this purpose, we will argue as in the proof of proposition 1.4.1 (but

now letting ε→ 0+).

We first notice thatvε(T )→ v(T ) weakly in H and

(ν + cνLε(ϕε))1/2D(vε)→ (ν + cνϕ)1/2D(v) weakly in L2(Q)2×2.

(1.39)

Then, we multiply the regularized motion equation (1.22) by vε and we integrate

over Ω × (0, T ). Using Green’s formula, the fact that ∇ · vε = 0 and Holder’s and

Young’s inequalities, we deduce that

1

2‖vε(T )‖2

2,Ω +

∫∫Q

(ν + cνLε(ϕε)) |D(vε)|2 dx dt =

∫∫Q

f · vε dx dt+1

2‖v0‖2

2,Ω.

From (1.34), we get

limε→0

[1

2‖vε(T )‖2

2,Ω +

∫∫Q

(ν + cνLε(ϕε)) |D(vε)|2 dx dt]

=

∫∫Q

f · v dx dt+1

2‖v0‖2

2,Ω.

On the other hand, v is a solution to (1.5), whence

1

2‖v(T )‖2

2,Ω +

∫∫Q

(ν + cνϕ) |D(v)|2 dx dt =

∫∫Q

f · v dx dt+1

2‖v0‖2

2,Ω

and

limε→0

[1

2‖vε(T )‖2

2,Ω +

∫∫Q

(ν + cνLε(ϕε)) |D(vε)|2 dx dt]

=1

2‖v(T )‖2

2,Ω +

∫∫Q

(ν + cνϕ) |D(v)|2 dx dt.(1.40)

From (1.39), (1.40) and the a.e. convergence of ϕε and kε, the desired strong conver-

gence of D(vε) is ensured.

41

A consequence is that

gε→ (ν + cνϕ)|D(u)|2 strongly in L1(Q). (1.41)

Now, it can be shown that (kε) is a Cauchy sequence in C0([0, T ];L1(Ω)) and,

moreover,

limε→0+

∫∫Q

(T − t) |cνϕε∇TR(kε)− cνϕ∇TR(k)|2 dx dt = 0

for every R > 0. In particular, TR(kε) converges strongly to TR(k) in the space

L2(0, T ′;H10 (Ω)) for every R > 0 and every T ′ < T . All this is implied by (1.25),

(1.26) and (1.41), but is not immediate; For more details, we refer for instance to [57,

Appendix E].

This shows that there exists a subsequence, still indexed with ε, such that we

have the following for any β ∈ W 2,∞(R) such that Supp β′ ⊂ [−R,R]:

kε → k and β(kε)→ β(k) weakly in Lq(0, T ;W 1,q0 (Ω)) ∩ Lτ (Q), (1.42)

TR(kε)→ TR(k) strongly in L2(0, T ;H10 (Ω)). (1.43)

Furthermore, by multiplying (1.20) by β′(kε), we also see that

β(kε)t + vε∇β(kε)− ckϕε∆β(kε) + β′(kε)k2ε

ϕε

+ ckϕε∇β′(kε) · ∇kε = β′(kε)gε in Q. (1.44)

Let us multiply (1.44) by a test function η ∈ C1([0, T ];H10 (Ω)) ∩ L∞(Q) such

that η(x, t) ≡ 0 in a neighborhood of T and let us integrate over Q. After some

integrations by parts, using (1.20) and observing the properties of η, we get:

−∫∫

Q

β(kε) ηt dx dt+

∫∫Q

(κ+ cκϕε)∇β(kε) · ∇η dx dt

+

∫∫Q

(vε · ∇)β(kε) · η dx dt−∫

Ω

β(k0) η(x, 0) dx

=

∫∫Q

β′(kε)

[gε −

k2ε

ϕε− (κ+ cκϕε)∇β′(kε) · ∇kε

]· η dx dt. (1.45)

Thanks to (1.41) and (1.42)–(1.43), we can take ε → 0 in this identity. This

gives (1.9) for functions η of this kind. By a standard density argument, we de-

duce (1.9) for all η ∈ C1([0, T ];H10 (Ω)) ∩ L∞(Q) with η(x, T ) ≡ 0.

This ends the proof of Theorem 1.2.1.

42

Remark 1.4.1 The existence of weak-renormalized solutions to other related sys-

tems has been established in other papers; see for instance [5, 26, 43]; see also [38]

for the case of a viscous, compressible and heat conducting fluid.

Remark 1.4.2 If we neglect convection and we omit the transport term (u · ∇)u in

the motion equation (1.5), the argument used in the proof of Theorem 1.2.1 remains

valid for N = 3. On the other hand, the uniqueness of the weak-renormalized solution

is unknown even when N = 2 and the coefficients cν and cκ vanish.

Remark 1.4.3 It is readily seen that the previous proof of Theorem 1.2.1 does not

work in the case N = 3. Indeed, the strong convergence in L2(Q)3×3 of the gradients

of the approximate velocity fields is out of scope; this is a major difficulty even

for similar approximations to the Navier-Stokes equations. Unfortunately, we do

need this convergence to take limits in the equation for kε if we are looking for a

weak-renormalized solution in the sense of Definition 1.2.1.

In the 3D case, it seems appropriate to reformulate the problem in terms of other

variables; see [44] for some partial results; see also [16], where a related 3D problem

with Fourier-Navier (slip) conditions on v has been solved satisfactorily.

Chapter 2

On the Control of a Simplified k-ε

Model of Turbulence

44

On the Control of a Simplified k-ε Model of

Turbulence

Pitagoras P. de Carvalho, Enrique Fernandez-Cara and Juan Lımaco

Abstract. This paper deals with the control of a kind of turbulent flows.

We consider a simplified k − ε model with distributed controls, locally

supported in space. We proof that the system is partially locally null-

controllable, in the sense that the velocity field can be driven exactly to

zero if the initial state is small enough. The proof relies on an argument

where we have concatenated several techniques: fixed-point formulation,

linearization, energy and Carleman estimates, local inversion, etc. Ths

result can be viewed as a nontrivial step towards the control of turbulent

fluids.

2.1 Introduction

Let Ω ⊂ RN be a bounded connected open set with a regular boundary ∂Ω (N = 2

or N = 3) and let ω ⊂ Ω be a (small) nonempty open set.

Let T > 0 be given and let us consider the cylindrical domain Q = Ω × (0, T ),

with lateral boundary Σ = ∂Ω× (0, T ). In the sequel, (· , ·) and ‖ · ‖ stand for the L2

scalar product and norm in Ω, respectively. We will denote by C a generic positive

constant; sometimes, we will indicate the data on which it depends. On the other

hand, n(x) will stand for the outwards unit normal vector to Ω at the point x ∈ ∂Ω.

We will investigate the local null controllability properties of a simplified model

of turbulence of the k − ε kind.

In order to present and justify the state system, let us start from the usual k− ε

45

model:

vt + (v · ∇)v −∇ · ((ν + cνk2

ε)Dv) +∇(p− 2

3k) = u1ω in Q,

∇ · v = 0 in Q,

kt + v · ∇k −∇ · ((κ+ c0k2

ε)∇k) = cν

k2


εt + v · ∇ε−∇ · ((κ′ + c′0k2

ε)∇ε) = cηk|Dv|2 − a

ε2

kin Q,

v = 0 on Σ,

∂k

∂n= 0 and

∂ε

∂n= 0 on Σ,


(2.1)

Here, v = v(x, t), p = p(x, t), k = k(x, t) and ε = ε(x, t) are the “averaged”

velocity field, pressure, turbulent kinetic energy and rate of turbulent energy dissi-

pation of a fluid whose particles are located in Ω during the time interval (0, T ); v0,

k0 and ε0 are the initial conditions at time t = 0; 1ω is the characteristic function of

ω; ν, cν , κ, c0, κ′, c′0, cη and a are positive constants; see [54].

The quantity νT := cνk2/ε is the so called turbulent viscosity, and Dv stands for

the symmetrized gradient of v, that is, Dv := ∇v +∇Tv. Recall that νT appears in

the averaged motion PDE as a consequence of the following Boussinesq hypothesis:

v′ ⊗ v′ = −νtDv −2

3k Id.,

where, for any z = z(x, t), z denotes the average of z and v′ is the turbulent pertu-

bation of the velocity field; for more details, see for instance [25], [50] and [55].

The following vector spaces, usual in the context of incompressible fluids, will be

used along the paper:

H = w ∈ L2(Ω)N : ∇ · w = 0 in Ω, w · n = 0 on ∂Ω,

and

V = w ∈ H10 (Ω)N : ∇ · w = 0 in Ω.

We will denote by A : D(A) 7→ H the Stokes operator. By definition, one has

D(A) = H2(Ω)N ∩ V ; Aw = P (−∆w) ∀w ∈ D(A),

46

where P : L2(Ω)N 7→ H is the usual orthogonal projector.

We will work with a simplified version of (2.1). To this purpose, we introduce

φ0 :=k2

ε

and we assume that φ0 only depends on t. As a consequence, after some straightfor-

ward computations,we get the following PDEs and additional conditions for v, q :=

p− 2

3k and k:

vt + (v · ∇)v −∇ · ((ν + cνφ0)Dv) +∇q = u1ω in Q,

∇ · v = 0 in Q,

kt + (v · ∇)k −∇ · ((κ+ c0φ0)∇k) +k2

φ0

= cνφ0|Dv|2 in Q,

v = 0 on Σ,

∂k

∂n= 0 on Σ,

v(x, 0) = v0(x) and k(x, 0) = k0(x) in Ω.

(2.2)

Additionally, φ0 must satisfy the ODE problem:φ0,t=

[2cν−cη|Ω|

(∫Ω

|Dv|2

kdx

)− 2c0

|Ω|

(∫Ω

|∇k|2

k2dx

)]φ2

0+a−2

|Ω|

(∫Ω

k dx

),

φ0(0) = φ00,

where φ00 is a positive initial datum.

For simplicity, we will consider the previous system with cη = 2cν and a ≥ 2.

Accordingly, one hasφ0,t = −2c0

|Ω|

(∫Ω

|∇k|2

k2dx

)φ2

0 +a− 2

|Ω|

(∫Ω

k dx

)in (0, T ),

φ0(0) = φ00.

(2.3)

In (2.2)–(2.3), u is the control and (v, q, k, φ0) is the state. When N = 2, for

any v0 ∈ H, any nonnegative bounded C1 function φ0 and any u ∈ L2(ω × (0, T ))N ,

(2.2)–(2.3), possesses exactly one strong solution (v, q), with

v ∈ L2(0, T ;D(A)) ∩ C0([0, T ];V ), vt ∈ L2(0, T ;H). (2.4)

47

When N = 3, this is true if v0 and u are sufficiently small in their respective spaces.

These assertions can be deduced arguing (for instance) as in [27].

Definition 2.1.1 Let k0 and φ00 be given, with

k0 ∈ H1(Ω), k0 ≥ 0 a.e., φ00 ∈ R+. (2.5)

It will be said that (2.2)–(2.3) is partially locally null-controllable at time T if there

exists ε > 0 such that, for any v0 ∈ V with

‖v0‖H10≤ ε,

there exist controls u ∈ L2(ω × (0, T ))N and associated states (v, q) satisfying (2.4)

and

v(x, T ) = 0 in Ω. (2.6)

The main result in this paper is the following:

Theorem 2.1.1 For any k0 and φ0 satisfying (2.5) and any T > 0, the nonlinear

system (2.2)–(2.3) is partially locally null-controllable at T .

For the proof, we will have to employ several different techniques, all them usual

in this context nowadays. In particular, we will rewrite the partial null controllability

problem as a fixed-point equation and we will apply an Inverse Mapping Theorem

in Hilbert spaces and then Schauder’s Fixed-Point Theorem. The arguments are

inspired by the work of Fursikov and Imanuvilov; see [45], [46] and [47].

This paper is organized as follows. In Section 2, we prove some technical results,

needed to establish the null contollability of (2.2)–(2.3). In Section 3, we give the

proof of Theorem 2.1.1. Then, some additional comments and questions are presented

in Sections 4.

2.2 Preliminary results

In this section, we set µ := ν+cνΦ0(t), where Φ0 is a nonnegative function in C1([0, 1]).

We will consider the linear system

vt − (ν + cνΦ0)∆v +∇q = u1ω + f in Q,

∇ · v = 0 in Q,

v = 0 on Σ,

v(x, 0) = v0(x) in Ω

(2.7)

48

and the adjoint

−ϕt − (ν + cνΦ0)∆ϕ+∇π = F in Q,

∇ · ϕ = 0 in Q,

ϕ = 0 on Σ,

ϕ(x, T ) = ϕT (x) in Ω.

(2.8)

2.2.1 Some Carleman estimates

We will need some (well known) results from Fursikov and Imanuvilov [45]; see

also [40]. Also, it will be convenient to introduce a new non-empty open set ω0, with

ω0 b ω. The following technical lemma, due to Fursikov and Imanuvilov [45], is

fundamental:

Lemma 2.2.1 There exists a function η0 ∈ C2(Ω) satisfying:η0(x) > 0 ∀x ∈ Ω, η0(x) = 0 ∀x ∈ ∂Ω and

|∇η0(x)| > 0 ∀x ∈ Ω \ ω0.

Let τ = τ(t) be a function satisfying

τ ∈ C∞([0, T ]), τ > 0 in (0, T ), τ(t) =

t if t ≤ T

4,

T − t if t ≥ 3T

4

and let us introduce the weightsα(x, t) :=

eλ(‖η0‖∞+m0+1) − eλ(η0(x)+m0)

τ(t)8, ξ(x, t) :=

eλ(η0(x)+m0)

τ(t)8,

ρ(x, t) := esα(x,t), ρ(t) := exp (smaxx∈ Ω

α(x, t)), ξ(t) := maxx∈Ω

ξ(x, t),(2.9)

where λ, s > 0 are real numbers and the constant m0 > 0 is fixed, sufficiently large

to have

|αt| ≤ Cξ9/8, |αtt| ≤ Cξ5/4 ∀λ > 0.

It is then easy to prove that there exists λ00 > 0 such that, for any λ ≥ λ00, one has

maxx∈ Ω

α0(x) ≤ 2 minx∈ Ω

α0(x). (2.10)

49

In the sequel, we will always take λ ≥ λ00.

The following global Carleman estimate holds for the solutions to (2.8):

Lemma 2.2.2 Let us assume that Φ0 is a nonnegative function in C1([0, T ]), with

‖Φ0‖C1([0,T ]) ≤ M . There exist positive constants λ0, s0 and C depending on

Ω, ω, T, ν, cν , and M such that, for any s ≥ s0 and λ ≥ λ0, any F ∈ L2(Q)

and any ϕT ∈ H, the associated solution to (2.8) satisfies∫∫Q

ρ −2(sξ)−1(|ϕt|2 + |∇π|2) dx dt+

∫∫Q

ρ−2(sξ)|∇ × ϕ|2 dx dt

+

∫∫Q

ρ−2((sξ)−1|∆ϕ|2 + |∇ϕ|2 + (sξ)2|ϕ|2) dx dt

≤ C

(∫∫Q

ρ−2|F |2 dx dt+

∫∫ω×(0,T )

ρ−2(sξ)3|ϕ|2 dx dt

). (2.11)

The proof is given in [53]; see also [52].

By combining the previous lemma and appropriate energy estimates, we can

deduce a second family of Carleman inequalities, with weights that do not vanish at

t = 0. More precisely, let us now introduce the function

`(t) :=

τ(T

2

)=T

2for 0 ≤ t ≤ T

2,

τ(t) = T − t for3T

4≤ t ≤ T

and the associated weightsα(x, t) :=

eλ(‖η0‖∞+m0+1) − eλ(η0(x)+m0)

`(t)8, ξ(x, t) :=

eλ(η0(x)+m0)

`(t)8,

ρ(x, t) := esα(x,t), ρ∗(t) := exp(smaxx∈Ω

α(x, t)), ξ∗(t) := maxx∈Ω

ξ(x, t).(2.12)

Then the functions ξ, ρ, ρ∗ and ξ∗ are strictly positive and bounded from below

in any set of the form Ω× [0, T − δ] with δ > 0.

Lemma 2.2.3 Let the assumptions in Lemma 2.2.2 be satisfied. There exist positive

constants λ0, s0 and C depending on Ω, ω, T , ν, cν and M such that, for any s ≥ s0

50

and λ ≥ λ0, any F ∈ L2(Q) and any ϕT ∈ H, the associated solution to (2.8)

satisfies∫∫Q

ρ−2∗ (sξ∗)

−1(|ϕt|2 + |∇π|2) dx dt+

∫∫Q

ρ−2(sξ)|∇ × ϕ|2 dx dt

+

∫∫Q

ρ−2

((sξ)−1|∆ϕ|2 + |∇ϕ|2 + (sξ)2|ϕ|2

)dx dt (2.13)

≤ C

(∫∫Q

ρ −2|F |2 dx dt+

∫∫ω×(0,T )

ρ −2(sξ)3|ϕ|2 dx dt

).

Proof: It is a consequence of (2.11) and the usual energy estimates for ϕ. Since the

main ideas are known and can be found in many papers, we will only give a sketch.

Let us take λ ≥ λ0 and s ≥ s0 and let us set

Γ(ϕ, π; a, b) :=

∫∫Ω×(a,b)

ρ−2∗ (sξ∗)

−1(|ϕt|2 + |∇π|2) dx dt

+

∫∫Ω×(a,b)

ρ −2((sξ)−1|∆ϕ|2 + |∇ϕ|2 + (sξ)2|ϕ|2

)dx dt

+

∫∫Ω×(a,b)

ρ −2(sξ)|∇ × ϕ|2 dx dt (2.14)

and

S(ϕ;F ) :=

∫∫Q

ρ −2|F |2 dx dt+

∫∫ω×(0,T )

ρ −2(sξ)3|ϕ|2 dx dt.

Then Γ(ϕ, π; 0, T ) = Γ(ϕ, π; 0, T/2) + Γ(ϕ, π;T/2, T ) and, from Lemma 2.2.2, we

clearly have

Γ(ϕ, π;T/2, T ) ≤ CS(ϕ;F ). (2.15)

We must prove that

Γ(ϕ, π; 0, T/2) ≤ C(S(ϕ;F )

and we know that

Γ(ϕ, π; 0, T/2) ≤ C

∫ T/2

0

(‖ϕt‖2 + ‖∆ϕ‖2 + ‖∇ϕ‖2 + ‖ϕ‖2) dt.

Let us check that this integral can be bounded as in (2.15).

51

From (2.8), it is readily seen that

−1

2

d

dt‖ϕ‖2 + µ‖∇ϕ‖2 ≤ µ

2‖∇ϕ‖2 + C‖F‖2 in (0, T ),

whence

‖ϕ(·, t1)‖2 +

∫ t2

t1

‖∇ϕ(·, s)‖2 ds ≤ ‖ϕ(·, t2)‖2 + C

∫ t2

t1

(‖F (·, s)‖2

for all 0 ≤ t1 ≤ t2 ≤ T. In particular, taking t1 ∈ [0, T/2] and t2 ∈ [T/2, 3T/4] and

integrating with respect to t1 and then with respect to t2, we see that∫ T/2

0

‖ϕ(·, t)‖2 dt ≤ C

(∫ 3T/4

T/2

‖ϕ(·, s)‖2 ds+

∫ 3T/4

0

‖F (·, s)‖2 ds

)≤ CS(ϕ;F ).

Also, ∫ T/2

0

‖∇ϕ(·, t)‖2 dt ≤ ‖ϕ(·, t2)‖2 + C

∫ 3T/4

0

‖F (·, s)‖2 ds,

for all t2 ∈ [T/2, 3T/4] and, integrating with respect to t2 in this interval, we deduce

an estimate of the integral of ‖∇ϕ‖2 in (0, T/2):∫ T/2

0

‖∇ϕ(·, t)‖2 dt ≤ C

(∫ 3T/4

T/2

‖ϕ(·, s)‖2 ds+

∫ 3T/4

0

‖F (·, s)‖2 ds

)≤ CS(ϕ;F ).

A similar argument holds for the integral of ‖∆ϕ‖2. Indeed, one has

−1

2

d

dt‖∇ϕ‖2 + µ‖∆ϕ‖2 = (F,−∆ϕ) ≤ µ

2‖∆ϕ‖2 + C‖F‖2 in (0, T ).

Arguing as before, we see that∫ T/2

0

‖∆ϕ(·, t)‖2 dt ≤ ‖∇ϕ(·, t2)‖2 + C

∫ 3T/4

0

‖F (·, s)‖2 ds

for all t2 ∈ [T/2, 3T/4] and, after integration with respect to t2, we see that∫ T/2

0

‖∆ϕ(·, t)‖2 dt ≤ C

(∫ 3T/4

T/2

‖∇ϕ(·, s)‖2 ds+

∫ 3T/4

0

‖F (·, s)‖2 ds

)≤ CS(ϕ;F ).

52

Finally, using that ‖ϕt‖2 = (µ∆ϕ+ F,−ϕt), we deduce that∫ T/2

0

‖ϕt‖2 dt ≤ CS(ϕ;F ).

As a consequence, Γ(ϕ, π; 0, T/2) ≤ CS(ϕ;F ) and the proof is done. 2

From now on, we will fix λ and s as in Lemma 2.2.3 and we will consider the

corresponding functions α, α, ρ, ξ, etc. given by (2.9) and (2.11). Also, we will

introduce the function

η := ρ ξ−1. (2.16)

An immediate consequence of Lemma 2.2.3 and this definition is the following:

Corollary 2.2.1 There exist positive constants λ, s and C only depending on Ω, ω,

T , ν, cν and M such that, for any F ∈ L2(Q) and any ϕT ∈ H, the corresponding

solution to (2.8) satisfies∫∫Q

η−2|ϕ|2 dx dt ≤ C

(∫∫Q

ρ−2|F |2 dx dt+

∫∫ω×(0,T )

η−2|ϕ|2 dx dt

).

2.2.2 The null controllability of the linear system (2.7)

We will have to impose some specific conditions to f and v0 in order to drive the

solution to (2.7) exactly to zero. The following result holds:

Proposition 2.2.1 Let us assume that v0 ∈ H and ηf ∈ L2(Q)N . Then, we can

find control-states (v, q, u) for (2.7) satisfying

v ∈ L2(0, T ;V ) ∩ C0([0, T ];H), u ∈ L2(ω × (0, T ))N (2.17)

and ∫∫Q

ρ2|v|2 dx dt +

∫∫ω×(0,T )

η2|u|2 dx dt < +∞. (2.18)

In particular, one has (2.6).

53

The proof relies on (2.12) and can be easily obtained arguing as in [45]. The key

idea is to consider the extremal problem

Minimize

∫∫Q

ρ2|v|2 dx dt+

∫∫ω×(0,T )

η2|u|2 dx dt

Subject to u ∈ L2(ω × (0, T ))N , (v, q, u) satisfies (2.7).

(2.19)

There exists exactly one solution to (2.19) that, thanks to (2.13), is the desired

control-state and satisfies (2.17) and (2.18).

It is also true that, in a certain sense, the solution to (2.19) depends continuously

on the data (f, v0). In particular, if the (fn, v0n) satisfy

‖v0n − v0‖ → 0 and

∫∫Q

η2|fn − f |2 dxdt→ 0,

then the vn and un furnished by Proposition 2.2.1 satisfy∫∫Q

ρ 2|vn − v|2 dx dt→ 0 and

∫∫ω×(0,T )

η 2|un − u|2 dx dt→ 0, (2.20)

where u and v correspond to (f, v0) . For brevity we omit the details, that can be

found in [41].

2.2.3 Some additional estimates

The state found in Proposition 2.2.1 satisfies some additional properties that will

be needed below, in Section 3. Thus, it will be shown in this section that not only v

but also ∇v, ∆v, and vt belong to some specific weighted L2 spaces.

Let us introduce the spatially homogeneous weights

ρ(t) := exp(sminx∈ Ω

α(x, t)), ζ(t) := ρ(t)`(t)12 and γ(t) := ρ(t)`(t)33/2. (2.21)

From (2.12), (2.16) and (2.21), it is easy to see that

ζ ≤ Cη and |ζζt| ≤ Cρ 2. (2.22)

The following results hold:

54

Proposition 2.2.2 Let the hypotheses in Proposition 2.2.1 be satisfied and let (v, q, u)

be the solution to (2.19). Thenmax[0,T ]

∫Ω

ζ2|v|2 dx+

∫∫Q

ζ2|∇v|2 dx dt≤C(‖v0‖2+

∫∫Q

ρ 2|v|2 dx dt

+

∫∫Q

η2|f |2 dx dt+

∫∫ω×(0,T )

η2|u|2 dx dt) (2.23)

Proof: To get this result, we multiply the PDE in (2.7) by ζ2v and we integrate in

Ω. We obtain: ∫Ω

ζ2(vt − µ∆v +∇q) · v dx =

∫Ω

ζ2(u1ω + f) · v dx.

Remember that µ = ν + cνΦ0(t).

In view of the inequalities in (2.22), the following estimates hold:∫Ω

ζ2u1ω · v dx ≤ C

(∫ω

η2|u|2 dx)1/2(∫

Ω

ζ4η−2|v|2 dx)1/2

≤ 1

2

∫ω

η2|u|2 dx+ C

∫Ω

ρ 2|v|2 dx ,

∫Ω

ζ2f · v dx ≤ 1

2

∫Ω

η 2|f |2 dx+ C

∫Ω

ρ 2|v|2 dx ,

∫Ω

ζ2vt · v dx =1

2

d

dt

∫Ω

ζ2|v|2 dx−∫

Ω

ζ ζt|v|2 dx

≥ 1

2

d

dt

∫Ω

ζ2|v|2 dx− C∫

Ω

ρ 2|v|2 dx.

On the other hand,∫Ω

ζ2∇q · v dx dt = 0 and

∫Ω

ζ2(−∆v) · v dx =

∫Ω

ζ2|∇v|2 dx.

Therefore,

1

2

d

dt

∫Ω

ζ2|v|2 dx+ µ

∫Ω

ζ2|∇v|2 dx ≤C(∫

ω

η2|u|2 dx

+

∫Ω

ρ2|v|2 dx+

∫Ω

η2|f |2 dx).

55

Now, integrating in time, the estimate in (2.23) is easily found. 2

Proposition 2.2.3 Let the hypotheses in Proposition 2.2.1 be satisfied and let (v, q, u)

be the solution to (2.19). Let us assume that v0 ∈ V . Then one hasmax[0,T ]

∫Ω

γ2|∇v|2 dx+

∫∫Q

γ2(|vt|2 + |∆v|2) dx dt ≤ C

(‖v0‖2

H10

+

∫∫Q

ρ2|v|2 dx dt+

∫∫Q

η2|f |2 dx dt+

∫∫ω×(0,T )

η2|u|2 dx dt).

(2.24)

Proof: Recall that, under the assumption v0 ∈ V, if Φ0 is a nonnegative C1 function

in [0, T ] and f ∈ L2(Q)N , the solution (v, q) to (2.7) satisfies

v ∈ L2(0, T ;D(A)) ∩ C0([0, T ];V ), vt ∈ L2(0, T ;H),

where A : D(A) 7→ H is the Stokes operator.

Let us multiply the PDE in (2.7) by γ2vt and let us integrate in Ω. The following

holds: ∫Ω

γ2(vt − µ∆v +∇q) · vt dx =

∫Ω

γ2(u1ω + f) · vt dx.

From the definition of γ, ζ and η we see that

γ ≤ Cζ ≤ Cη ; |γγt| ≤ Cζ2.

Consequently, for any small ε > 0, we find that∫Ω

γ2u1ω · vt dx ≤ ε

∫Ω

γ2|vt|2 dx+ Cε

∫ω

η2|u|2 dx,

∫Ω

γ2f · vt dx ≤ ε

∫Ω

γ2|vt|2 dx+ Cε

∫Ω

η2|f |2 dx

and, also,

−∫

Ω

γ2∆v · vt dx =1

2

d

dt

∫Ω

γ2|∇v|2 dx−∫

Ω

γγt|∇v|2 dx

≥ 1

2

d

dt

∫Ω

γ2|∇v|2 dx− C∫

Ω

ζ2|∇v|2 dx.

56

On the other hand, the integral involving the pressure vanishes. Therefore, the

following is found integrating in time:∫∫Q

γ2|vt|2 dx dt+ max[0,T ]

∫Ω

γ2|∇v|2 dx ≤ C

(‖v0‖2

H10 (Ω)

+

∫∫Q

ζ2|∇v|2 dx dt+∫∫

Q

η2|f |2 dx dt+∫∫

ω×(0,T )

η2|u|2 dx dt). (2.25)

This furnishes the estimates of γ2|vt|2 and γ2|∇v|2 in (2.24).

In order to estimate the weighted integral of |∆v|2, we multiply the PDE in (2.7)

by γ2Av (recall that A is the Stokes operator). After integration in Ω, we have:∫Ω

γ2(vt − µ∆v +∇q) · Av dx =

∫Ω

γ2(u1ω + f) · Av dx.

Observe that∫Ω

γ2u1ω · Av dx ≤ ε

∫Ω

γ2|∆v|2 dx+ Cε

∫ω

η2|u|2 dx,∫Ω

γ2f · Av dx ≤ ε

∫Ω

γ2|∆v|2 dx+ Cε

∫Ω

η2|f |2 dx,∫Ω

γ2vt · Av dx =

∫Ω

γ2vt · (−∆v) dx ≤ ε

∫Ω

γ2|∆v|2 dx+ Cε

∫Ω

γ2|vt|2 dx

for any small ε > 0,∫Ω

γ2∇q · Av dx = 0 and

∫Ω

γ2(−∆v) · Av dx =

∫Ω

γ2|∆v|2 dx.

Integrating in time, we now see that∫∫Q

γ2|∆v|2 dx dt ≤ C

(∫∫ω×(0,T )

η 2|u|2 dx dt

+

∫∫Q

η2|f |2 dx dt+

∫∫Q

γ2|vt|2 dx dt).

From (2.25) and this last inequality, we deduce (2.24). 2

57

2.3 Proof of Theorem 2.1.1

In this section, we will prove the partial local null controllability of (2.2).

Let us denote by L2(η2;Q) the Hilbert space formed by the measurable functions

f = f(x, t) such that ηf ∈ L2(Q), that is,

‖f‖2L2(η2;Q) :=

∫∫Q

η2|f |2 dx dt < +∞.

Let Φ0 ∈ C1([0, T ]) be given, with Φ0 ≥ 0 and ‖Φ0‖C1([0,T ]) ≤M . Let us set

L(Φ0)v := vt − (ν + cνΦ0)∆v (2.26)

and let us introduce the Hilbert spaces

W (Φ0) :=

(v, q, u) : ρv ∈ L2(Q)N , γv ∈ L2(0, T ;D(A)),

ηu ∈ L2(ω × (0, T ))N , q ∈ L2(0, T ;H1(Ω)), (2.27)∫Ω

q dx = 0 a.e. η(L(Φ0) +∇q − u1ω

)∈ L2(Q)N

and

Z := L2(η2;Q)N × V . (2.28)

We will use the following Hilbertian norms in W (Φ0) and Z:

‖(v, q, u)‖2W (Φ0) := ‖ρ v‖2

L2(Q) + ‖γv‖2L2(0,T ;D(A)) + ‖η u‖2

L2(ω×(0,T ))

+ ‖q‖2L2(0,T ;H1(Ω)) + ‖η (L(Φ0) +∇q − u1ω)‖2

L2(Q)N

(2.29)

and

‖(f, v0)‖2Z := ‖f‖2

L2(η2;Q) + ‖v0‖2H1

0. (2.30)

Note that, if (v, q, u) ∈ W (Φ0), then vt ∈ L2(Q)N , whence v : [0, T ] 7→ V is

(absolutely) continuous and, in particular, we have v(· , 0) ∈ V and

‖v(· , 0)‖V ≤ C(M)‖(v, q, u)‖W (Φ0) ∀ (v, q, u) ∈ W (Φ0).

Furthermore, in view of Propositions 2.2.2 and 2.2.3, one also has ζv ∈ L2(0, T ;V )∩L∞(0, T ;H) and γv ∈ L2(0, T ;D(A)) ∩ L∞(0, T ;V ), with norms bounded again by

C(M)‖(v, q, u)‖W (Φ0) .

58

In the first part of this section, we will assume that the turbulent viscosity is

given and we will try to drive the (averaged) velocity field exactly to zero. Then, we

will use a fixed-point argument to prove that the whole system (2.2)–(2.3) is partially

locally null-controllable.

Thus, let us consider the mapping A : W (Φ0) 7→ Z, given as follows:

A(v, q, u) :=(vt + (v · ∇)v − (ν + cνΦ0)∆v +∇q − u1ω , v(· , 0)

). (2.31)

We will check that there exists ε > 0 such that, if (f, v0) ∈ Z and ‖(f, v0)‖Z ≤ ε,

the equation

A(v, q, u) = (f, v0), (v, q, u) ∈ W (Φ0), (2.32)

possesses at least one solution. In particular, this will show that the nonlinear system

vt + (v · ∇)v − (ν + cνΦ0)∆v +∇q = u1ω + f in Q,

∇ · v = 0 in Q,

v = 0 on Σ,

v(x, 0) = v0(x) in Ω

(2.33)

is locally null controllable and, furthermore, the state-controls (v, q, u) can be chosen

in W (Φ0).

We will apply the following version of the Inverse Mapping Theorem in infinite

dimensional spaces that can be found for instance in [1]. In the following statement,

Br(0) and Bε(ζ0) are open balls respectively of radius r and ε.

Theorem 2.3.1 Let Y and Z be Banach spaces and let H : Br(0) ⊂ Y 7→ Z be a C1

mapping. Let us assume that the derivative H ′(0) : Y 7→ Z is an epimorphism and

let us set ζ0 = H(0). Then there exist ε > 0, a mapping S : Bε(ζ0) ⊂ Z 7→ Y and a

constant K > 0 satisfying:S(z) ∈ Br(0) and H(S(z)) = z ∀z ∈ Bε(ζ0),

‖S(z)‖Y ≤ K‖z −H(0)‖Z ∀z ∈ Bε(ζ0).(2.34)

Notice that, in this result, usually known as Liusternik’s Inverse Function Theo-

rem, S is the inverse to the right of H. In order to show that Theorem 2.3.1 can be

applied in this setting, we will use several lemmas.

59

Lemma 2.3.1 Let A : W (Φ0) 7→ Z be the mapping defined by (2.31). Then, A is

well defined and continuous.

Proof: First, note that, in view of (2.21) and (2.10) we have

η2 ≤ Cζγ3 ≤ Cγ6. (2.35)

Notice that

‖A(v, q, u)‖2Z =

∫∫Q

η2|vt + (v · ∇)v − (ν + cνΦ0)∆v +∇q − u1ω|2 dx dt

+ ‖v(·, 0)‖2H1

0.

The following holds:∫∫Q

η2|vt + (v · ∇)v − (ν + cνΦ0)∆v +∇q − u1ω|2 dx dt

≤ 2

∫∫Q

η2|vt − (ν + cνΦ0)∆v +∇q − u1ω|2 dx dt

+ 2

∫∫Q

η2|(v · ∇)v|2 dx dt

:= M1 +M2 . (2.36)

From the definitions of the space W (Φ0) and the norm ‖(v, q, u)‖W (Φ0), it follows

that

M1 ≤ C‖(v, q, u)‖2W (Φ0) .

On the other hand, taking into account that for any w ∈ D(A) one has

‖∇w‖L3 ≤ C‖∇w‖1/2‖∆w‖1/2

and

‖(w · ∇)w‖2 ≤ C‖w‖2L6‖∇w‖2

L3 ≤ C‖∇w‖3‖∆w‖,

we see that

M2 ≤ C‖ζv‖1/2

L2(0,T ;V )‖γv‖L∞(0,T ;V )‖γv‖1/2

L2(0,T ;D(A))

≤ C‖(v, q, u)‖2W (Φ0) . (2.37)

This shows that A is well defined. On the other hand, that A is continuous is

easy to prove using similar arguments; for brevity, we omit the details. 2

60

Lemma 2.3.2 The mapping A : W (Φ0) 7→ Z is continuously differentiable.

Proof: We will present the proof when N = 3. The proof for N = 2 is similar.

Let us first see that A is G-differentiable at any (v, q, u) ∈ W (Φ0) and let us

compute the G-derivative A′(v, q, u).

Thus, let us fix (v, q, u) and let us fix (v′, q′, u′) ∈ W (Φ0) and σ > 0. Let us

denote by A1 and A2 the components of A:

A1(v, q, u) := vt + (v · ∇)v − (ν + cνΦ0)∆v +∇q − u1ω, A2(v, q, u) := v(·, 0).

We have:

1

σ

[A1((v, q, u) + σ(v′, q′, u′))−A1(v, q, u)

]= v′t+(v′ · ∇)v+σ(v′ · ∇)v′−(ν + cνΦ0)∆v′+∇q′−u′1ω+(v · ∇)v′. (2.38)

Let us introduce the linear mapping DA : W (Φ0) 7→ Z, with DA = (DA1, DA2)

and

DA1(v′, q′, u′) = v′t+(v · ∇)v′+(v′ · ∇)v−(ν+cνΦ0)∆v′+∇q′−u′1ω , (2.39)

and

DA2(v′, q′, u′) = v′(·, 0). (2.40)

From the definition of the spaces W (Φ0), Z and (2.39)–(2.40), it becomes clear

that DA ∈ L(W (Φ0);Z). Furthermore,

1

σ

[A((v, q, u) + σ(v′, q′, u′))−A(v, q, u)

]→ DA(v, q, u)(v′, q′, u′)

strongly in Z as σ → 0.

Indeed, using the estimate of M2 in (2.37), we get

1

σ

∥∥A1((v, q, u) + σ(v′, q′, u′))−A1(v, q, u)−DA1(v, q, u)(v′, q′, u′)∥∥L2(η2;Q)

= ‖σ(v′ · ∇)v′‖L2(η2;Q) ≤ Cσ ‖(v′, q′, u′)‖2W (Φ0) → 0,

while A2 is linear and continuous from W (Φ0) into V .

61

Thus, A is G-differentiable at any (v, q, u) ∈ W (Φ0), with G-derivative

A′(v, q, u) = DA . (2.41)

Now, let us prove that the mapping (v, q, u) 7→ A′(v, q, u) is continuous from

W (Φ0) into L(Φ0;Z) . As a consequence, in view of classical results, we will have

that A is not only G-differentiable but also F -differentiable and C1 and we will get

the desired result, see for instance Theorem 1-2, p. 21 in [68].

Thus, let assume that (vn, qn, un)→ (v, q, u) in W (Φ0) and let us check that

‖(DA(vn, qn, un)−DA(v, q, u)) (v′, q′, u′)‖2Z ≤ εn‖(v′, q′, u′)‖2

W (Φ0), (2.42)

for all (v, q, u) ∈ W (Φ0) for some εn → 0.

Arguing as in (2.36), the following holds:

‖(DA1(vn, qn, un)−DA1(v, q, u)) (v′, q′, u′)‖2L2(η2;Q)

≤ 2

∫∫Q

η2|(v′ · ∇)(vn − v)|2 dx dt+ 2

∫∫Q

η2|((vn − v) · ∇)v′|2 dx dt

≤ 2

∫ T

0

ζγ3‖∇v′‖2‖∇(vn − v)‖ ‖∆(vn − v)‖ dt (2.43)

+ 2

∫ T

0

ζγ3‖∇(vn − v)‖2‖∇v′‖ ‖∆v′‖ dt

≤ Cε1,n ‖(v′, q′, u′)‖2W (Φ0) ,

where

ε1,n :=

∫ T

0

ζγ‖∇(vn − v)‖ ‖∆(vn − v)‖ dt+

(sup[0,T ]

γ(t)‖∇(vn − v)‖)2

.

Noting that ε1,n → 0 as n→∞ and recalling again that A2 is linear and contin-

uous, we deduce that (2.42) is satisfied. 2

Lemma 2.3.3 Let A : W (Φ0) 7→ Z be the mapping defined by (2.31). Then the

linear mapping A′(0, 0, 0) is an epimorphism.

Proof: Let us fix (f, v0) ∈ Z. From Proposition 2.2.1, we know that there exists

(v, q, u) satisfying (2.7), (2.17) and (2.18). From the usual regularity results for

62

Stokes-like systems, we have v ∈ L2(0, T ;D(A)) and q ∈ L2(0, T ;H1(Ω)). Conse-

quently (v, q, u) ∈ W (Φ0) and, in view of (2.39), (2.40) and (2.41),

A′(0, 0, 0)(v′, q′, u′) = (f, v0),

that is,

vt − (ν + cνΦ0)∆v +∇q = u1ω + f in Q,

∇ · v = 0 in Q,

v = 0 on Σ,

v(x, 0) = v0(x) on Ω.

This shows that A′(0, 0, 0) is surjective and ends to proof. 2

In accordance with Lemmas 2.3.1, 2.3.2 and 2.3.3, we can apply Theorem 2.3.1

and deduce that, at least when (f, v0) belongs to a neighborhood of the origin in Z of

the form Bε(M)(0, 0), the equation (2.32) possesses a solution (v, q, u) = S(f, v0; Φ0),

with

‖(v, q, u)‖W (Φ0) ≤ K(M)‖(f, v0)‖Z . (2.44)

In particular, (2.33) is locally null-controllable.

Now, let φ00 and k0 satisfy (2.5). Let us set

M = 2

(φ00 +

(a− 2)T

|Ω|‖k0‖L1(Ω)

), b1 =

(|Ω|

2(a− 2)cνT

)1/2

,

β0 =κα2φ00

κα2 +2c0φ00

|Ω|

(T expT ‖k0‖2 + c2

νM2(1 + T expT )

) .

(2.45)

Let us assume that

‖v0‖H10≤ ε(M) (2.46)

and let us introduce the closed convex set

G =

(v, φ) ∈ L4(0, T ;V )× C0([0, T ]) : β0 ≤ φ ≤M,∫∫Q

|Dv|2 dx dt ≤ b21 ,

∫∫Q

|Dv|4 dx dt ≤ 1

and the mapping B : G 7→ L4(0, T ;V )× C0([0, T ]), given as follows:

63

• First, to each (v, φ) ∈ G, we associate the solution to the semilinear parabolic

system kt + v · ∇k − (κ+ c0φ)∆k +

k2

φ= cνφ|Dv|2 in Q,

∂k

∂n= 0 on Σ,

k(x, 0) = k0(x) on Ω.

• Secondly, we associate to k the solution φ0 to the ODE problem (2.3).

• Then, we associate to φ0 the quadruple (v, q, u, φ0), where (v, q, u) = S(0, v0;φ0).

• Finally, we set B(v, φ) = (v, φ0).

The mapping B is well defined, continuous and compact. Indeed, if we introduce

the Hilbert space

Y := v ∈ L2(0, T ;D(A)) : vt ∈ L2(Q)N

and we regard B as a mapping with values in Y × C1([0, T ]), it is obviously well

defined and continuous. Since Y → L4(0, T ;V ) with a compact embedding, the

compactness of B is ensured.

On the other hand, if ‖v0‖H10

is sufficiently small, B maps G into itself.

Indeed, from (2.3) it is immediate that

φ0(t) ≤ φ00 +(a− 2)T

|Ω|‖k‖L∞(0,T ;L1(Ω))

≤ φ00 +(a− 2)T

|Ω|(‖k0‖L1 + cνMb2

1

)= M

and

64

φ(t) ≥ φ00

1 +2c0φ00

|Ω|

∫∫Q

|∇k|2

|α + k|2dx dt

≥ φ00

1 +c0φ00

|Ω|α2κ

(‖k0‖2 + ‖k‖2

L2(Q) + c2νM

2

∫∫Q

|Dv| dx dt)

≥ β0

for all t ∈ [0, T ]. Moreover, since we have (2.44) with f = 0, there exists ε, only

depending on Ω, ω, T, ν, cν and M such that, whenever ‖v0‖H10≤ ε, one has∫∫

Q

|Dv|2 dx dt ≤ b21 and

∫∫Q

|Dv|4 dx dt ≤ 1 .

As a consequence, we can apply Schauder’s Theorem to B and deduce the exis-

tence of a fixed point. This proves that (2.2)–(2.3) is partially locally null-controllable

and ends the proof.

2.4 Some additional comments and questions

The partially globally null controllability of (2.2)–(2.3) is an open question. It

does not seem easy to solve and the answer is unknown even for the Navier-Stokes

system with Dirichlet boundary conditions on v. Indeed, the smallness assumption

on the data in Theorem 2.1.1 is clearly necessary if one tries to apply Theorem 2.3.1

or another result playing the same role. To prove a global result, we should have to

make use of a global inverse mapping theorem, but this needs much more complicate

estimates, that do not seem affordable.

Note that, with other or with no boundary conditions, global null controllability

results have been established for Navier-Stokes and Boussinesq fluids by Coron [29]

and Coron and Fursikov [30] in the two-dimensional case, Fursikov and Imanuvilov

[46] and Coron, Marbach and Sueur [31] in the three-dimensional case. Accordingly,

it is reasonable to expect results of the same kind when the PDEs in (2.2) are

completed, for instance, with Navier-slip or periodic boundary conditions.

Another open question, in part connected to the previous one, concerns the partial

exact controllability to the trajectories.

65

It is said that (2.2)–(2.3) is partially locally exactly controllable to trajectories

at time T if, for any solution (v, q) corresponding to a control u, there exists ε > 0

such that, if

‖v0 − v(·, 0)‖H10≤ ε,

we can find controls u ∈ L2(ω × (0, T ))N and associated states (v, q) satisfying

v(x, T ) = v(x, T ) in Ω. (2.47)

The previous property was established for the Navier-Stokes and Boussinesq flu-

ids respectively in [49] and [39]. However, to our knowledge, it is unknown whether

it holds for (2.2)–(2.3). If one tries to apply arguments as those above, one finds at

once a major difficulty: one is led to a system similar to (2.8), where linear nonlocal

terms appear, for which observability estimates are not clear at all.

Let us also indicate that it would be interesting to see whether the arguments in

[19] and [41] can be applied in this context to establish the partially local null con-

trollability of (2.2)–(2.3) with N − 1 scalar controls; when N = 3, the controllability

of (2.2)–(2.3) with one single scalar control is also a problem to consider, in view of

the results in [28].

The main result in this paper may be viewed as a first step in the controllability

analysis of k− ε models of turbulence. In this direction, let us indicate that it would

be satisfactory to prove the local null controllability of (2.1). However, this seems

difficult at present.

Another interesting aspect is the computation of “good” null controls of (2.2)–

(2.3). If would be interesting to address an efficient iterative algorithm, able to

produce a sequel of controls that converge to a null control for this system. This will

be the goal of a forthcoming paper.

Chapter 3

On the Computation of Nash and

Pareto Equilibria for some

Bi-Objective Control Problems

67

On the Computation of Nash and Pareto

Equilibria for some Bi-Objective Control Problems

Pitagoras P. de Carvalho and Enrique Fernandez-Cara

Abstract. This article is concerned with the numerical solution of some

multi-objective optimal control problems for systems governed by linear

and semilinear parabolic equations. More precisely, for such problems, we

look for Nash and Pareto equilibria, which respectively correspond to ap-

propriate noncooperative and cooperative strategies. First, we study the

linear case and then some semilinear problems. In order to compute the

solutions, we combine finite difference methods for the time discretiza-

tion, finite element methods for the space discretization and fixed-point

algorithms for the iterative solution of the discrete control problems. We

also illustrate these techniques with several numerical experiments.

3.1 Introduction

In a classical single-objective optimal control problem for a system governed by a

differential equation, there are input controls v acting on the equation trying to

minimize a cost functional v 7→ J(v). When there is no constraint on the control

and the functional satisfies suitable assumptions, there exists a unique solution u to

the control problem, which is determined by the optimality condition

J ′(u) = 0.

In a multi-objective control problem, there are usually several controls acting on

the equation and several cost functionals. In contrast with the single-objective case,

we can give several different definitions of “good” or “optimal” controls, depending on

the characteristics of the problem. They lead to what we call equilibria, that can be

cooperative (when the controls collaborate to achieve the goals) and noncooperative

(in the opposite case).

68

Since the concepts and arguments have origin in game theory and economics, the

notion of player is often used. Thus, for an extremal problem with p objectives or

functionals Ji to minimize, a Nash strategy reduces to the search of a set of p players

or controls vi, each of them optimizing Ji with respect to the i-th variable. Again, if

the Ji are regular enough and no constraint is imposed, the vi can be characterized

in terms of the derivatives of Ji:

∂Ji∂vi

(v1, . . . , vp) = 0, i = 1, . . . , p.

This way, each player has to optimize his/her assigned criterion and accepts that

the other criteria are fixed by the other players. When no player can further improve

his/her criterion, we say that the system has reached a Nash equilibrium state.

On the other hand, in theoretical economics, a Pareto equilibrium is a state of

allocation of resources such that it is impossible to get an individual improvement

of the functional values without making at least one individual worse off. Such

equilibria are obviously cooperative and, in general, not unique. In the framework

of a multi-objective control problem with p regular functionals Ji depending on p

controls vi, in the absence of constraints, Pareto equilibria must satisfy

p∑i=1

λiJ′i(v1, . . . , vp) = 0,

for some λi ≥ 0 with∑p

i=1 λi = 1.

3.2 Formulation of the Problems

Let Ω ⊂ RN be a bounded connected open set with regular boundary (N = 1, 2 or 3)

and assume that Γ1, Γ2 ⊂ ∂Ω, with Γ1 ∩ Γ2 = ∅ and ∂Ω = Γ1 ∪ Γ2. We will use the

notation Q := Ω× (0, T ), Σ1 := Γ1× (0, T ) and Σ2 := Γ2× (0, T ). In the sequel, we

will denote by (· , ·) and ‖ · ‖ the L2 scalar product and norm in Ω. We will denote

by C a generic positive constant; sometimes, we will indicate the data on which it

depends. Also, n = n(x) will stand for the outward unit normal vector to Ω at the

points x ∈ ∂Ω.

This paper is concerned with the numerical solution of some multi-objective op-

timal control problems that use Nash and Pareto strategies. To fix ideas, we will

69

consider systems with only two controls, although the arguments and results that

follow can be easily extended to cover a larger amount.

The state equation will be given by a linear or semilinear heat PDE completed

with appropriate boundary and initial conditions:

yt −∆y = f + v11O1 + v21O2 in Q,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x) in Ω

(3.1)

or

yt −∆y + F (y) = f + v11O1 + v21O2 in Q,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x) in Ω.

(3.2)

Here, we assume that the function F : R 7→ R is (globally) Lipschitz-continuous

and the right hand side f and the initial data y0 are prescribed.

In (3.1) and (3.2), O1,O2 ⊂ Ω are the control domains, with O1 ∩ O2 = ∅(both are supposed to be small); 1O1 and 1O2 are the corresponding characteristic

functions; the controls are v1 and v2.

Let O1,d,O2,d ⊂ Ω be open sets, representing prescribed observations domains.

We will consider the following functionals for the problems (3.1) and (3.2):

Ji(v1, v2) :=1

2

∫∫Oi,d×(0,T )


∫∫Oi×(0,T )

|vi|2 dx dt, i = 1, 2, (3.3)

where the µi are positive constants and the yi,d = yi,d(x, t) are given functions.

The multi-objective control problems considered in this paper are the following:

1. Nash multi-objective problem: Find controls vi ∈ L2(Oi × (0, T )) (i =

1, 2) satisfying J1(v1, v2) ≤ J1(v1, v2) ∀v1 ∈ L2(O1 × (0, T )),

J2(v1, v2) ≤ J2(v1, v2) ∀v2 ∈ L2(O2 × (0, T )).(3.4)

70

Note that, in the linear case, that is, when the state equation is (3.1), in view

of the strict convexity of v1 7→ J1(v1, v2), the (unique) v1 satisfying (3.4)1 is

characterized by the equality

∂J1

∂v1

(v1, v2) = 0.

Similarly, the (unique) v2 satisfying (3.4)2 is characterized by

∂J2

∂v2

(v1, v2) = 0.

Consequently, in the linear case, a Nash equilibrium is a pair (v1, v2) that solves

the linear system ∂J1

∂v1

(v1, v2) = 0,

∂J2

∂v2

(v1, v2) = 0.

(3.5)

In the semilinear case, (3.5) is in general only a necessary condition for a

pair (v1, v2) to be a Nash equilibrium.

2. Pareto multi-objective problem: Find controls vi ∈ L2(Oi × (0, T ))

(i = 1, 2), such that there is no (v1, v2) 6= (v1, v2) satisfyingvi ∈ L2(Oi × (0, T )) (i = 1, 2),

J1(v1, v2) ≤ J1(v1, v2) and J2(v1, v2) ≤ J2(v1, v2),

with strict inequality for at least one Ji.

(3.6)

Any couple (v1, v2) satisfying this property is called a Pareto equilibrium. In

the linear case, it is not difficult to prove that (v1, v2) is a Pareto equilibrium

if and only if there exists λ ∈ (0, 1) with(λ∂J1

∂v1

+ (1− λ)∂J2

∂v1

)(v1, v2) = 0,

(λ∂J1

∂v2

+ (1− λ)∂J2

∂v2

)(v1, v2) = 0.

(3.7)

On the other hand, in the semilinear case, (3.7) is as before only a necessary

condition.

71

The aim of this paper is to present efficient strategies for the numerical solution

of these multi-objective control problems. We will strict our considerations to the

resolution of (3.5) and (3.7). Obviously, these problems are very important from the

theoretical and practical viewpoints and appear frequently in the applications; for

some previous works on the subject (and also their connection to hierarchic control),

see for instance [3, 4, 32, 33, 51, 56, 59, 65, 66].

Remark 3.2.1 A special case appears when

O1,d ∩ O2,d 6= ∅.

This leads to a competitionwise problem, with each control or player trying to reach

different and possibly contradictory goals over a common domain. It is reasonable

to expect that this is the case where the behavior of the solution y associated to the

equilibrium (v1, v2) is most difficult to forecast.

This paper is organized as follows. In Section 3.3, we will present a strategy for

the computation of Nash equilibria for the linear system (3.1), as in [65] and [66].

In Section 3.4, we will adapt the ideas to the computation of Pareto equilibria. In

Sections 3.5 and 3.6, we will consider semilinear systems of the kind (3.2). Here, we

will indicate how Nash and Pareto equilibria can be found. Section 3.7 deals with

some numerical experiments. Finally, some technical results have been collected in

Section 3.8.

3.3 Computation of Nash Equilibria for (3.1)

3.3.1 A Formulation Equivalent to (3.5)

Let v1 and w2 be given, with v1, w2 ∈ L2(O2 × (0, T )) and let δv1 be a small pertur-

bation of v1. Then, with obvious notation, we have:

δ1J1(v1, w2) =

∫∫O1,d×(0,T )

[y(v1, w2)− y1,d] δ1y dx dt

+ µ1

∫∫O1×(0,T )

v1 δv1 dx dt+ . . .

72

where δ1y is the solution to

(δ1y)t −∆(δ1y) = (δv1)1O1 in Q,

δ1y = 0 on Σ1,

∂

∂n(δ1y) = 0 on Σ2,

δ1y(x, 0) = 0 in Ω

(3.8)

and the dots contain higher order terms.

Now, let φ1 = φ1(x, t) be a function defined in Q. Let us assume for the mo-

ment that φ1 possesses first-order time derivatives and up to second-order spatial

derivatives in C0(Q).

Then, multiplying in (3.8) by φ1 and integrating by parts, we easily obtain∫Ω

φ1(x, T ) δ1y(x, T ) dx+

∫∫Q

(−φ1,t −∆φ1) δ1y dx dt

−∫∫

Σ1

φ1∂

∂n(δ1y) dΓ dt+

∫∫Σ2

∂φ1

∂nδ1y dΓ dt

=

∫∫O1×(0,T )

φ1 δv1 dx dt.

(3.9)

Now, let φ1 be the solution to the following backwards in time (adjoint) system:

−φ1,t −∆φ1 = [y(v1, w2)− y1,d]1O1,din Q,

φ1 = 0 on Σ1,

∂φ1

∂n= 0 on Σ2,

φ1(x, T ) = 0 in Ω.

Then, we can deduce that (3.9) also holds for this φ1 and, consequently,⟨∂J1

∂v1

(v1, w2), δv1

⟩=

∫∫O1×(0,T )

(µ1v1 + φ11O1) δv1 dx dt.

Since δv1 is arbitrary in L2(O1 × (0, T )), we have proved that

∂J1

∂v1

(v1, w2) = µ1v1 + φ11O1 .

73

Thus, the first equality in (3.5) is equivalent to say that the state y associated to v1

and v2 must satisfy

yt −∆y = f − 1

µ1

φ11O1 + v21O2 in Q,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x) in Ω,

where φ1 solves

−φ1,t −∆φ1 = [y − y1,d]1O1,din Q,

φ1 = 0 on Σ1,

∂φ1

∂n= 0 on Σ2

φ1(x, T ) = 0 in Ω.

Arguing in a similar way, we see that the second equality in (3.5) can be also

rewritten in terms of y, v1 and an adequate second adjoint variable φ2.

Putting all together, we obtain the following reformulation of (3.5):

yt −∆y = f − 1

µ1

φ11O1 −1

µ2

φ21O2 in Q,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x) in Ω,

(3.10)

−φi,t −∆φi = [y − yi,d]1Oi,d in Q,

φi = 0 on Σ1,

∂φi∂n

= 0 on Σ2,

φi(x, T ) = 0 in Ω,

(3.11)

74

vi = − 1

µiφi∣∣Oi×(0,T )

, i = 1, 2. (3.12)

This is the optimality system we have to solve in order to compute a Nash equi-

librium for (3.1) with respect to the functionals (3.3).

3.3.2 A Fixed-Point Algorithm for Solving (3.10)–(3.12)

Let us introduce the notation Vi := L2(Oi × (0, T )) and V := V1 × V2. The scalar

product in V is given by

((v1, v2), (v1, v2))V :=

∫∫O1×(0,T )

v1v1 dx dt+

∫∫O2×(0,T )

v2v2 dx dt.

In the sequel, it will be assumed that µ1 and µ2 are sufficiently large. Under this

assumption, it is not difficult to check that (3.5) can be written in the form

a(v, v) = L(v) ∀v ∈ V, v ∈ V, (3.13)

where a : V × V 7→ R is a bilinear, continuous, symmetric, and V -elliptic form and

L : V 7→ R is a linear continuous form (they are explicitly given in Appendix; note

that the V -ellipticity of a(· , ·) is implied by the fact that µ1 and µ2 are sufficiently

large).

Therefore, as a consequence of Lax-Milgram Theorem (see for instance Corol-

lary 5.8 in [13]), (3.5) possesses a unique solution. This solution can be computed

by the following algorithm of the fixed-point kind:

ALG 1:

a) Choose v0 = (v01, v

02) ∈ V.

b) Then, for given n ≥ 0 and vn = (vn1 , vn2 ) ∈ V, compute the solution yn to (3.10)

with controls vi = vni , then the solutions φni to the systems (3.11) with y = yn

and finally the vn+1i from (3.12) with φni .

It is not difficult to see that, it µ1 and µ2 are sufficiently large, the previous iterates

converge to the unique Nash equilibrium. This is justified in detail in Appendix.

75

3.3.3 Reduction to Finite Dimension

In this and the following section, we will describe and use some approximate spaces

and schemes. The reduction of (3.10)–(3.12) to finite dimension can be performed

as follows in two steps:

• Step 1: Approximation in time. We consider the time discretization step ∆t,

defined by ∆t = T/M , where M is a positive integer. Then, if we set tm := m∆t for

all m, we have:

0 = t0 < t1 < t2 < · · · < tM = T.

Now, we approximate V1 and V2 respectively by

V ∆t1 :=

(L2(O1)

)Mand V ∆t

2 :=(L2(O2)

)M.

Accordingly, we can interpret the elements of V ∆ti as controls in Vi that are continuous

and piecewise constant in time.

• Step 2: Approximation in space. From now on, we will assume that Ω is a

polygonal domain of R2. We will also assume that the Oi,d and Oi are polygonal. We

introduce a triangulation Th of Ω, where h is the largest length of the edges of the

triangles of Th and the Oi,d and Oi are unions of triangles. Next, we approximate

the space L2(0, T ;H1(Ω)

)by W∆t

h , where

W∆th := (Wh)

M , Wh :=z ∈ C 0(Ω) : z

∣∣K∈ P1(K) ∀K ∈ Th

and P1(K) is the space of the polynomial functions on K of degree ≤ 1; thus,

dim(P1(K)) = 3 and dim(Wh) = Nh, where Nh is the number of vertices of Th. In

this second step, we first approximate Vi by the finite dimensional space V ∆ti,h , defined

as follows:

V ∆ti,h := (Vi,h)

M , Vi,h :=z ∈ C 0(Oi) : z

∣∣K∈ P1(K)) ∀K ∈ Oi

;

then, we set V ∆th := V ∆t

1,h × V ∆t2,h . Finally, we consider the space

z ∈ L2(0, T ;H1(Ω)) : z(·, t)∣∣Γ1

= 0 a.e. in (0, T )

and its finite dimensional version W∆th,0, where

W∆th,0 = (Wh,0)M , Wh,0 =

z ∈ Wh : z

∣∣Γ1

= 0.

76

The state equation (3.1) and the adjoint systems (3.11) can be approximated in

time and space by incorporating (for instance) implicit Euler finite difference and

P1-Lagrange finite element techniques. This allows to compute a state y∆th and two

adjoint states φ∆ti,h for each control pair v = (v1, v2) ∈ V ∆t

h .

In accordance with these definitions, we can now approximate (3.5) by a finite

dimensional problem: ∂J∆t

1,h

∂v1

(v1, v2) = 0,

∂J∆t2,h

∂v2

(v1, v2) = 0,

(3.14)

where the J∆ti,h are the finite dimensional versions of the Ji induced by time and space

approximation.

3.3.4 Fixed-Point Solution of the Discretized Linear Prob-

lem

We can solve the approximate problem (3.14) by the following fixed-point algorithm:

ALG 1bis:


02) ∈ V ∆t

h and introduce an approximation y0,h ∈ Wh,0 to y0.

b) Then, for given n ≥ 0 and vn = (vn1 , vn2 ) ∈ V ∆t

h , compute the approximate state

ynh by solving

yn,0h = yh,0,∫Ω

(1

∆t

(yn,m+1h − yn,mh )z +∇yn,m+1

h · ∇z)dx

=

∫Ω

(f(x, tm+1) + vn1 (x, tm+1)1O1 + vn2 (x, tm+1)1O2

)z dx,

∀z ∈ Wh,0, yn,m+1h ∈ Wh,0, m = 0, 1, ...,M − 1,

(3.15)

77

the approximate adjoint states φni,h by solving

φn,Mi,h = 0,∫Ω

(− 1

∆t

(φn,m+1i,h − φn,mi,h )z +∇φn,mi,h · ∇z

)dx

=

∫Oi,d

(ymh − yi,d(x, tm)

)z dx,

∀z ∈ Wh,0, φn,mi,h ∈ Wh,0, m = M − 1, M − 2, ... , 0

(3.16)

and, finally, set

vn+1i = − 1

µiφ∆ti,h

∣∣∣∣Oi×(0,T )

, i = 1, 2.

3.4 Computation of Pareto Equilibria for (3.1)

This section is devoted to characterize and compute Pareto equilibrium pairs for

(3.1) with respect to the Ji. Specifically, we will solve (3.7) for any λ ∈ (0, 1); this

will provide a whole family of equilibria where, eventually, we can perform a further

choice.

3.4.1 A Formulation Equivalent to (3.7) and a Fixed-Point

Algorithm

From the arguments in Section 3.3.1, it is clear that, for any λ ∈ (0, 1) and any (v1, v2) ∈V , one has:(

λ∂J1

∂v1

+ (1− λ)∂J2

∂v1

)(v1, v2) = λ(µ1v1 + φ11O1) + (1− λ)φ21O2

and (λ∂J1

∂v2

+ (1− λ)∂J2

∂v2

)(v1, v2) = λφ11O1 + (1− λ)(µ2v2 + φ21O2),

where the φi solve the adjoint systems (3.11).

78

Consequently, (3.7) is equivalent to the optimality system

yt −∆y = f + v11O1 + v21O2 in Q,

y = 0 on Σ1

∂y

∂n= 0 on Σ2

y(x, 0) = y0(x) in Ω,

(3.17)

−φi,t −∆φi = [y − yi,d]1Oi,d in Q,

φi = 0 on Σ1,

∂φi∂n

= 0 on Σ2,

φi(x, T ) = 0 in Ω,

(3.18)

v1 = − 1

µ1

(φ1 +

1− λλ

φ2

)∣∣∣O1×(0,T )

, v2 = − 1

µ2

( λ

1− λφ1 + φ2

)∣∣∣O2×(0,T )

. (3.19)

Let us fix λ ∈ (0, 1). Again, we will assume that µ1 and µ2 are sufficiently large.

As in the case of Nash equilibria, we can rewrite (3.17)–(3.19) in the form

a(v, v) = L(v) ∀v ∈ V, v ∈ V, (3.20)

where a : V ×V 7→ R is bilinear, continuous, symmetric and V -elliptic and L : V 7→ R

is a linear continuous form.

Therefore, as a consequence of the Lax-Milgram Theorem, for each λ ∈ (0, 1),

(3.7) has a unique solution. In order to compute this solution, we can use the

following fixed-point algorithm:

ALG 2:


02) ∈ V .


with vi = vni , then the solutions φni to the adjoint systems (3.18) with y = yn

and, finally, compute the vn+1i using (3.19) with φi = φni .

79

3.4.2 Finite Dimensional Approximation and Solution

As in Section 3.3.3, let us fix ∆t = T/M , the corresponding tm and a triangulation

Th of Ω such that the Oi,d and Oi are unions of triangles.

Conserving the notation introduced in Section 3.3.3, we see that, in practice, the

task reduces to solve a system of the form(λ∂J∆t

1,h

∂v1

+ (1− λ)∂J∆t

2,h

∂v1

)(v1, v2) = 0,

(λ∂J∆t

1,h

∂v2

+ (1− λ)∂J∆t

2,h

∂v2

)(v1, v2) = 0.

(3.21)

This can be achieved through the following fixed-point iterates:

ALG 2bis:


02) ∈ V ∆t

h and fix an approximation y0,h ∈ Wh,0 to y0.



ynh by solving (3.15), the approximate adjoint states φni,h by solving (3.16) and

finally set

vn+11 = − 1

µ1

(φn1,h +

1− λλ

φn2,h

)∣∣∣O1×(0,T )

,

and

vn+12 = − 1

µ2

( λ

1− λφn1,h + φn2,h

)∣∣∣O2×(0,T )

.


In this and the following section, we will consider the more general case of a semilinear

parabolic system. The aim is again to characterize and compute equilibrium pairs.

Note that, now, (3.5) is only a necessary condition (not sufficient in general) for

(v1, v2) to be a Nash equilibrium. However, our goal will be to solve this system; this

will suffice in practice many times.

80

Arguing as in the linear case, we can rewrite (3.5) equivalently as a coupled opti-

mality system. Thus, after some straightforward computations, we find the following:

yt −∆y + F (y) = f + v11O1 + v21O2 in Q,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x) in Ω,

(3.22)

−φi,t −∆φi + F ′(y)φi = [y − yi,d]1Oi,d in Q,

φi = 0 on Σ1,

∂φi∂n

= 0 on Σ2,

φi(x, T ) = 0 in Ω,

(3.23)

vi = − 1

µiφi

∣∣∣Oi×(0,T )

, i = 1, 2. (3.24)

Once more, let us assume that µ1 and µ2 are large enough. Then, (3.22)–(3.24)

can be rewritten as a nonlinear fixed-point equation in V which possesses at least

one solution; more details are given in Appendix.

In the sequel, we will present two iterative algorithms for the solution of (3.22)–

(3.24). For simplicity, let us assume that F is C1 (and globally Lipschitz-continuous)

and let us introduce G : R 7→ R, with

G(s) :=

F (s)− F (0)

sif s 6= 0,

F ′(0) otherwise.

Note that F (s) ≡ G(s) · s + F (0). Consequently, for any (v1, v2) ∈ V , the PDE in

(3.2) can be rewritten in the form

yt −∆y +G(y)y = f − F (0) + v11O1 + v21O2 .

The first algorithm is the following:

ALG 3 :

81


02) ∈ V . Set y−1(x, t) ≡ y0(x).

b) Then, for given n ≥ 0, yn−1 and vn = (vn1 , vn2 ) ∈ V , do the following:

b1) Set yn,0 = yn−1

b2) Then, for given ` ≥ 0 and yn,`, compute the solution yn,`+1 to the system

yt −∆y +G(yn,`)y = f − F (0) + vn11O1 + vn21O2 in Q ,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x) in Ω.

(3.25)

After convergence, denote by yn the last computed yn,`+1.

b3) Compute the adjoint states φni by solving the systems (3.23) with y = yn.

b4) Finally, set

vn+1i = − 1

µiφni

∣∣∣Oi×(0,T )

, i = 1, 2. (3.26)

Our second algorithm is a simplification of ALG 3 where we simultaneously

iterate to solve the semilinear PDE in (3.22) and upgrade the controls:

ALG 4 :


02) ∈ V and set y−1(x, t) ≡ y0(x).

b) Then, for given n ≥ 0, yn−1 and vn = (vn1 , vn2 ) ∈ V , compute yn by solving

yt −∆y +G(yn−1)y = f − F (0) + vn11O1 + vn21O2 in Q ,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x) in Ω,

(3.27)

compute the adjoint states φni by solving the systems (3.23) with y = yn and

finally compute the vn+1i from (3.26).

82

Arguing as in Sections 3.3.3, 3.3.4 and 3.4.2, we can introduce finite dimensional

approximations to (3.22)–(3.24) with finite difference and finite element techniques.

Accordingly, we can present finite dimensional versions of ALG 3 and ALG 4:

ALG 3bis:


02) ∈ V ∆t

h , fix an approximation y0,h ∈ Wh,0 to y0 and set

y−1h (x, t) ≡ y0,h(x).

b) Then, for given n ≥ 0, yn−1h and vn = (vn1 , v

n2 ) ∈ V ∆t

h , do the following:

b1) Set yn,0 = yn−1h .

b2) Then, for given ` ≥ 0 and yn,`h , compute yn,`+1h as follows:

yn,`+1,0h = y0,h,∫Ω

[ 1

∆t

(yn,`+1,m+1h − yn,`+1,m

h

)z +∇yn,`+1,m+1

h · ∇z

+G(yn,`,m+1h )yn,`+1,m+1

h z]dx

=

∫Ω

(f(x, tm+1)− F (0) + vn1 (x, tm+1)1O1

+ vn2 (x, tm+1)1O2

)z dx

∀z ∈ Wh,0, yn,`+1,m+1h ∈ Wh,0, m = 0, 1, ...,M − 1.

(3.28)

After convergence, denote ynh the last computed yn,`+1h .

b3) Compute the approximate adjoint states φni,h as follows:

φn,Mi,h = 0,∫Ω

[− 1

∆t

(φn,m+1i,h − φn,mi,h

)z +∇φn,mi,h · ∇z + F ′(ynh(x, tm))φn,mi,h z

]dx

=

∫Oi,d

(ynh(x, tm)− yi,d(x, tm)

)z dx

∀z ∈ Wh,0, φn,mi,h ∈ Wh,0, m = M − 1,M − 2, ..., 0.

(3.29)

83

b4) Finally, set

vn+1i = − 1

µiφni,h

∣∣∣Oi×(0,T )

, i = 1, 2. (3.30)

ALG 4bis:


02) ∈ V ∆t

h , fix an approximation y0,h ∈ Wh,0 to y0 and set

y−1h (x, t) ≡ y0,h(x).

b) Then, for given n ≥ 0, yn−1 and vn = (vn1 , vn2 ) ∈ V ∆t

h , compute ynh by solving

yn,,0h = y0,h,∫Ω

[ 1

∆t

(yn,m+1h − yn,mh

)z +∇yn,m+1

h · ∇z

+G(yn−1h (x, tm+1))yn,m+1

h z]dx

=

∫Ω

(f(x, tm+1)− F (0) + vn1 (x, tm+1)1O1

+ vn2 (x, tm+1)1O2

)z dx

∀z ∈ Wh,0, yn,m+1h ∈ Wh,0, m = 0, 1, ...,M − 1,

(3.31)

compute the φni,h by solving the linear systems (3.29) and finally compute the

vn+1i from (3.30).


As in Section 3.5, we can try to solve a system that can be viewed as a necessary

condition of the Pareto equilibrium property. More precisely, we fix λ ∈ (0, 1) and

we try to find a solution to (3.7).

Arguing as before, we get the following equivalent formulation:

yt −∆y + F (y) = f + v11O1 + v21O2 in Q,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x) in Ω,

(3.32)

84

−φi,t −∆φi + F ′(y)φi = [y − yi,d]1Oi,d in Q,

φi = 0 on Σ1,

∂φi∂n

= 0 on Σ2,

φi(x, T ) = 0 in Ω,

(3.33)

v1 = − 1

µ1

(φ1 +

1− λλ

φ2

)∣∣∣O1×(0,T )

, v2 = − 1

µ2

( λ

1− λφ1 + φ2

)∣∣∣O2×(0,T )

. (3.34)

Assuming that µ1 and µ2 are large enough, it is possible rewrite (3.32)–(3.34) as

a nonlinear fixed-point equation in V that possesses at least one solution; see the

details in Appendix.

On the other hand, we can introduce fixed-point iterates for the solution of (3.32)–

(3.34) as follows:

ALG 5:


02) ∈ V and set y−1(x, t) ≡ y0(x).


b1) Set yn,0 = yn−1.

b2) Then, for given ` ≥ 0 and yn,`, compute the solution yn,`+1 to (3.25).



b4) Finally, set

vn+11 = − 1

µ1

(φn1 +

1− λλ

φn2

)∣∣∣O1×(0,T )

vn+12 = − 1

µ2

( λ

1− λφn1 + φn2

)∣∣∣O2×(0,T )

.

(3.35)

ALG 6:


02) ∈ V and set y−1(x, t) ≡ y0(x)

85

b) Then, for given n ≥ 0, yn−1 and vn = (vn1 , vn2 ) ∈ V , compute yn by solving (3.27),

compute the φni by solving the systems (3.23) and finally compute the vn+1i

from (3.35).

To end this section, let us present the finite dimensional versions of ALG 5

and ALG 6, where we have conserved the notation introduced in Section 3.3.3.

ALG 5bis:


02) ∈ V ∆t

h , fix as approximation yh,0 ∈ Wh,0 to y0 and set

y−1h (x, t) ≡ y0,h(x).


n2 ) ∈ V ∆t


b1) Set yn,0h = yn−1h .

b2) Then, for given ` ≥ 0 and yn,`h , compute yn,`+1h from (3.28).

After convergence, denote by ynh the last computed yn,`+1h .

b3) Compute the φni,h from (3.29).

b4) Finally, compute the vn+1i from (3.35).

ALG 6bis:


02) ∈ V ∆t

h , fix an approximation yh,0 ∈ Wh,0 to y0 and set

y−1(x, t) ≡ y0,h(x).


n2 ) ∈ V ∆t

h , compute ynh from (3.31),

compute φni,h from (3.29) and finally compute the vn+1i from (3.35).

3.7 Some Numerical Experiments

This section is devoted to present the results of several numerical experiments; they

have been performed with the FreeFem++ library; see:

http://www.freefem.org/ff++.

86

3.7.1 Description of the Problems

The domain Ω and the sets O1, O2, O1,d and O2,d are the following:

Ω = B((0, 0); 6), O1 = (3, 4)× (1, 3), O2 = (3, 4)× (−3,−1),

O1,d = O2,d = (1, 2)× (−3, 3).

This set will be denoted by Od in the sequel. We take T = 2, Γ1 = ∂Ω and, accord-

ingly, Γ2 = 0. The functions f , y0 and F (for semilinear problems) are respectively

given by

f(x, t) ≡ 1O with O = (1, 2)× (−3, 3), y0(x) ≡ 1, F (s) ≡ s(1 + sin s).

On the other hand, the target functions yi,d will be

y1,d(x, t) ≡ x21 − x2

2, y2,d(x, t) ≡ x1x2.

The data f, y1,d and y2,d are depicted in Fig. 3.1–3.3.

Figure 3.1 : The function f

87

Figure 3.2: The function y1,d Figure 3.3: The function y2,d

The sets Ω,O, O1, O2 and Od and the initial mesh are displayed in Fig. 3.4.

Figure 3.4 - The domain and subdomains and the initial mesh

88

Figure 3.5: The initial mesh. Number of

vertices: 1228. Number of triangles: 2404

Figure 3.6: The final adapted mesh.

Number of vertices: 1460. Number of tri-

angles: 2781

The number of time steps in M = 40 (this gives ∆t = 0.05).

In the following sections, we present several Tests for the algorithms in Sections 3

to 6. We have always taken µ1 = µ2 and this common value has been denoted by µ.

The stopping criteria have been

‖(vn+11 , vn+1

2 )− (vn1 , vn2 )‖

‖(vn+11 , vn+1

2 )‖≤ ε (3.36)

for ALG 1bis and ALG 2bis,

‖yn,`+1 − yn,`)‖‖(yn,`+1‖

≤ ε and (3.36), (3.37)

for ALG 3bis and ALG 5bis and

‖(vn+11 , vn+1

2 )− (vn1 , vn2 )‖+ ‖yn+1 − yn‖

‖(vn+11 , vn+1

2 )‖+ ‖yn+1‖≤ ε (3.38)

for ALG 4bis and ALG 6bis, with ε = 10−5.

89

When dealing with the computation of Nash equilibria, we have considered several

values of µ, ranging from µ = 0.15 to µ = 9.5. As expected, we have seen that the

lower µ is the less favorable situation is found; more explanations can be found

in Appendix. On the other hand, the computations of Pareto equilibria have been

carried out for fixed µ and various λ.

3.7.2 Computation of Nash and Pareto Equilibria in the Lin-

ear Case

Test 1: ALG 1bis for (the numerical approximation of) (3.1)

We take µ = 0.15, 0.2, ... , 8.5, 9.5. For µ = 0.15, the state at time t = T and the

adjoint states at time t = 0 are displayed in Fig. 3.7–3.9. For µ = 9.5, the state at

t = T can be viewed in Fig. 3.10.

Figure 3.7: ALG 1bis. The computed

state at t = T for µ = 0.15

Figure 3.8: ALG 1bis. The first adjoint

state at t = 0 for µ = 0.15

In order to illustrate the behaviour of ALG 1bis, we present in Fig. 3.11 the

number of iterates needed to get (3.36) as a function of µ.

All these computations, as well as those in the following paragraphs, have been

made in combination with mesh adaptation techniques. To give an idea of the related

work, we present in Fig. 4.2 the final (adapted) mesh found and used for µ = 4.5.

90

Test 2: ALG 2bis for (3.1)

We take µ = 4.5 and λ = 0.05, 0.10, ... , 0.95. The computed states at time t = T

are displayed in Fig. 3.12, 3.13, 3.14 and 3.15, respectively for λ = 0.05, 0.25, 0.75

and 0.95. Also, the number of iterates required by (3.36) as a function of λ is shown

in Fig. 3.16.

3.7.3 Computation of Nash and Pareto Equilibria in the

Semilinear Case

Test 3: ALG 3bis and ALG 4bis for (3.2)

We take here various µ, ranging again from 0.15 to 9.5. The number of iterates

needed to get (3.37) or (3.38) is given in Fig. 3.17 and 3.18.

Figure 3.9: ALG 1bis. The second ad-

joint state at t = 0 for µ = 0.15

Figure 3.10: ALG 1bis. The state at

t = T for µ = 9.5

91

Figure 3.11 - ALG 1bis. Iterates vs. µ

Test 4: ALG 5bis and ALG 6bis for (3.2)

We take λ as before. For completeness, we present in Fig. 3.19 and 3.20 the computed

states at t = T respectively for λ = 0.5 and λ = 0.95.

The behaviour of these algorithms is illustrated in Fig. 3.21 and 3.22.

The L2 norms of the computed controls v1 and v2 are depicted in Fig. 3.23. The

values found for the cost functionals J1 and J2 are shown (at logarithmic scale)

in Fig. 3.24.

3.8 Appendix

3.8.1 Existence and uniqueness of a solution to (3.5)

Let us consider the mapping

(v1, v2) ∈ V 7→(∂J1

∂v1

(v1, v2),∂J2

∂v2

(v1, v2)

). (3.39)

Obviously, it is affine, that is, there exists a linear continuous mapping A ∈ L (V ;V )

and there exists b ∈ V such that(∂J1

∂v1

(v1, v2),∂J2

∂v2

(v1, v2)

)= A (v1, v2)− b ∀(v1, v2) ∈ V.

Let us identify the mapping A .

92


state at t = T for λ = 0.05


state at t = T for λ = 0.25


t = T for λ = 0.75


t = T for λ = 0.95

93

Figure 3.16 - ALG 2bis. Iterates vs. λ

Figure 3.17: ALG 3bis. Iterates vs. µ Figure 3.18: ALG 4bis. Iterates vs. µ

For every (v1, v2) ∈ V , one has

A (v1, v2) =(µ1v1 + φ11O1 , µ2v2 + φ21O2

),

where φi is the solution to

−φi,t −∆φi = y1Oi,d in Q,

φi = 0 on Σ1,

∂φi∂n

= 0 on Σ2,

φi(x, T ) = 0 in Ω,

(3.40)

94


t = T for λ = 0.5


t = T for λ = 0.95

Figure 3.21: ALG 5bis. Iterates vs. λ Figure 3.22: ALG 6bis. Iterates vs. λ

95

Figure 3.23: ALG 6bis. Norms of v1 and

v2 vs. λ

Figure 3.24: ALG 6bis. J1 and J2 vs. λ

and y is the solution to

yt −∆y = v11O1 + v21O2 in Q,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = 0 in Ω.

(3.41)

Proposition 3.8.1 The mapping A is linear, continuous and symmetric. Further-

more, if µ1 and µ2 are sufficiently large (depending on Ω and T ), A is strongly

positive, that is, there exists α0 > 0 such that

(A (v1, v2), (v1, v2))V ≥ α0‖(v1, v2)‖2V ∀(v1, v2) ∈ V. (3.42)

Proof: The argument is given in [65], Proposition 4.1. For completeness, let us

recall the proof of strong positiveness.

Note that, for any (v1, v2) ∈ V ,

(A (v1, v2), (v1, v2))V = µ1

∫∫O1×(0,T )

|v1|2 dx dt+ µ2

∫∫O2×(0,T )

|v2|2 dx dt

+

∫∫O1×(0,T )

φ1 v1 dx dt+

∫∫O2×(0,T )

φ2 v2 dx dt.(3.43)

96

Let z1 (resp. z2) be the solution to (3.41) with v2 = 0 (resp. with v1 = 0). Then∫∫Oi×(0,T )

φi vi dx dt =

∫∫Oi×(0,T )

φi (zi,t −∆zi) dx dt =

∫∫Oi×(0,T )

y zi dx dt. (3.44)

Consequently, there exists C0 depending on Ω and T such that∣∣∣∣∫∫Oi×(0,T )

φi vi dx dt

∣∣∣∣ ≤ C0‖(v1, v2)‖V ‖vi‖L2(Oi×(0,T )), i = 1, 2. (3.45)

From (3.43) and (3.45), we see at once that, if min(µ1, µ2) >√

2C0, then

(A (v1, v2), (v1, v2))V ≥ (min(µ1, µ2)−√

2C0)‖(v1, v2)‖2V ∀(v1, v2) ∈ V,

when (3.42) holds. 2

Remark 3.8.1 From (3.43) and (3.44), we also observe that, if the sets O1,d and O2,d

coincide and we set Od = O1,d, then

(A (v1, v2), (v1, v2))V ≥ min(µ1, µ2)‖(v1, v2)‖2V +

∫∫Od×(0,T )

|y|2 dx dt ,

∀(v1, v2) ∈ V.

Therefore, in this case, A is always strongly positive, regardless of the sizes of

the µi.

Let us identify b, that is, the constant part of the affine mapping (3.39). One

has:

b =(φ11O1 , φ21O2

),


−φi,t −∆φi =(y − yi,d

)1Oi,d in Q,

φi = 0 on Σ1,

∂φi∂n

= 0 on Σ2,

φi(x, T ) = 0 in Ω

(3.46)

97

and y is the solution of

yt −∆y = f in Q,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x) in Ω.

(3.47)

Now, if we define a(· , ·) : V × V 7→ R by

a(v, w) := (A (v), w)V ∀v, w ∈ V,

and L : V 7→ R by

L(v) := (b, v)V , ∀v ∈ V,

we deduce that (3.5) is equivalent to (3.13).

From Proposition 3.8.1, we have that the mapping a(· , ·) is bilinear, continuous

and symmetric. Moreover, if µ1 and µ2 are large enough, it is also V -elliptic. On

the other hand, L is linear and continuous. Hence, the existence and uniqueness of

solution to (3.13) is ensured if the µi are sufficiently large.

Remark 3.8.2 Let us fix λ ∈ (0, 1) and let us consider the system (3.43). Arguing

as before, it becomes clear that there exists C(Ω, T, λ) such that, if min(µ1, µ2) >

C(Ω, T, λ), this system has exactly one solution.

3.8.2 The convergence of ALG 1

Let us recall the iterates: for any given n ≥ 0 and vn = (vn1 , vn2 ) ∈ V , we compute

yn, φni and then vn+1 = (vn+11 , vn+1

2 ) by solving the linear systems

ynt −∆yn = f + vn11O1 + vn21O2 in Q,

yn = 0 on Σ1,

∂yn

∂n= 0 on Σ2,

yn(x, 0) = y0(x) in Σ1,

98

−φni,t −∆φni = [yn − yi,d]1Oi,d in Q,

φni = 0 on Σ1,

∂φni∂n

= 0 on Σ2,

φni (x, T ) = 0 in Σ1,

vn+1i = − 1

µiφni∣∣Oi×(0,T )

, i = 1, 2.

Obviously, this can be written in the form

vn+1 = Bvn + h,

where B : V 7→ V are defined as follows:

• Bv = (B1v,B2v), Biv = − 1


and φi is the solution to (3.40), where

y is the solution to (3.41).

• h = (h1, h2), hi = − 1


and φi is the solution to (3.46), where y is

the solution to (3.47).

It is easy to check that B ∈ L(V ;V ) and

‖Bv‖V ≤C(Ω, T )

min(µ1, µ2)‖v‖V ∀v ∈ V.

Hence, if µ1 and µ2 are large enough, the mapping v 7→ Bv+ h is a contraction and

the sequence furnished by ALG 1 converges to the unique solution to (3.5).

3.8.3 Existence and uniqueness of a solution to (3.22)–(3.24)

In this section, we assume that F : R 7→ R is globally Lipschitz-continuous and

|F (z1)− F (z2)| ≤ CF |z1 − z2| ∀z1, z2 ∈ R.

99

Note that the couple (v1, v2) solves (3.22)–(3.24) if and only if it is a fixed-point of

the nonlinear mapping Λ : V 7→ V , whereΛ(v) = (Λ1(v),Λ2(v)), Λi(v) = − 1


,

φi is the solution to (3.23) for i = 1, 2,


It is not difficult to check that there exists C(Ω, T, CF , ‖f‖L2(Q)) such that, if

min(µ1, µ2) > C(Ω, T, CF , ‖f‖L2(Q)),

the mapping Λ is a contraction. Indeed, from the usual parabolic energy estimates,

it is clear that

‖Λ(v)− Λ(v)‖V ≤C(Ω, T )

min(µ1, µ2)‖(φ1, φ2)− (φ1, φ2)‖L2(Q)×L2(Q)

≤ C(Ω, T, CF )

min(µ1, µ2)‖y − y‖L2(Q)

≤C(Ω, T, CF , ‖f‖L2(Q))

min(µ1, µ2)‖v − v‖V

for all v, v ∈ V , where the notation is self-explanatory.

As a consequence, we find that, if µ1 and µ2 are large enough, (3.22)–(3.24)

possesses a unique solution. In other words, (3.5) is uniquely solvable.

Remark 3.8.3 For any fixed λ ∈ (0, 1), we can consider the system (3.32)–(3.34).

Arguing in a similar way, we deduce that there exists C(Ω, T, CF , ‖f‖L2(Q), λ) such

that, for greater values of min(µ1, µ2), this system possesses exactly one solution.

Chapter 4

On the Computation of Nash and

Pareto Equilibria for some

Bi-Objective Optimal Control

Problems for the Wave Equation

101

On the Computation of Nash and Pareto

Equilibria for some Bi-Objective Optimal Control

Problems for the Wave Equation

Pitagoras P. de Carvalho, Enrique Fernandez-Cara and Juan Lımaco

Abstract. In this manuscript we discuss the numerical implementation

of a systematic method for solving some multi-objective optimal control

problems for wave equations. More precisely, for such problems, we look

for Nash and Pareto equilibria, which respectively correspond to appro-

priate noncooperative and cooperative strategies. The numerical meth-

ods described here consist in a combination of: finite element approxima-

tions for the space discretization; explicit finite difference schemes for the

time discretization; a preconditioned fixed-point algorithm for the solu-

tion of the discrete control problems. The efficiency of the computational

methodology is illustrated by the results of numerical experiments.

4.1 Introduction

In a classical mono-objective control problem for a system modeled by a differential

equation or system, we usually find a state equation or system and one control

with the mission of achieving a predetermined goal. Usually (but not always), the

goal is to minimize a cost functional in a prescribed family of admissible controls.

A more interesting situation arises when several objectives are considered. It can

also be expectable to have more than one control acting on the equation. In these

case, we are led to consider multi-objective control problems. In contrast with the

mono-objective case, various strategies for the choice of good controls can appear,

depending of the characteristics of the problem.

These strategies lead to what we call equilibria, that can be cooperative (when

the controls collaborate to achieve the goals) and noncooperative (in the opposite

case).

102

Since the concepts and arguments have origin in game theory and economics, the

notion of player is often used. Thus, for an extremal problem with p objectives or

functionals cost Ji to minimize, the noncooperative optimization strategy proposed

by Nash in [62] reduces to the search of a set of p players or controls vi, each of them

optimizing Ji with respect to the i-th variable. Again, if the Ji are regular enough

and no constraint is imposed, the vi can be characterized in terms of the derivatives

of Ji:∂Ji∂vi

(v1, . . . , vp) = 0, i = 1, . . . , p.

On the other hand, in theoretical economics, the cooperative optimization strat-

egy proposed by Pareto in [64] is a state of allocation of resources such that it is

impossible to get an individual improvement of the functional values without mak-

ing at least one individual worse off. In the framework of a multi-objective control

problem with p regular functionals cost Ji depending on p controls vi, in the absence

of constraints, Pareto equilibria must satisfy

p∑i=1

λiJ′i(v1, . . . , vp) = 0,

for some λi ≥ 0 with

p∑i=1

λi = 1.

Of course, there are several strategies for multiobjective optimization, an example

is the Stackelberg hierarchical-cooperative strategy in [67], where we can introduce

goal variations, such as Stackelberg-Nash, Stackelberg-Pareto, etc.

Some previous results on strategies for the control of differential equations are de-

veloped by:

• A. Ramos and R. Glowinski in [65], in this paper the authors develop numer-

ically optimal control results, using Nash strategy for multi-objective linear

problems.

• J. L. Lions in [59], in this work the author gives some results about the Pareto

strategies.

• M. O. Bristeau and R. Glowinski in [14], in this work the authors compare

Pareto and Nash strategies by using genetic algorithms to compute numerically

the solutions corresponding to these strategies.

103

The main goal of this paper is to discuss the numerical implementation of a

method for the optimal control of systems modeled by time dependent partial dif-

ferential equations, more specifically optimal control results in multi-objective wave

equations. In this paper, we shall consider a specific but important example since it

concerns the optimal control using Nash and Pareto strategies in linear and semilin-

ear wave equations.

4.2 The Problems and Their Motivations

Let Ω ⊂ RN be a bounded connected open set with regular boundary (N = 1, 2 or 3)

and assume that Γ1, Γ2 ⊂ ∂Ω, with Γ1 ∩ Γ2 = ∅ and ∂Ω = Γ1 ∪ Γ2. We will use the

notation Q := Ω× (0, T ), Σ1 := Γ1× (0, T ) and Σ2 := Γ2× (0, T ). In the sequel, we

will denote by (· , ·) and ‖ · ‖ the L2 scalar product and norm in Ω. We will denote

by C a generic positive constant; sometimes, we will indicate the data on which it

depends. Also, n = n(x) will stand for the outward unit normal vector to Ω at the

points x ∈ ∂Ω.

The problem that we consider are equations of state that be given by a linear or

semilinear wave PDE completed with appropriate boundary and initial conditions:

ytt −∆y = f1O + v11O1 + v21O2 in QT ,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x), yt(x, 0) = y1(x) in Ω

(4.1)

or

ytt −∆y + F (y) = f1O + v11O1 + v21O2 in QT ,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x), yt(x, 0) = y1(x) in Ω.

(4.2)

where y = y(x; t) is the state, F : R 7→ R is (globally) Lipschitz-continuous and

the right hand side f and the initial data y0 are prescribed.

104

In problems (4.1) and (4.2), O1, O2 ⊂ Ω are the control domains, with O1 ∩O2 = ∅ (both are supposed to be small); 1O1 and 1O2 are the characteristic

functions and the controls are v1 and v2.

Finally, we consider the following functionals cost for the problems (4.1) and

(4.2):

Ji(v1, v2) :=1

2

∫∫Oi,d×(0,T )


∫∫Oi×(0,T )

|vi|2 dx dt, i = 1, 2, (4.3)

where the µi are positive constants, yi,d = yi,d(x, t) are given functions andO1,d, O2,d ⊂Ω be open sets, representing observations domains for the controls v1 and v2.

The strategies for the multi-objective control problems considered in this paper

are the following:

1. Nash Equilibria

For every vi ∈ L2(Oi×(0, T )) i = 1, 2, we consider the following optimal control

problem:

Find v1(v2) ∈ L2(O1 × (0, T )) and v2(v1) ∈ L2(O2 × (0, T )), satisfyingJ1(v1(v2), v2) ≤ J1(v1, v2) ∀v1 ∈ L2(O1 × (0, T )),

J2(v1, v2(v1)) ≤ J2(v1, v2) ∀v2 ∈ L2(O2 × (0, T )).(4.4)

Note that, in the linear case, that is, when the state equation is (4.1), in view

of the strict convexity of v1 7→ J1(v1, v2), the (unique) v1(v2) satisfying (4.4)1

is characterized by the equality

∂J1

∂v1

(v1(v2), v2) = 0.

Similarly, the (unique) v2(v1) satisfying (4.4)2 is characterized by

∂J2

∂v2

(v1, v2(v1)) = 0.

Consequently, in the linear case, a Nash equilibrium is a pair (v1, v2) ∈ L2(O1×(0, T )) × L2(O2 × (0, T )) (where v1 = v1(v2) and v2 = v2(v1)) that solves the

105

coupled (optimality) system ∂J1

∂v1

(v1, v2) = 0,

∂J2

∂v2

(v1, v2) = 0.

(4.5)

Obviously in the semilinear case, (4.5) is in general only a necessary condition

for a pair (v1, v2) to be a Nash equilibrium.

2. Pareto Equilibria

Now, for every vi ∈ L2(Oi × (0, T )) (i = 1, 2), we want to find controls

vi ∈ L2(Oi×(0, T )) (i = 1, 2), such that there is no (v1, v2) 6= (v1, v2), satisfyingJ1(v1, v2) ≤ J1(v1, v2) and J2(v1, v2) ≤ J2(v1, v2),

with strict inequality for at least one Ji.(4.6)

Any couple (v1, v2) satisfying this property is called a Pareto equilibrium. In

the linear case, it is not difficult to prove that (v1, v2) is a Pareto equilibrium

if there exists λ ∈ (0, 1) with(λ∂J1

∂v1

+ (1− λ)∂J2

∂v1

)(v1, v2) = 0,

(λ∂J1

∂v2

+ (1− λ)∂J2

∂v2

)(v1, v2) = 0.

(4.7)

On the other hand, in the semilinear case, (4.7) is, as before, only a necessary

condition.

In this paper is presented efficient strategies for the numerical solution of these

multi-objective control problems. Obviously, these problems are very important from

the theoretical and practical viewpoints and appear frequently in the applications; for

some previous works on the subject (and also their connection to hierarchic control),

see for instance [14, 3, 4, 32, 33, 51, 56, 59, 65, 66].

Observation 4.2.1 A special remark should be made in case of

O1,d ∩ O2,d 6= ∅.

106

In this case, we must have a competition problem, with each control or player trying

to achieve different and possibly contradictory goals in a common domain. It is rea-

sonable to expect that this is the case where the behavior of the solution y associated

to the equilibrium (v1, v2) is most difficult to forecast. 2

The outline of the development of this article is as follows: In Section 4.3, we

will present a strategy for the computation of Nash equilibria for the linear system

(4.1), as in [65] and [66]. In Section 4.4, we will adapt the ideas to the computation

of Pareto equilibria. In Sections 4.5 and 4.6, we will consider semilinear systems of

the kind (4.2). Here, we will indicate how Nash and Pareto equilibria can be found.

Section 4.7 deals with some numerical experiments. Finally, some technical results

have been collected in Section 4.8.


4.3.1 Equivalent Formulation of the Optimality System (4.5)

Let v1 and w2 be given, with v1, w2 ∈ L2(O2 × (0, T )) and let δ1v1 be a small

perturbation of v1. Then, with obvious notation, we have:

δ1J1(v1, w2) = α1

∫∫O1,d×(0,T )

[y(v1, w2)− y1,d] δ1y dx dt

+ µ1

∫∫O1×(0,T )

v1δ1v1 dx dt+ ... , (4.8)

where δ1y is the solution to

(δ1y)tt −∆(δ1y) = (δ1v1)1O1 in Q,

δ1y = 0 on Σ1,

∂

∂n(δ1y) = 0 on Σ2,

δ1y(x, 0) = 0, (δ1y)t(x, 0) = 0 in Ω

(4.9)

and the dots contain higher order terms.

Now, let φ1 = φ1(x, t) be a function defined in Q. Let us assume for the moment

that φ1 possesses up to second-order derivatives in time and space in C0(Q).

107

Then, multiplying in (4.9) by φ1 and integrating by parts, we easily obtain∫Ω

[φ1 δ1yt−φ1,t δ1y

]dx∣∣∣t=Tt=0

+

∫∫Q

(∂2φ1

∂t2−∆φ1

)δ1y dx dt

−∫

Σ1

φ1

(∂δ1Y

∂n

)dΓ dt+

∫Σ2

(∂φ1

∂n

)δ1y dΓ dt (4.10)

=

∫∫O1×(0,T )

φ1 δ1v1 dx dt.

Now, let φ1 be the solution to the following backwards in time (adjoint) system:

φ1,tt −∆φ1 = [y(v1, w2)− y1,d]1O1,din Q,

φ1 = 0 on Σ1,

∂φ1

∂n= 0 on Σ2,

φ1(x, T ) = 0, φ1,t(x, T ) = 0 in Ω .

Then, we can deduce that (4.10) also holds for this φ1 and, consequently,⟨∂J1

∂v1

(v1, w2) , δ1v1

⟩=

∫O1×(0,T )

(µ1v1 + φ11O1) δ1v1 dx dt.

Since δ1v1 is arbitrary in L2(O1 × (0, T )), we have proved that

∂J1

∂v1

(v1, w2) = µ1v1 + φ11O1 .

Thus, the first equality in (4.5) is equivalent to say that the state y associated to v1

and v2 must satisfy

ytt −∆y = f − 1

µ1

φ11O1 + v21O2 in Q,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x), yt(x, 0) = y1(x) in Ω,

108

where v1 = − 1

µ1

φ11O1 and φ1 solves the system

φ1,tt −∆φ1 = [y − y1,d]1O1,din Q,

φ1 = 0 on Σ1,

∂φ1

∂n= 0 on Σ2,

φ1(x, T ) = 0, φ1,t(x, T ) = 0 in Ω.

Arguing in a similar way, we see that the second equality in (4.5) can also be

rewritten an adequate second adjoint variable φ2

(with v2 = − 1

µ1

φ21O2

).

Finally, we obtain the following reformulation of (4.5):

ytt −∆y = f − 1

µ1

φ11O1 −1

µ2

φ21O2 in Q,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x), yt(x, 0) = y1(x) in Ω,

(4.11)

φi,tt −∆φi = [y − yi,d]1Oi,d in Q,

φi = 0 on Σ1,

∂φi∂n

= 0 on Σ2,

φi(x, T ) = 0, φi,t(x, T ) = 0 in Ω,

(4.12)

vi = − 1

µiφi

∣∣∣Oi×(0,T )

, i = 1, 2 . (4.13)

This is the optimality system we have to solve in order to compute a Nash equi-

librium for (4.1) with respect to the functionals (4.3).

4.3.2 A Fixed-Point Algorithm for Solving (4.11)–(4.13)

Let us introduce the notations Vi := L2(Oi × (0, T )) and V := V1 × V2.

109

The scalar product in V is given by

((v1, v2), (v1, v2))V :=

∫∫O1×(0,T )

v1v1 dx dt+

∫∫O2×(0,T )

v2v2 dx dt .

We consider the mapping

(v1, v2) ∈ V 7→(∂J1

∂v1

(v1, v2),∂J2

∂v2

(v1, v2)

)∈ V .

From the linearity of (4.1) and the structures of J1 and J2, there exist A ∈ L (V ;V )

and b ∈ V such that(∂J1

∂v1

(v1, v2),∂J2

∂v2

(v1, v2)

)= A (v1, v2)− b ∀(v1, v2) ∈ V. (4.14)

For every (v1, v2) ∈ V , one has

A (v1, v2) =(µ1v1 + φ11O1 , µ2v2 + φ21O2

),

where A : V × V → R is a linear, continuous and symmetric. The equilibrium is

obtained when the equation in (4.14) is zero.

More details on A , b, and result of existence and uniqueness of solution for (4.5),

see Appendix.

This solution can be computed by the following algorithm of the fixed-point kind:

ALG 1:


02) ∈ V.


with controls vi = vni , then the solutions φni to the systems (4.12) with y = yn

and finally vn+1i from (4.13), with φi = φni .

It is not difficult to see that, if µ1 and µ2 are sufficiently large, the previous

iterates converge to the unique Nash equilibrium. This is justified in detail in the

Appendix.

110

4.3.3 Reduction to Finite Dimension

In this and the following section, we will describe and use some approximate spaces

and schemes. The reduction of (4.11)–(4.13) to finite dimension can be performed

as follows in two steps:

• Step 1: Approximation in time.

We consider the time discretization step ∆t, defined by ∆t = T/M , where M is

a positive integer. Then, if we set tm := m∆t, we have

0 < t1 < t2 < · · · < tM = T.

Now, we approximate V1 and V2 respectively by

V ∆t1 :=

(L2(O1)

)Mand V ∆t

2 :=(L2(O2)

)M.

Accordingly, we can interpret the elements of V ∆ti as controls in Vi that are

continuous and piecewise constant in time.

• Step 2: Approximation in space.

From now on, we will assume that Ω is a polygonal domain of R2. We will

also assume that the Oi,d and Oi are polygonal. We introduce a triangulation Th

of Ω, where h is the largest length of the edges of the triangles of Th and the

Oi,d and Oi are unions of triangles. Next, we approximate the set of solutions

W = w ∈ L∞(0, T ;H10 (Ω)) : wt ∈ L∞(0, T ;L2(Ω)) by W∆t

h , where

W∆th := (Wh)

M , Wh :=z ∈ C 0(Ω) : z

∣∣K∈ P1(K) ∀K ∈ Th

and P1(K) is the space of the polynomial functions of degree≤ 1; thus, dim(P1(K)) =

3 and dim(Wh) = Nh, where Nh is the number of vertices of Th. In this second

step, we first approximate Vi by V ∆ti,h , defined as follows:

V ∆ti,h = (Vi,h)

M , Vi,h :=z ∈ C 0(Oi) : z

∣∣K∈ P1(K) ∀K ∈ Oi

;

then, we set V ∆th := V ∆t

1,h × V ∆t2,h . Finally, we consider the finite-dimensional version

W∆th,0 , where

W∆th,0 = (Wh,0)N , Wh,0 =

z ∈ Wh : z

∣∣Γ1

= 0.

111

The state equation (4.1) and the adjoint systems (4.12) can be approximated in time

and space incorporating (for instance) implicit Euler finite differences in time and

spatial P1-Lagrange finite element techniques. This allows to compute a state y∆th

and two adjoint states φ∆ti,h for each control pair v = (v1, v2) ∈ V ∆t

h .

In accordance with these definitions, we can approximate (4.5) by a finite-dimensional

minimization problem: ∂J∆t

1,h

∂v1

(v1, v2) = 0,

∂J∆t2,h

∂v2

(v1, v2) = 0,

(4.15)

where the J∆ti,h are the finite-dimensional versions of the Ji induced by time and space

approximation.

4.3.4 Fixed-Point Solution of the Discretized Linear Prob-

lem

We can solve the approximate problem for (4.15) by the following fixed-point algo-

rithm:

ALG 1bis:


02) ∈ V ∆t

h and introduce an approximation (y0,h, y1,h) ∈ Wh,0×Wh,0 to (y0, y1).



yn,0h , yn,1h , ... as follows :

yn,0h = yh,0, yn,1h = yn,0h + ∆t · yh,1 ,∫Ω

[1

(∆t)2

(yn,m+1h − 2yn,mh + yn,m−1

h

)z +∇yn,m+1

h · ∇z]dx

=

∫Ω

(f(x, tm+1) + vn1 (x, tm+1)1O1 + vn2 (x, tm+1)1O2

)z dx,

∀z ∈ Wh,0, yn,m+1h ∈ Wh,0, m = 1, ...,M − 1,

(4.16)

112

the approximate adjoint states φn,Mi,h , φn,M−1i,h , ... by solving

φn,Mi,h = 0 , φn,M−1i,h = 0 ,∫

Ω

[1

(∆t)2

(φn,m+1i,h − 2φn,mi,h + φn,m−1

i,h )z +∇φn,m−1i,h · ∇z

]dx

=

∫Oi,d

(yn,m−1h − yi,d(x, tm−1)

)z dx,

∀z ∈ Wh,0, φn,mi,h ∈ Wh,0, m = M, M − 1, ... , 1, i = 1, 2

(4.17)

and, finally, set

vn+1i = − 1

µiφ∆ti,h

∣∣∣∣Oi×(0,T )

, i = 1, 2.


This section is devoted to characterize and compute Pareto equilibra for (4.1) with

respect to the functionals Ji. Specifically, we will solve (4.7) for any λ ∈ (0, 1); this

will produce a whole family of equilibria where, eventually, we can perform a further

choice.

4.4.1 A Formulation Equivalent to (4.7) and a Fixed-Point

Algorithm

From the arguments in Section 4.3.1, it is clear that, for any λ ∈ (0, 1) and any

(v1, v2) ∈ V one has:(λ∂J1

∂v1

+ (1− λ)∂J2

∂v1

)(v1, v2) = λ(µ1v1 + φ11O1) + (1− λ)φ21O2

and (λ∂J1

∂v2

+ (1− λ)∂J2

∂v2

)(v1, v2) = λφ11O1 + (1− λ)(µ2v2 + φ21O2),

where the φi solve the adjoint systems (4.12), with i = 1, 2.

113

Consequently, (4.7) is equivalent to the optimality system

ytt −∆y = f + v11O1 + v21O2 in Q,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x), yt(x, 0) = y1(x) in Ω,

(4.18)

φi,tt −∆φi = [y(v1, w2)− yi,d]1Oi,d in Q,

φi = 0 on Σ1,

∂φi∂n

= 0 on Σ2,

φi(x, T ) = 0 , φi,t(x, T ) = 0 in Ω,

(4.19)

v1 = − 1

µ1

(φ1+

(1− λ)

λφ2

)∣∣∣∣∣O1×(0,T )

, v2 = − 1

µ2

(λ

(1− λ)φ1+φ2

)∣∣∣∣∣O2×(0,T )

. (4.20)

for µ1 and µ2 are sufficiently large.

Let us fix λ ∈ (0, 1). As in the case of Nash equilibria, we consider A ∈ L (V ;V )

and b ∈ V such that for all (v1, v2) ∈ V ,((λ∂J1

∂v1

+ (1− λ)∂J2

∂v1

)(v1, v2),

(λ∂J1

∂v2

+ (1− λ)∂J2

∂v2

)(v1, v2)

)= A (v1, v2)− b ,

(4.21)

where A : V × V → R is a linear, continuous and symmetric. The equilibrium is

obtained when the equation in (4.21) equals zero.

Therefore, as a consequence of the Lax-Milgram Theorem, for each λ ∈ (0, 1),

the problem (4.7) has a unique solution. In order to compute this solution, we can

use the fixed-point algorithm as follows:

ALG 2:


02) ∈ V .


with vi = vni , then the solutions φni to the adjoint systems (4.19) with y = yn

and, finally, compute the vn+1i using (4.20) with φi = φni .

114

4.4.2 Finite Dimensional Approximation and Solution

As in Section 4.3.3, let us fix ∆t = ∆t = T/M , the corresponding tm and a triangu-

lation Th of Ω, such that the Oi,d and the Oi (i = 1, 2) are unions of triangles.

Conserving the notation introduced in Section 4.3.3, we see that in practice, the

task reduces to solve an approximate system to (4.7) of the form

(λ∂J∆t

1,h

∂v1

+ (1− λ)∂J∆t

2,h

∂v1

)(v1, v2) = 0,

(λ∂J∆t

1,h

∂v2

+ (1− λ)∂J∆t

2,h

∂v2

)(v1, v2) = 0.

(4.22)

This can be achieved through the following fixed-point iterates:

ALG 2bis:


02) ∈ V ∆t

h and fix an approximation (y0,h , y1,h) ∈ Wh,0 ×Wh,0

to (y0 , y1).


h , compute the approximate states

yn,0h , yn,1h , ... by solving (4.16), the approximate adjoint states φn,Mi,h , φn,M−1i,h , ...

by solving (4.17) and, finally, setvn+1

1 = − 1

µ1

(φn1,h +

1− λλ

φn2,h

)∣∣∣O1×(0,T )

,

vn+12 = − 1

µ2

( λ

1− λφn1,h + φn2,h

)∣∣∣O2×(0,T )

.


In this and the following section, we will consider the more general case of a semilinear

system. The aim is again to characterize and compute equilibrium pairs.

Note that, now, (4.5) is only a necessary condition (not sufficient in general) for

(v1, v2) to be a Nash equilibrium. However, our goal will be to solve this system; this

will suffice in practice many times.

115

Arguing as in the linear case, we can rewrite (4.5) equivalently as a coupled opti-

mality system. Thus, after some straightforward computations, we find the following:

ytt −∆y + F (y) = f + v11O1 + v21O2 in Q,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x) and yt(x, 0) = y1(x) in Ω,

(4.23)

φi,tt −∆φi + F ′(y)φi = [y − yi,d]1Oi,d in Q,

φi = 0 on Σ1,

∂φi∂n

= 0 on Σ2,

φi(x, T ) = 0 and φi,t(x, T ) = 0 in Ω,

(4.24)

vi = − 1

µiφi

∣∣∣Oi×(0,T )

, i = 1, 2. (4.25)

Once more, let us assume that µ1 and µ2 are large enough. Then, (4.23)–(4.25)

can be rewritten as a nonlinear fixed-point equation in V which possesses at least

one solution; more details are given in Appendix.

In the sequel, we will present two iterative algorithms for the solution of (4.23)–

(4.25). For simplicity, let us assume that F is C1 (and globally Lipschitz-continuous)

and let us introduce G : R 7→ R, with

G(s) :=

F (s)− F (0)

sif s 6= 0,

F ′(0) otherwise.

Note that F (s) ≡ G(s)s + F (0). Consequently, for any (v1, v2) ∈ V , the PDE in

(4.2) can be rewritten in the form

ytt −∆y +G(y)y = f − F (0) + v11O1 + v21O2 .

The first algorithm is the following:

116

ALG 3 :


02) ∈ V . Set y−1(x, t) ≡ y0(x) + ty1(x).



b2) Then, for given ` ≥ 0 and yn,`, compute the solution yn,`+1 to the system

ytt −∆y +G(yn,`)y = f − F (0) + vn11O1 + vn21O2 in Q ,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x) and yt(x, 0) = y1(x) in Ω.

(4.26)



b4) Finally, set

vn+1i = − 1

µiφni

∣∣∣Oi×(0,T )

, i = 1, 2. (4.27)

Our second algorithm is a simplification of ALG 3 where we simultaneously

iterate to solve the semilinear PDE in (4.23) and upgrade the controls with:

ALG 4 :


02) ∈ V and set y−1(x, t) ≡ y0(x) + ty1(x).

b) Then, for given n ≥ 0, yn−1 and vn = (vn1 , vn2 ) ∈ V , compute yn by solving

ytt −∆y +G(yn−1)y = f − F (0) + vn11O1 + vn21O2 in Q ,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,


(4.28)

compute the adjoint states φni (i = 1, 2), by solving the systems (4.24) with

y = yn and finally compute the vn+1i from (4.27).

117

Arguing as in Sections 4.3.3, 4.3.4 and 4.4.2, we can introduce finite dimensional

approximations to (4.23)–(4.25) with finite difference and finite element techniques.

Accordingly, we can present finite dimensional versions of ALG 3 and ALG 4:

ALG 3bis:


02) ∈ V ∆t

h , fix an approximation (y0,h, y1,h) ∈ Wh,0 ×Wh,0

to (y0, y1), and set y−1h (x, t) ≡ y0,h(x) + ty1,h(x).


n2 ) ∈ V ∆t


b1) Set yn,0 = yn−1h .

b2) Then, for given ` ≥ 0 and yn,`h , compute yn, `+1, 0h , yn, `+1, 1

h , ... as follows:

yn, `+1, 0h = y0,h, yn, `+1, 1

h = y0,h + ∆t · y1,h ,∫Ω

[(yn,`+1,m+1h − 2yn,`+1,m

h + yn,`+1,m−1h

(∆t)2

)z +∇yn,`+1,m+1

h · ∇z

+ G(yn,`,m+1h )yn,`+1,m+1

h z

]dx

=

∫Ω

(f(x, tm+1)− F (0) + vn1 (x, tm+1)1O1 + vn2 (x, tm+1)1O2

)z dx

∀z ∈ Wh,0, yn,`+1,m+1h ∈ Wh,0, m = 1, ...,M − 1.

(4.29)

After convergence, denote ynh the last computed yn,`+1h .

b3) Compute the approximate adjoint states φn,Mi,h , φn,M−1i,h , ... as follows:

φn,Mi,h = 0, φn,M−1i,h = 0,∫

Ω

[(φn,m+1i,h − 2φn,mi,h + φn,m−1

i,h

(∆t)2

)z +∇φn,m−1

i,h · ∇z

+ F ′(ynh(x, tm−1))φn,m−1i,h · z

]dx

=

∫Oi,d

(ynh(x, tm−1)− yi,d(x, tm−1)

)z dx

∀z ∈ Wh,0, φn,m−1i,h ∈ Wh,0, m = M − 1, ..., 1.

(4.30)

118

b4) Finally, set

vn+1i = − 1

µiφni,h

∣∣∣Oi×(0,T )

, i = 1, 2. (4.31)

ALG 4bis:


02) ∈ V ∆t

h , fix an approximation (y0,h, y1,h) ∈ Wh,0 ×Wh,0 to

(y0, y1) and set y−1h (x, t) ≡ y0,h(x) + ty1,h(x).


n2 ) ∈ V ∆t

h , compute yn,0h , yn,1h , ... as

follows

yn,0h = y0,h, yn,1h = y0,h + ∆t · y1,h ,∫Ω

[(yn,m+1h − 2yn,mh + yn,m−1

h

(∆t)2

)z +∇yn,m+1

h · ∇z

+ G(yn−1,m+1h )yn,m+1

h z

]dx

=

∫Ω

(f(x, tm+1)− F (0) + vn1 (x, tm+1)1O1 + vn2 (x, tm+1)1O2

)z dx

∀z ∈ Wh,0, yn,m+1h ∈ Wh,0, m = 1, ...,M − 1,

(4.32)

compute the φn,Mi,h , φn,M−1i,h , ... by solving the linear systems (4.30) and finally

compute the vn+1i from (4.31).


As in Section 4.5, we can try to solve a system that can be viewed as a necessary

condition of the Pareto equilibrium property. More precisely, we fix λ ∈ (0, 1) and

we try to find a solution to (4.7).

Arguing as before, we get the following equivalent formulation:

ytt −∆y + F (y) = f + v11O1 + v21O2 in Q,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,


(4.33)

119

φi,tt −∆φi + F ′(y)φi = [y − yi,d]1Oi,d in Q,

φi = 0 on Σ1,

∂φi∂n

= 0 on Σ2,

φi(x, T ) = 0 and φi,t(x, T ) = 0 in Ω,

(4.34)

v1 = − 1

µ1

(φ1 +

1− λλ

φ2

)∣∣∣O1×(0,T )

, v2 = − 1

µ2

( λ

1− λφ1 + φ2

)∣∣∣O2×(0,T )

. (4.35)

Again, assuming that µ1 and µ2 are large enough, it is possible rewrite (4.33)–

(4.35) as a nonlinear fixed-point equation in V that possesses at least one solution;

see the details in the Appendix.

On the other hand, we can introduce fixed-point iterates for the solution of (4.33)–

(4.35) as follows:

ALG 5:





b2) Then, for given ` ≥ 0 and yn,`, compute the solution yn,`+1 to (4.26).



b4) Finally, set vn+1

1 = − 1

µ1

(φn1 +

1− λλ

φn2

)∣∣∣O1×(0,T )

,

vn+12 = − 1

µ2

( λ

1− λφn1 + φn2

)∣∣∣O2×(0,T )

,

(4.36)

with µ1 and µ2 sufficiently large.

120

ALG 6:



b) Then, for given n ≥ 0, yn−1 and vn = (vn1 , vn2 ) ∈ V , compute yn by solving (4.28),

compute the φni by solving the systems (4.24) and finally compute the vn+1i

from (4.36), with i = 1, 2.

To end this section, let us present finite dimensional versions of ALG 5 and ALG 6,

where we have conserved the notation introduced in Section 4.3.3.

ALG 5bis:


02) ∈ V ∆t

h , fix an approximation (y0,h, y1,h) ∈ Wh,0 ×Wh,0 to

(y0, y1) and set y−1h (x, t) ≡ y0,h(x) + ty1,h(x).


n2 ) ∈ V ∆t


b1) Set yn,0h = yn−1h .

b2) Then, for given ` ≥ 0 and yn,`h , compute yn, `+1, 0h , yn, `+1, 1

h , ... from (4.29).

After convergence, denote by ynh the last computed yn,`+1h .

b3) For (i = 1, 2), compute the φn,Mi,h , φn,M−1i,h , ... from (4.30).

b4) Finally, with (i = 1, 2), compute the vn+1i from (4.36).

ALG 6bis:


02) ∈ V ∆t

h , fix an approximation (yh,0, yh,1) ∈ Wh,0 ×Wh,0 to

(y0, y1) and set y−1(x, t) ≡ y0,h(x) + ty1,h(x).


n2 ) ∈ V ∆t

h , compute yn,0h , yn,1h , ... from

(4.32), compute φn,Mi,h , φn,M−1i,h , ... from (4.30) and finally compute the vn+1

i from

(4.36).

121

4.7 Some Numerical Results

In the present section, we apply the fixed-point methods for present some nu-

merical experiments; they have been performed with the FreeFem++ library; see

http://www.freefem.org/ff++.

4.7.1 Description of the Problems

For the numerical results, initially consider that the initial data y0 and y1 are

given in each of the problems, and T = 1.5 fixed.

The domain Ω and the sets O1, O2, O1,d and O2,d are the following:

Ω = B((0, 0); 7), O1 = (3, 5)× (2, 4), O2 = (3, 5)× (−2,−4),

O1,d = O2,d = (0.5, 3)× (−2, 2),

this set will be denoted by Od in the sequel.

The domain set for Ω, O, O1, O2 and Od are discretized in triangles and repre-

sented in the figures (4.1) and (4.2) below:

Figure 4.1: The initial mesh. Figure 4.2: The final adapted mesh.

The figures (4.1) and (4.2) above also indicate the process of adaptation of the

mesh after the process of iterations.

122

We take Γ1 = ∂Ω and, accordingly, Γ2 = 0. The functions f and F (for semilinear

problems) are respectively given by

f(x, t) ≡ 1O with O = (0.5, 3)× (−2, 2), F (s) ≡ s(1 + sin s).

We also have the target functions, which are defined by yi,d , where

y1,d(x, t) ≡ x21 − x2

2 and y2,d(x, t) ≡ x1x2.

The data f, y1,d and y2,d are fixed, and depicted in Fig. 4.3–4.5.

Figure 4.3 - The function f

The number of time steps is M = 40 (this gives ∆t = 0.05).

In the following Sections, we present several tests for the algorithms in in the

previous Sections. We have always taken µ1 = µ2 and this common value has been

denoted by µ.

123

Figure 4.4: The function y1,d Figure 4.5: The function y2,d

The stopping criteria have been

‖(vn+11 , vn+1

2 )− (vn1 , vn2 )‖

‖(vn+11 , vn+1

2 )‖≤ ε (4.37)

for ALG 1bis and ALG 2bis,

‖yn,`+1 − yn,`‖‖yn,`+1‖

≤ ε and (4.37), (4.38)

for ALG 3bis and ALG 5bis and

‖(vn+11 , vn+1

2 )− (vn1 , vn2 )‖+ ‖yn+1 − yn‖

‖(vn+11 , vn+1

2 )‖+ ‖yn+1‖≤ ε (4.39)

for ALG 4bis and ALG 6bis, with ε = 10−5.

When dealing with the computation of Nash equilibria, we have considered several

values of µ, ranging from µ = 1.5 to µ = 15.5. As expected, we have seen that the

lower µ is the less favorable situation is found; more explanations can be found in

the Appendix. On the other hand, the computations of Pareto equilibria have been

carried out for fixed µ and various λ.

Below, we also have some tables that compare the computation of data in some

experiments.

124

4.7.2 Computation of Nash and Pareto Equilibria in the Lin-

ear Case

Test 1: ALG 1bis for (the numerical approximation of) (4.1)

A comparison of some tests of µ = 0.5, ... , 15.5 is found in the Table 4.1. The

higher the value of µ, the less effort the controls perform to control the problem.

The graphical comparison between final states for µ = 1.5 and µ = 15.5 in the time

t = T , can be viewed in the Figures 4.6 and 4.7. For µ = 1.5 the adjoint states at

time t = 0 are displayed in Fig. 4.8 and 4.9.





Results for ALG 1bis

µ Internal iterates2∑i=1

||vi− vni ||L2(Oi×(0,T )) < ε

0.5 17 ε = 2.80427× 10−6

1.5 9 ε = 4.50827× 10−6

5.5 6 ε = 2.16248× 10−6

10.5 5 ε = 2.81227× 10−6

15.5 4 ε = 3.15548× 10−7

Table 4.1: Computational results for y1,d = y2,d = 1, T = 1.5 and y1,0 = y0,0 = 1.

125

Figure 4.8: ALG 1bis. The first adjoint

state at t = 0 for µ = 1.5

Figure 4.9: ALG 1bis. The second ad-

joint state at t = 0 for µ = 1.5


We take µ = 15.5 fixed and λ = 0.1, ... , 0.9. A comparison of computed data in

this algorithm at time t = T are displayed in the Table 4.2. The graphical comparison

between final states for λ = 0.1, 0.5 and 0.9, are given in the Figures 4.10, 4.11 and

4.12, respectively.


λ Internal iterates2∑i=1

||vi− vni ||L2(Oi×(0,T )) < ε

0.1 10 ε = 8.17931× 10−6

0.3 7 ε = 3.86862× 10−6

0.5 6 ε = 7.13911× 10−6

0.7 6 ε = 7.24884× 10−6

0.9 10 ε = 3.5709× 10−6

Table 4.2: Computational results for y1,d = y2,d = 1, µ = 15.5, T = 1.5 and

y1,0 = y0,0 = 1.

126

Figure 4.10: ALG 2bis. The computed state at t = T for λ = 0.1 and µ = 15.5


state at t = T for λ = 0.5 and µ = 15.5



4.7.3 Computation of Nash and Pareto Equilibria in the

Semilinear Case


Now we have that a comparison of computed data for this algorithm at time t =

T are displayed in the Table 4.3. We take here various µ, ranging from 0.5 to 15.5.

We show the result of one experiment related to ALG 3bis, given in Figure 4.13.

127

Figure 4.13: ALG 3bis. The computed state at t = T , y0 = y1 = 0.1 and µ = 1.5


µ Iterates ||y − yn||L2(Ω×(0,T )) < ε2∑i=1

||vi− vni ||L2(Oi×(0,T )) < ε

0.5 7 ε = 3.52008× 10−6 ε = 7.14639× 10−6

1.5 5 ε = 1.86570× 10−6 ε = 4.68156× 10−6

5.5 5 ε = 3.97616× 10−7 ε = 2.18891× 10−7

10.5 5 ε = 2.49649× 10−7 ε = 1.37965× 10−8

15.5 5 ε = 2.41124× 10−7 ε = 4.25188× 10−9

Table 4.3: Computational results for y1,d = y2,d = 1, T = 1.5 and y1,0 = y0,0 = 0.1.


In this experiment, let’s fixe µ = 15.5 and vary λ in (0.05, ... , 0.95). For

completeness, we present in Fig. 4.14, 4.15 and 4.16 the computed states at t = T

respectively for λ = 0.1, λ = 0.5 and λ = 0.9.

128

Figure 4.14: ALG 5bis. The computed state at t = T for λ = 0.1 and µ = 15.5






λ Iterates ||y − yn||L2(Ω×(0,T )) < ε2∑i=1

||vi− vni ||L2(Oi×(0,T )) < ε

0.05 11 ε = 1.95101× 10−6 ε = 5.12751× 10−7

0.1 7 ε = 1.63680× 10−6 ε = 4.70910× 10−7

0.5 6 ε = 7.67620× 10−6 ε = 4.51259× 10−7

0.9 8 ε = 1.03443× 10−6 ε = 3.57903× 10−7

0.95 10 ε = 1.43962× 10−6 ε = 5.66113× 10−7

Table 4.4: Computational results for y1,d = y2,d = 1, T = 1.5 and y1,0 = y0,0 = 1.

129

4.8 Appendix

4.8.1 Existence and uniqueness of a solution to (4.5)

Let us consider the mapping

(v1, v2) ∈ V 7→(∂J1

∂v1

(v1, v2),∂J2

∂v2

(v1, v2)

)∈ V . (4.40)

From the linearity of (4.1) and the structures of J1 and J2, there exist A ∈ L (V ;V )

and b ∈ V such that(∂J1

∂v1

(v1, v2),∂J2

∂v2

(v1, v2)

)= A (v1, v2)− b ∀(v1, v2) ∈ V.

Let us identify the mapping A . For every (v1, v2) ∈ V , one has

A (v1, v2) =(µ1v1 + φ11O1 , µ2v2 + φ21O2

),

where the φi are given by

φi,tt −∆φi = y1Oi,d in Q,

φi = 0 on Σ1,

∂φi∂n

= 0 on Σ2,

φi(x, T ) = 0, φi,t(x, T ) = 0 in Ω

(4.41)

and y is the solution to

ytt −∆y = v11O1 + v21O2 in Q,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = 0, yt(x, 0) = 0 in Ω.

(4.42)

Proposition 4.8.1 The mapping A is linear, continuous and symmetric. Further-

more, if µ1 and µ2 are sufficiently large (depending on Ω and T ), A is strongly

positive, that is, there exists α0 > 0 such that

(A (v1, v2), (v1, v2))V ≥ α0‖(v1, v2)‖2V ∀(v1, v2) ∈ V. (4.43)

130

Proof: The argument is adapted from [65], Proposition 4.1. For completeness, let

us recall the proof of strong positiveness.

Note that, for any (v1, v2) ∈ V ,

(A (v1, v2), (v1, v2))V = µ1

∫∫O1×(0,T )

|v1|2 dx dt+ µ2

∫∫O2×(0,T )

|v2|2 dx dt

+

∫∫O1×(0,T )

φ1 v1 dx dt+

∫∫O2×(0,T )

φ2 v2 dx dt.(4.44)

Let z1 (resp. z2) be the solution to (4.42) with v2 = 0 (resp. with v1 = 0). Then∫∫Oi×(0,T )

φi vi dx dt =

∫∫Oi×(0,T )

φi (zi,tt −∆zi) dx dt =

∫∫Oi×(0,T )

y zi dx dt. (4.45)

Consequently, there exists C0 depending on Ω and T such that∣∣∣∣∫∫Oi×(0,T )

φi vi dx dt

∣∣∣∣ ≤ C0‖(v1, v2)‖V ‖vi‖L2(Oi×(0,T )), i = 1, 2. (4.46)

From (4.44) and (4.46), we see at once that, if min(µ1, µ2) >√

2C0, then

(A (v1, v2), (v1, v2))V ≥[min(µ1, µ2)−

√2C0

]‖(v1, v2)‖2

V ∀(v1, v2) ∈ V,

whence (4.43) holds. 2

Remark 4.8.1 From (4.44) and (4.45) we also observe that, if the sets O1,d and O2,d

coincide and we set Od = Oi,d and α1 = α2 = α, then

(A (v1, v2), (v1, v2))V ≥ min(µ1, µ2)‖(v1, v2)‖2V +

∫∫Od×(0,T )

|y|2 dx dt

for all (v1, v2) ∈ V .

Therefore, in this case, A is always strongly positive, regardless of the sizes of the µi.

2

Let us identify b, that is, the constant part of the affine mapping (4.40). One

has:

b =(φ11O1 , φ21O2

),

131


φi,tt −∆φi = αi(y − yi,d

)1Oi,d in Q,

φi = 0 on Σ1,

∂φi∂n

= 0 on Σ2,

φi(x, T ) = 0, φi,t(x, T ) = 0 in Ω

(4.47)

and y is the solution of

ytt −∆y = f in Q,

y = 0 on Σ1,

∂y

∂n= 0 on Σ2,

y(x, 0) = y0(x), yt(x, 0) = y1(x) in Ω .

(4.48)

Now, if we define a(· , ·) : V × V 7→ R by

a(v, w) := (A v, w)V ∀v, w ∈ V,

and L : V 7→ R by

L(v) := (b, v)V ∀v ∈ V,

we deduce that (4.5) is equivalent to

a(v, w) = L(v), ∀v ∈ V. (4.49)

From Proposition 4.8.1, we have that the mapping a(· , ·) is bilinear, continuous

and symmetric. Moreover, if µ1 and µ2 are large enough, it is also V -elliptic. On the

other hand, L is linear and continuous. Hence, from the well known Lax-Milgram

Theorem, the existence and uniqueness of solution to (4.49) is ensured if the µi are

sufficiently large.

Remark 4.8.2 Let us fix λ ∈ (0, 1) and let us consider the system (4.44). Arguing

as before, it becomes clear that there exists C(Ω, T, λ) such that, if min(µ1, µ2) >

C(Ω, T, λ), this system has exactly one solution. 2

132

4.8.2 The convergence of ALG 1

Let us recall the iterates: for any given n ≥ 0 and vn = (vn1 , vn2 ) ∈ V , we compute

yn, φni and then vn+1 = (vn+11 , vn+1

2 ) by solving the linear systems

yntt −∆yn = f + vn11O1 + vn21O2 in Q,

yn = 0 on Σ1,

∂yn

∂n= 0 on Σ2,

yn(x, 0) = y0, ynt (x, 0) = y1 on Ω.

φni,tt −∆φni = [yn − yi,d]1Oi,d in Q,

φn = 0 on Σ1,

∂φn

∂n= 0 on Σ2,

φn(x, T ) = 0, φnt (x, T ) = 0 on Ω.

vn+1i = − 1

µiφni∣∣Oi×(0,T )

, i = 1, 2.

Obviously, this can be written in the form

vn+1 = Bvn + h,

where B : V 7→ V is defined as follows:

• Bv = (B1v,B2v), Biv = − 1


and φi is the solution to (4.41), where


• h = (h1, h2), hi = − 1


and φi is the solution to (4.47), where y is

the solution to (4.48).

It is easy to check that B ∈ L(V ;V ) and

‖Bv‖V ≤C(Ω, T )

min(µ1, µ2)‖v‖V ∀v ∈ V.

Hence, if µ1 and µ2 are large enough, the mapping v 7→ Bv+ h is a contraction and

the sequence furnished by ALG 1 converges to the unique solution to (4.5).

133

4.8.3 Existence and uniqueness of a solution to the semilin-

ear system (4.23)–(4.25)

In this section, we assume that F : R 7→ R is globally Lipschitz-continuous and

|F (z1)− F (z2)| ≤ CF |z1 − z2| ∀z1, z2 ∈ R.

Note that the couple (v1, v2) solves (4.23)–(4.25) if and only if it is a fixed-point of

the nonlinear mapping Λ : V 7→ V , whereΛ(v) = (Λ1(v),Λ2(v)), Λi(v) = − 1


,

φi is the solution to (4.24) for i = 1, 2,

and y is the solution to (4.23).

It is not difficult to check that there exists C(Ω, T, CF , ‖f‖L2(Q)) such that, if

min(µ1, µ2) > C(Ω, T, CF , ‖f‖L2(Q)),

the mapping Λ is a contraction. Indeed, from the usual energy estimates, it is clear

that

‖Λ(v)− Λ(v)‖V ≤2∑i=1

∫∫Oi×(0,T )

1

µi2|(φ1, φ2)− (φ1, φ2)|2 dx dt

≤ C(Ω, T )

min(µ1, µ2)‖(φ1, φ2)− (φ1, φ2)‖L2(Q)×L2(Q)

≤ C(Ω, T, CF )

min(µ1, µ2)‖y − y‖L2(Q)

≤C(Ω, T, CF , ‖f‖L2(Q))

min(µ1, µ2)‖v − v‖V

for all v, v ∈ V , where the notation is self-explanatory.

As a consequence, we find that, if µ1 and µ2 are large enough, (4.23)–(4.25)

possesses a unique solution. In other words, (4.5) is uniquely solvable.

Remark 4.8.3 For any fixed λ ∈ (0, 1), we can consider the system (4.33)–(4.35).

Arguing in a similar way, we deduce that there exists C(Ω, T, CF , ‖f‖L2(Q), λ) such

that, for greater values of min(µ1, µ2), this system possesses exactly one solution. 2

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