A variational approach to constrained controllability for distributed systems

J. Math. Anal. Appl. 416 (2014) 805–823

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Journal of Mathematical Analysis and Applications

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A variational approach to constrained controllability fordistributed systems

Larbi BerrahmouneDépartement de Mathématiques, Ecole Normale Supérieure de Rabat, Université Mohammed V Agdal,BP 5118, Rabat, Morocco

a r t i c l e i n f o a b s t r a c t

Article history:Received 3 May 2013Available online 11 March 2014Submitted by H. Frankowska

Dedicated to Abdelhaq El Jai

Keywords:Distributed systemsVariational approachConstrained exact controllabilitySaturation

We consider linear distributed control systems of the form y′(t) = Ay(t) +Bu(t) where A generates a strongly continuous semigroup (etA)t�0 on an infinitedimensional Hilbert space Y . We suppose that the control operator B is boundedfrom the (Hilbert) control space U to Y . Taking into account eventual controlconstraint (such as saturation), we study the problem of exact controllability byusing a variational approach. Applications to hyperbolic-like systems of the formz′′(t) + Az(t) = Bu(t) are treated.

© 2014 Elsevier Inc. All rights reserved.

1. Introduction and preliminary results

The notion of controllability is one of the most fundamental issues associated with the control of asystem. Here, we consider this notion when the system under investigation is distributed. There has beena very intensive research in this area, especially in the context of systems described by partial differentialequations. We refer to the survey papers [3,14,18] for an introduction to the case of such systems. We recallthat in some functional setting, the controllability is equivalent to the observability problem which concernsthe possibility of recovering full estimates on the solutions of the uncontrolled adjoint system. The maintools which are available to analyze it are: multiplier techniques, moment problems, nonharmonic Fourierseries, Hilbert uniqueness method (HUM), microlocal analysis and Carleman inequalities. Moreover, in [11],a variational principle has been used in order to build the appropriate control by exploiting the observabilityproperty. It consists on using a quadratic functional whose minimizer gives the control steering the systemto the desired state. This procedure has been fruitfully exploited later for various systems, including thosemodeled by ordinary differential equations (see [12]). Unlike the unconstrained controllability which hasbeen well understood, to the author’s knowledge, few works have been devoted to the realistic case ofconstrained controllability. This problem is important since all real world applications involve actuators

E-mail address: [email protected].

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with amplitude and rate limitations. It is our objective to consider the problem of exact controllabilityfor distributed systems with (eventual) constrained control. To this end, we use a generalized variationalapproach based on the minimization of an appropriate convex functional, generalizing the quadratic onerelated with the unconstrained case. We mention that such a method has been revealed efficient in treatingdistributed systems with switching constraints on the control (see [19]). The interest of such a methodcan be summarized as follows. (i) The variational approach is constructive since its inherent optimalitycondition will give a method to build the appropriate control. (ii) The method is general and covers theunconstrained controllability and the constrained one as well. (iii) The variational approach can be adaptedto various types of constraints. As applications, we shall consider the problem of exact controllability forthe hyperbolic-like systems with saturating control. We recall also that for multivariable systems describedby ordinary differential equations, the constrained controllability problem has been studied and there arewell established results in this area by means of tools inappropriate in the context of distributes systems(see, for instance, [9,15]). We shall mention some of them later.

Let Y be a Hilbert space (state space) with inner product 〈. , .〉 and norm ‖.‖ and let U be a secondHilbert space (control space) with inner product 〈. , .〉U and norm ‖.‖U . Also, let A be the infinitesimalgenerator of a linear C0-semigroup on Y denoted by etA. Finally, let B denote the linear control operatorwhich is supposed to be bounded from U to Y . We consider the abstract control system

{y′(t) = Ay(t) + Bu(t),y(0) = y0.

(1.1)

It is well-known that for u ∈ L2(0, T ;U) and y0 ∈ Y , the weak solution is given by the variation of constantformula

y(t) = etAy0 +t∫

0

e(t−s)ABu(s) ds. (1.2)

In particular, given a prescribed time T0 > 0, we shall be concerned with the final state

yu(T0) = eT0Ay0 +T0∫0

e(T0−s)ABu(s) ds (1.3)

and the following space of reachable states from y0 in time T0

R(T0; y0) ={yu(T0)

∣∣ u ∈ L2(0, T0;U)}. (1.4)

Then the notions related with exact controllability are defined as follows.

Definition 1.1. The system (1.1) (or the pair (A,B)) is exactly controllable in time T0 if given any initialand desired states y0, yd, there exists u ∈ L2(0, T0;U) such that the solution of (1.1) satisfies y(T0) = yd.

Definition 1.2. The system (1.1) (or the pair (A,B)) is null controllable in time T0 if given any initialstate y0, the set of reachable states R(T0, y0) contains 0.

The following remarks precise the fact that, in contrast to the framework of multivariable systems, thenotion of controllability is more complex in the context of distributed systems.

L. Berrahmoune / J. Math. Anal. Appl. 416 (2014) 805–823 807

Remark 1.1.

(a) Unlike the case of multivariable systems where exact and null controllability are equivalent, we have todistinguish between them when dealing with distributed systems. It is well-known that these notionsare equivalent for reversible systems such as the ones modeled by hyperbolic-like systems. Moreover,this fact is not true for parabolic-like systems such as those described by the heat equation model. Formore details on this fact, we refer to [13].

(b) One can easily deduce that exact controllability is equivalent to R(T0, 0) = Y .(c) For any T0 > 0, the pair (A,B) is never exactly controllable in time T0 if the control operator B is

compact (see [16]). In particular, the exact controllability will never occur when the system is controlledby finitely many controllers in the sense that U = R

p for some integer p.(d) For any T0 > 0, the pair (A,B) is never exactly controllable in time T0 if the semigroup (etA)t�0 is

compact (see [17]). Hence, the exact controllability is impossible for parabolic-like systems such as theones modeled by the heat equation in bounded domains.

(e) It is well-known that exact controllability and null controllability are equivalent respectively to thefollowing observability inequalities (see [14])

T0∫0

∥∥B∗e(T0−t)A∗ϕ0

∥∥2Udt � c‖ϕ0‖2 for all ϕ0 ∈ Y, (1.5)

T0∫0


∥∥2Udt � c

∥∥eT0A∗ϕ0

∥∥2 for all ϕ0 ∈ Y, (1.6)

for some positive constant c.(f) In the context of distributed systems, the observability property (1.5) is rather verified for reversible

systems. Indeed, it is shown in [3] that if (1.5) holds and if, for each t � 0, the range of etA is densein Y , then etA can be extended to a strongly continuous group on 0 < t < ∞.

Moreover, the characterization (1.5) can be reduced to the following criteria which will be useful in thesequel.

Proposition 1.1. The pair (A,B) is exactly controllable in time T0 if, and only if, one of the followingequivalent properties holds

inf‖ϕ0‖=1

T0∫0


∥∥2Udt � c > 0, (1.7)

lim inf‖ϕ0‖→∞

∫ T00 ‖B∗e(T0−t)A∗

ϕ0‖2U dt

‖ϕ0‖2 � c > 0. (1.8)

The characterization results above are useful from a theoretical point of view but they do not giveconstructive methods in order to find a control steering the system from the initial state y0 to a desiredone yd. The variational principle introduced in [11] and developed in [12] gives an alternative constructivemethod. It uses the so-called Hilbert Uniqueness Method (HUM) developed in [10]. In our context, the mainingredient of the method is based on the following characterization result which can be adapted from theframework of HUM (see, for instance, [12]).


Proposition 1.2. A desired state yd is reached from y0 in time T0 by a control u ∈ L2(0, T0;U) if, and onlyif,

T0∫0

⟨u(t), B∗e(T0−t)A∗

ϕ0⟩Udt− 〈yd, ϕ0〉 +

⟨y0, e

T0A∗ϕ0

⟩= 0 for all ϕ0 ∈ Y. (1.9)

Relation (1.9) may be considered as an optimality condition for critical points of the quadratic functionalJq given by

Jq(ϕ0) = 12

T0∫0


∥∥2Udt− 〈yd, ϕ0〉 +

⟨y0, e

T0A∗ϕ0

⟩. (1.10)

We note that the choice of Jq is appropriate when the control is unconstrained. Our goal consists on givingan extension of the variational approach to the realistic situation where the control is subject to someconstraint. To this end, we precise the framework of constrained controllability. We consider a prespecifiedsubset U ⊂ U and we introduce the following set of reachable states with constraints

RU (T0, y0) ={yu(T0)

∣∣ u ∈ L2(0, T0;U), u(t) ∈ U a.e. on (0, T0)}. (1.11)

Then we define the constrained exact controllability notions as follows.

Definition 1.3. The system (1.1) is globally U-exactly controllable at y0 in time T0 if Y = RU (T0, y0). It issaid to be globally U-exactly controllable in time T0 if Y = RU (T0, y0) for any y0 ∈ Y .

Definition 1.4. The system (1.1) is locally U-exactly controllable at y0 in time T0 if there exists an opensubset V ⊂ Y , containing y0 such that V ⊂ RU (T0, y0).

Following the framework treated in the finite dimensional systems literature, we can enlarge the notionof U-exact controllability by varying the final time. To this end, we consider the union of the sets RU (T, y0)for all T ∈ (0,∞), denoted by RU (y0), and we introduce:

Definition 1.5. The system (1.1) is globally U-exactly controllable at y0 in finite time if Y = RU (y0). It issaid to be globally U-exactly controllable in finite time if Y = RU (y0) for any y0 ∈ Y .

In the same way, we introduce the constrained null controllability as follows. The system (1.1) is U-nullcontrollable in time T0 at y0 if there exists a control u ∈ L2(0, T0;U) such that yu(T0) = 0 and u(t) ∈ Ualmost everywhere in (0, T0). The system (1.1) is U-null controllable in time T0 if it is U-null controllablein time T0 at any y0 ∈ Y . The system (1.1) is locally U-null controllable in time T0 if there exists an opensubset V ⊂ Y , containing the origin, such that (1.1) is U-null controllable in time T0 at any y0 ∈ V . Considerthe set CT0 of initial states for which the system (1.1) is U-null controllable in time T0. The union of CT0 forall T0 ∈ (0,∞), denoted by C, is called the null controllable set. Then the system is said to be locally U-nullcontrollable in finite time if there exists an open subset V ⊂ Y , containing the origin, such that V ⊂ C. Itis said to be globally U-null controllable in finite time if C = Y . Let us mention the following results whichare available in the finite dimensional systems literature in the case where U is a bounded neighborhoodof 0 (see, for instance, [15]):

(i) The system (1.1) is globally U-exactly controllable at 0 in finite time if, and only if, the pair (A,B) is(exactly) controllable and the matrix A has no eigenvalues with negative real part.


(ii) The system (1.1) is globally U-null controllable in finite time if, and only if, the pair (A,B) is (exactly)null controllable and the matrix A has no eigenvalues with positive real part.

(iii) The system (1.1) is globally U-exactly controllable in finite time if, and only if, the pair (A,B) is(exactly) controllable and all the eigenvalues of the matrix A are purely imaginary.

It is our objective to address problems of (eventually) constrained exact controllability with a generalizedvariational approach. Despite the fact that the saturating control problem was our main motivation, weshall consider at the outset a variational principle for general convex functionals. The plan of the paperis as follows. In Section 2, we present a general variational approach allowing us to treat eventual controlconstraints. In Section 3, we consider the case of control saturation and we treat hyperbolic-like systems.Moreover, applications to the wave and plate equations are given.

2. Variational approach to exact controllability

2.1. General variational approach

In this subsection, we treat the exact controllability problem with a general variational approach. To thisend, let U ⊂ U be a prespecified (eventually) constraint subset so that the unconstrained controllabilitywill correspond to the case where U = U . In order to introduce the tools of the variational approach, weconsider a differentiable convex functional Ψ : U → R

+ and we suppose that Ψ is locally Lipschitz in thesense that for any R > 0, there exists kR > 0 such that

∣∣Ψ(u) − Ψ(v)∣∣ � kR‖u− v‖U (2.1)

for all u, v ∈ U such that ‖u‖U � R, ‖v‖U � R. Then we introduce the functional given by

Jc(ϕ0) =T0∫0

Ψ(B∗e(T0−t)A∗

ϕ0)dt− 〈yd, ϕ0〉 +

⟨y0, e

T0A∗ϕ0

⟩. (2.2)

Furthermore, we set

UΨ = Ψ ′(U). (2.3)

A canonical identification of U with its dual allows us to consider UΨ as a subset of U . Then, as we shallsee, the following result can be viewed as a general convex version of the characterization (1.8).

Theorem 2.1. Suppose that Ψ is convex, differentiable and locally Lipschitz. Assume also that UΨ ⊂ U and


∫ T00 Ψ(B∗e(T0−t)A∗

ϕ0) dt‖ϕ0‖

= δ > 0. (2.4)

(i) If δ < ∞, then for every initial and desired states y0, yd ∈ Y such that

∥∥yd − eT0Ay0∥∥ < δ, (2.5)

there exists some control u ∈ L2(0, T0;U), with values in U and steering y0 to yd.(ii) If δ = ∞, then the system (1.1) is globally U-exactly controllable in time T0.


In both cases, the control driving y0 to yd is given by

u(t) = Ψ ′(B∗e(T0−t)A∗ϕ0

)for all 0 < t < T0, (2.6)

where ϕ0 satisfies

Jc(ϕ0) = minϕ0

Jc(ϕ0). (2.7)

Proof. The functional Jc is clearly convex. On the other hand, Jc is also differentiable. To this end, weintroduce the functional Kc given by

Kc(ϕ0) =T0∫0


ϕ0)dt. (2.8)

Let ϕ0 ∈ Y , R > 0 such that ‖ϕ0‖ � R. Given θ0 ∈ Y fixed, we consider η > 0 small enough so that‖ϕ0 + hθ0‖ � 2R whenever |h| � η. In order to explicit, as h → 0, the limit of the expression

Kc(ϕ0 + hθ0) −Kc(ϕ0)h

=T0∫0

Ψ(B∗e(T0−t)A∗(ϕ0 + hθ0)) − Ψ(B∗e(T0−t)A∗ϕ0)

hdt, (2.9)

we introduce

Θh(t) = Ψ(B∗e(T0−t)A∗(ϕ0 + hθ0)) − Ψ(B∗e(T0−t)A∗ϕ0)

h. (2.10)

Let M > 0, ωA � 0 such that

∥∥etA∗∥∥ � MeωAt for all t � 0. (2.11)

From the estimates


∥∥U�

∥∥B∗∥∥MReωAT0 for all 0 � t � T0, (2.12)∥∥B∗e(T0−t)A∗(ϕ0 + hθ0)

∥∥U� 2

∥∥B∗∥∥MReωAT0 for all 0 � t � T0, |h| � η, (2.13)

we obtain for some positive constant kR

∣∣Θh(t)∣∣ � kR

∥∥B∗∥∥∥∥e(T0−t)A∗θ0∥∥ for all 0 � t � T0, 0 < |h| � η. (2.14)

By the dominated convergence theorem, we get

limh→0

Kc(ϕ0 + hθ0) −Kc(ϕ0)h

=T0∫0

⟨Ψ ′(B∗e(T0−t)A∗

ϕ0), B∗e(T0−t)A∗

θ0⟩Udt, (2.15)

so that

J ′c(ϕ0)θ0 =

T0∫ ⟨Ψ ′(B∗e(T0−t)A∗

ϕ0), B∗e(T0−t)A∗

θ0⟩Udt− 〈yd, θ0〉 +

⟨y0, e

T0A∗θ0⟩. (2.16)

0


Moreover, Jc is coercive, i.e.


Jc(ϕ0) = ∞, (2.17)

if either (i) or (ii) is satisfied. Indeed, if (i) holds, then


Jc(ϕ0)‖ϕ0‖

� lim inf‖ϕ0‖→∞

∫ T00 Ψ(B∗e(T0−t)A∗

ϕ0) dt‖ϕ0‖

−∥∥yd − eT0Ay0

∥∥� δ −

∥∥yd − eT0Ay0∥∥ (2.18)

so that for ‖yd − eT0Ay0‖ < δ, we get


Jc(ϕ0)‖ϕ0‖

> 0. (2.19)

As for the case (ii), the coercivity can be easily obtained for any y0, yd ∈ Y . Therefore, the functional Jcachieves a minimum ϕ0 which is characterized by J ′

c(ϕ0) = 0. Thus

⟨J ′c(ϕ0), ϕ0

⟩= 0 for all ϕ0 ∈ Y. (2.20)

This yields for all ϕ0 ∈ Y ,

T0∫0

⟨Ψ ′(B∗e(T0−t)A∗

ϕ0), B∗e(T0−t)A∗

ϕ0⟩Udt +

⟨y0, e

T0A∗ϕ0

⟩− 〈yd, ϕ0〉 = 0. (2.21)

Hence, from Proposition 1.2, we conclude that the control given by (2.6) steers the state y0 to yd. Thiscompletes the proof of the theorem. �

From Theorem 2.1, we deduce easily the following local controllability results.

Corollary 2.2. Suppose that the assumptions of Theorem 2.1 hold and

0 < lim inf‖ϕ0‖→∞

∫ T00 Ψ(B∗e(T0−t)A∗

ϕ0) dt‖ϕ0‖

= δ < ∞. (2.22)

Then the system (1.1) is locally U-exactly controllable at 0 in time T0 and the control steering 0 to anydesired state yd is given by

u(t) = Ψ ′(B∗e(T0−t)A∗ϕ0

)for all 0 < t < T0, (2.23)

where ϕ0 satisfies

J0(ϕ0) = minϕ0

J0(ϕ0) (2.24)

and J0 is defined by

J0(ϕ0) =T0∫0


ϕ0)dt− 〈yd, ϕ0〉. (2.25)


In the same way, we can treat also the local null controllability problems as follows.

Corollary 2.3. Suppose that the assumptions of Corollary 2.2 hold. Then the system (1.1) is locally U-nullcontrollable in time T0 and the control driving y0 to 0 is given by

u(t) = Ψ ′(B∗e(T0−t)A∗ϕ0

)for all 0 < t < T0, (2.26)

where ϕ0 satisfies

J1(ϕ0) = minϕ0

J1(ϕ0), (2.27)

where J1 is defined by

J1(ϕ0) =T0∫0


ϕ0)dt +

⟨y0, e

T0A∗ϕ0

⟩. (2.28)

Proof. Let

r0 = δ

‖eT0A∗‖ . (2.29)

For any initial state y0 such that

‖y0‖ < r0, (2.30)

the functional J1 is clearly coercive. The local U-null controllability can be obtained by proceeding as inTheorem 2.1. �

The following result enhances us to see that our setting based on general convex functional covers thecase of quadratic functional treated in (HUM) framework.

Corollary 2.4. Suppose that the assumptions of Theorem 2.1 hold and for some positive function γ : R+ → R+

such that γ(r) → ∞ as r → ∞, we have

T0∫0


ϕ0)dt � γ

(‖ϕ0‖

)‖ϕ0‖ for all ϕ0 ∈ Y. (2.31)

Then the system (1.1) is globally U-exactly controllable in time T0 and for any initial and desired states y0,yd, the control driving y0 to yd is given by (2.6)–(2.7).

Remark 2.1. The functional Jc given by (2.2) is a generalization of Jq (given by (1.10)) since we have Jc = Jqfor

Ψ(u) = 12‖u‖

2U . (2.32)

In other words, the characterization (2.4) can be interpreted as a general convex version of (1.8).

The following result shows that under the circumstances where we have local U-exact controllability, wecan obtain global U-exact controllability in finite time.


Theorem 2.5. Suppose that the assumptions of Corollary 2.2 hold and etA is a group of isometries. Thenthe system (1.1) is globally U-exactly controllable in finite time. The control driving y0 to any desired stateyd is given by

u(t) = Ψ ′(B∗e(kT0−t)A∗ϕ0

)for all 0 < t < kT0, (2.33)

where ϕ0 satisfies

Jkc (ϕ0) = min

ϕ0Jkc (ϕ0), (2.34)

k is an integer such that

kδ > ‖y0‖ + ‖yd‖ (2.35)

and the functional Jkc is defined by

Jkc (ϕ0) =

kT0∫0

Ψ(B∗e(kT0−t)A∗

ϕ0)dt− 〈yd, ϕ0〉 +

⟨y0, e

kT0A∗ϕ0

⟩. (2.36)

Proof. Let yd be an arbitrary desired state in Y and consider a nonzero integer k satisfying (2.35). Thenwe consider the decomposition

kT0∫0


ϕ0)dt =

k−1∑j=0

(j+1)T0∫jT0


ϕ0)dt. (2.37)

By an elementary change of variables, we get for all j = 0, . . . , k − 1

(j+1)T0∫jT0


ϕ0)dt =

T0∫0

Ψ(B∗e((k−j)T0−t)A∗

ϕ0)dt

=T0∫0

Ψ(B∗e(T0−t)A∗(

e(k−1−j)T0A∗ϕ0

))dt, (2.38)

so that


∫ kT00 Ψ(B∗e(kT0−t)A∗

ϕ0) dt‖ϕ0‖

�k−1∑j=0


∫ T00 Ψ(B∗e(T0−t)A∗(e(k−1−j)T0A

∗ϕ0)) dt

‖ϕ0‖.

On the other hand, since etA∗ is a unitary group, we obtain for all j = 0, . . . , k − 1


∫ T00 Ψ(B∗e(T0−t)A∗(e(k−1−j)T0A

∗ϕ0)) dt

‖ϕ0‖= lim inf

‖ϕ0‖→∞

∫ T00 Ψ(B∗e(T0−t)A∗(e(k−1−j)T0A

∗ϕ0)) dt

‖e(k−1−j)T0A∗ϕ0‖

= lim inf∫ T00 Ψ(B∗e(T0−t)A∗

ϕ0) dt.

‖ϕ0‖→∞ ‖ϕ0‖


This yields


∫ kT00 Ψ(B∗e(kT0−t)A∗

ϕ0) dt‖ϕ0‖

� kδ. (2.39)

Furthermore, we have

kδ > ‖y0‖ + ‖yd‖

=∥∥ekT0Ay0

∥∥ + ‖yd‖

�∥∥yd − ekT0Ay0

∥∥. (2.40)

It follows from Theorem 2.1 that the control given by (2.33)–(2.36) steers y0 to yd in time kT0. Thiscompletes the proof of the theorem. �2.2. The case of constrained controllability

Despite the fact that condition (2.4) is difficult to check, it seems however an appropriate tool in pre-senting the problem of constrained exact controllability in a unified way. In this subsection, we supposeimplicitly that U = U and we shall give more explicit conditions on Ψ under which the unconstrained exactcontrollability implies the constrained one.

Theorem 2.6. Suppose that the pair (A,B) is exactly controllable in time T0. Moreover, suppose that Ψ isconvex, differentiable, locally Lipschitz and for some positive constants α, β

Ψ(u) � α inf(‖u‖U , ‖u‖2

U

)− β for all u ∈ U. (2.41)

Assume also that UΨ ⊂ U . Then for some 0 < η � ∞, there exists some control u ∈ L2(0, T0;U), with valuesin U , steering y0 to yd and given by (2.6)–(2.7), provided that the initial state y0 and the desired state ydsatisfy

∥∥yd − eT0Ay0∥∥ < η. (2.42)

Proof. Following Theorem 2.1, it is sufficient to prove that


∫ T00 Ψ(B∗e(T0−t)A∗

ϕ0) dt‖ϕ0‖

> 0. (2.43)

To this end, we introduce

I ={t ∈ (0, T0)

∣∣ ∥∥B∗e(T0−t)A∗ϕ0

∥∥U� 1

}, (2.44)

J ={t ∈ (0, T0)

∣∣ ∥∥B∗e(T0−t)A∗ϕ0

∥∥U> 1

}, (2.45)

and the function λ : U → R+ defined by

λ(u) = α inf(‖u‖U , ‖u‖2

U

). (2.46)


Then

T0∫0


ϕ0)dt �

T0∫0

λ(B∗e(T0−t)A∗

ϕ0)dt− βT0

� α

∫I


∥∥2Udt + α

∫J


∥∥Udt− βT0.

On the other hand, without loss of generality, we can assume that ‖ϕ0‖ > 1. By setting

ϕ0 = ϕ0

‖ϕ0‖, (2.47)

we get from above

∫ T00 Ψ(B∗e(T0−t)A∗

ϕ0) dt‖ϕ0‖

� α‖ϕ0‖∫I


∥∥2Udt

+ α

∫J


∥∥Udt− βT0

‖ϕ0‖.

From which we deduce easily

∫ T00 Ψ(B∗e(T0−t)A∗

ϕ0) dt‖ϕ0‖

� α

∫I


∥∥2Udt

+ α

∫J


∥∥Udt− βT0

‖ϕ0‖,

so that

∫ T00 Ψ(B∗e(T0−t)A∗

ϕ0) dt‖ϕ0‖

�T0∫0


ϕ0)dt− βT0

‖ϕ0‖. (2.48)

The proof of (2.43) can be reduced to


T0∫0


ϕ0)dt > 0. (2.49)

By contradiction, suppose that for some sequence (ϕn)n, we have

limn→∞

T0∫0


ϕn

)dt = 0, (2.50)

where ϕn = ϕn

‖ϕn‖ . Consider on (0, T0) the sequence (αn)n of non-negative functions given by

αn(t) = λ(B∗e(T0−t)A∗

ϕn

). (2.51)


This sequence is bounded on (0, T0) and (2.50) implies αn → 0 in L1(0, T0) as n → ∞. Hence there existsa subsequence, still denoted by (αn)n, such that

αn(t) → 0 a.e. on (0, T0). (2.52)

By considering the corresponding subsequence (βn)n defined by

βn(t) =∥∥B∗e(T0−t)A∗

ϕn

∥∥U, (2.53)

we get

βn(t) → 0 a.e. on (0, T0). (2.54)

We deduce by using the dominated convergence theorem that as n → ∞, βn → 0 in L2(0, T0) so that

limn→∞

T0∫0

∥∥B∗e(T0−t)A∗ϕn

∥∥2Udt = 0. (2.55)

This contradicts the exact controllability of the pair (A,B) characterized by

T0∫0

∥∥B∗e(T0−t)A∗ϕn

∥∥2Udt � c > 0 for all n. (2.56)

This completes the proof of the theorem. �Applying Theorem 2.5 and Theorem 2.6 we obtain easily:

Theorem 2.7. Suppose that the assumptions of Theorem 2.6 hold and etA is a group of isometries. Then thesystem (1.1) is globally U-exactly controllable in finite time and the control steering y0 to any desired stateyd is given by (2.33)–(2.36) for some integer k.

Remark 2.2. Theorem 2.6 yields both local U-exact controllability at 0 and local U-null controllability.Corollary 2.2 and Corollary 2.3 give the corresponding appropriate controls.

3. Controllability with saturating control

3.1. General results

Here our concern will focus on the practical situation where the control actuation is subject to thesaturation constraint (eventually after re-scaling)

∥∥u(t)∥∥U� 1 for all t � 0. (3.1)

Hence, we are led to take as subset of control constraint the unit ball

Us ={u ∈ U

∣∣ ‖u‖U � 1}. (3.2)


Then a heuristic way to get an appropriate control resulting from a convenient convex functional Ψ consistson choosing Ψ such that

∥∥Ψ ′(B∗e(T0−t)A∗ϕ0

)∥∥U� 1, (3.3)

where ϕ0 minimizes (2.2). Hence Ψ would satisfy

∥∥Ψ ′(u)∥∥U� 1 for all u ∈ U. (3.4)

A natural choice for the differential Ψ ′ which comes at mind is given by

Ψ ′(u) ={ u

‖u‖Uif ‖u‖U � 1,

u if ‖u‖U � 1.(3.5)

We mention also the following simpler one given by

Ψ ′(u) = u

1 + ‖u‖U. (3.6)

Inspired by the study of stabilization with saturating control in [2], where the appropriate feedback ismonotone and derived from convenient convex functionals, one can solve this problem in more general way.Let ρ : R+ → R be a continuous function with rρ(r) nondecreasing and

ρ(r) > 0 for all r � 0, (3.7)

ρ(r) � min(

1, 1r

)for all r > 0. (3.8)

Then we consider the functional Ψρ : U → R+ given by

Ψρ(u) =‖u‖U∫0

sρ(s) ds. (3.9)

The function t →∫ t

0 sρ(s) ds is clearly convex on R+. Thus Ψρ is convex on U . Moreover, it is easy to check

⟨Ψ ′ρ(u), v

⟩U

=⟨ρ(‖u‖U

)u, v

⟩U

for all u, v ∈ U, (3.10)

so that (3.8) yields (3.4) and Ψ ′ρ(U) ⊂ Us holds true. Then, as a functional to be minimized, we consider

Jρ(ϕ0) =T0∫0

Ψρ

(B∗e(T0−t)A∗

ϕ0)dt− 〈yd, ϕ0〉 +

⟨y0, e

T0A∗ϕ0

⟩. (3.11)

Then by applying Theorem 2.6, we obtain:

Theorem 3.1. Suppose that the pair (A,B) is exactly controllable in time T0. Then there exists some r0 > 0such that for any initial and desired states y0, yd satisfying ‖yd − eT0Ay0‖ � r0, there exists a saturationcontrol steering y0 to yd and given by

u(t) = ρ(∥∥B∗e(T0−t)A∗

ϕ0∥∥ )

B∗e(T0−t)A∗ϕ0, (3.12)

U


where ϕ0 satisfies

Jρ(ϕ0) = minϕ0

Jρ(ϕ0). (3.13)

Proof. Following Theorem 2.6, it is sufficient to prove that Ψρ satisfies (2.41). To this end, we shall establishsome estimates relative to Ψρ(u). Let

ρm = min0�s�1

ρ(s). (3.14)

In the case ‖u‖U � 1, we have

‖u‖U∫0

sρ(s) ds � ρm

‖u‖U∫0

s ds = ρm2 ‖u‖2

U . (3.15)

As for the case ‖u‖U > 1, since rρ(r) is nondecreasing, we get easily

‖u‖U∫0

sρ(s) ds =1∫

0

sρ(s) ds +‖u‖U∫1

sρ(s) ds

� ρm

1∫0

s ds + ρ(1)‖u‖U∫1

ds.

Hence

‖u‖U∫0

sρ(s) ds � ρm2 − ρ(1) + ρ(1)‖u‖U

� ρm‖u‖U − ρ(1)

� ρm2 ‖u‖U − ρ(1), (3.16)

so that from (3.15) and (3.16) we get

Ψρ(u) �{ ρm

2 ‖u‖2U − ρ(1) if ‖u‖U � 1,

ρm

2 ‖u‖U − ρ(1) if ‖u‖U � 1.(3.17)

This completes the proof of the theorem. �Remark 3.1. It is easy to see that (3.5) and (3.6) are contained in our setting since they correspond to theconvex functionals

Ψ0(u) = ρ0(‖u‖U

)u, ρ0(r) = min

(1, 1

r

), (3.18)

and

Ψ1(u) = ρ1(‖u‖U

)u, ρ1(r) = 1

1 + r, (3.19)

respectively. The assumptions on ρ(.) above are readily satisfied by ρ0 and ρ1.


Applying Theorem 2.7, we get:

Theorem 3.2. Suppose that the pair (A,B) is exactly controllable in time T0 and etA is a group of isometries.Then the system (1.1) is globally Us-exactly controllable in finite time: for any desired state yd ∈ Y , thereexists some T1 > T0 such that yd can be reached from y0 in time T1 by a saturation control. Moreover, thesaturation control is given by

u(t) = ρ(∥∥B∗e(T1−t)A∗

ϕ0∥∥U

)B∗e(T1−t)A∗

ϕ0, (3.20)

where ϕ0 satisfies

JT1ρ (ϕ0) = min

ϕ0JT1ρ (ϕ0), (3.21)

with

JT1ρ (ϕ0) =

T1∫0

Ψρ

(B∗e(T1−t)A∗

ϕ0)dt− 〈yd, ϕ0〉 +

⟨y0, e

T1A∗ϕ0

⟩. (3.22)

Remark 3.2. By considering either the case y0 = 0 or yd = 0, we obtain again either local Us-exactcontrollability at 0, or local Us-null controllability.

3.2. Applications to hyperbolic-like systems

3.2.1. General settingNote at the outset that in the case where the semigroup etA is a group of isometries, we shall focus our

study on the situation in which the system (1.1) can be recast as second order hyperbolic-like system. Thischoice is motivated also by the fact that the observability property is available in some significant casesthat we shall consider later. Let V, H be real Hilbert spaces such that V ⊂ H with dense and continuousembedding. Let us introduce A : V → V ′ the unique linear bounded operator characterized by

〈Av, w〉V′,V = 〈v, w〉V for all v, w ∈ V. (3.23)

Recall that, by identifying H with its dual, we have V ⊂ H ≡ H′ ⊂ V ′ and

〈v, w〉V′,V = 〈v, w〉H for all v ∈ V, w ∈ H. (3.24)

Furthermore, let U be the control space and B : U → H denote a bounded linear control operator. Then ashyperbolic-like system, we consider the one given by

{z′′(t) + Az(t) = Bu(t),z(0) = z0, z′(0) = z1.

(3.25)

Clearly, the system has a first order version similar to (1.1) if we set Y = V × H, U = {0} × U , B =( 0B),

y(t) = {z(t), z′(t)} and the operator A will be given by

D(A) ={{x0, x1} ∈ V ×H

∣∣ x1 ∈ V, Ax0 ∈ H}, A

({x0, x1}

)= {x1,−Ax0}. (3.26)


The observability property analogous to (1.5) is given by

T0∫0

∥∥B∗ϕ′(t)∥∥2U dt � c

(‖ϕ0‖2

V + ‖ϕ1‖2H)

for all ϕ0 ∈ V, ϕ1 ∈ H, (3.27)

for some positive constants T0, c, where ϕ is the solution of the uncontrolled adjoint system

{ϕ′′(t) + Aϕ(t) = 0,ϕ(T0) = ϕ0, ϕ′(T0) = ϕ1.

(3.28)

Moreover, in order to treat the controllability problem under the saturation constraint given by (3.2), weconsider the convex functional:

Jρ({ϕ0, ϕ1}

)=

T0∫0

Ψρ

(B∗ϕ′(t)

)dt−

⟨{zd0 , z

d1}, {ϕ0, ϕ1}

⟩V×H

+⟨{

z0, z1},{ϕ(0), ϕ′(0)}⟩

V×H (3.29)

where ϕ is the solution of (3.28). Then in the context of the system (3.25), Theorem 3.1 and Theorem 3.2can be formulated respectively as follows.

Theorem 3.3. Suppose that the observability property (3.27) holds. Then there exists some r0 > 0 such thatany desired state {zd0 , zd1} ∈ V ×H satisfying

∥∥{zd0 , zd1}− eT0A{z0, z1}∥∥V×H � r0,

can be reached from {z0, z1} by a saturation control in time T0. Moreover, the appropriate saturation controlis given by

u(t) = ρ(∥∥B∗ϕ′(t)

∥∥H)B∗ϕ′(t), (3.30)

where ϕ is the solution of (3.28) with terminal data {ϕ0, ϕ1} at time T0 such that

Jρ({ϕ0, ϕ1}

)= min

{ϕ0,ϕ1}Jρ

({ϕ0, ϕ1}

). (3.31)

Theorem 3.4. Suppose that the observability property (3.27) holds. Then the system (3.25) is globallyUs-exactly controllable in finite time: for any desired state {zd0 , zd1} ∈ V × H, there exists some T1 > T0such that {zd0 , zd1} can be reached from {z0, z1} by a saturation control in time T1. Moreover, the appropriatesaturation control is given by (3.30) where ϕ is the solution of

{ϕ′′(t) + Aϕ(t) = 0,ϕ(T1) = ϕ0, ϕ′(T1) = ϕ1,

(3.32)

with terminal data {ϕ0, ϕ1} at time T1 satisfying

JT1ρ

({ϕ0, ϕ1}

)= min

{ϕ0,ϕ1}JT1ρ

({ϕ0, ϕ1}

)(3.33)

where


JT1ρ

({ϕ0, ϕ1}

)=

T1∫0

Ψρ

(B∗ϕ′(t)

)dt−

⟨{zd0 , z

d1}, {ϕ0, ϕ1}

⟩V×H

+⟨{z0, z1},

{ϕ(0), ϕ′(0)

}⟩V×H. (3.34)

3.2.2. ExamplesThe observability inequality (3.27) has been obtained by various authors for different systems including

waves, plates, elasticity and thermo-elasticity. In what follows, Ω ⊂ RN is an open bounded domain with

sufficiently smooth boundary Γ . Taking into account the framework where the control operator is bounded,we shall focus here on systems modeled by the wave and plate equations with distributed control supportedon an internal subregion ω, where ω is a non-empty open subset of Ω with χω as characteristic function.The control space we shall consider is L2(Ω) so that the saturation constraint will be specified by the set

Uω ={u ∈ L2(Ω)

∣∣∣∫ω

∣∣u(x)∣∣2 dx � 1

}. (3.35)

Example 1 (Wave equation with internal distributed control). We consider the wave equation given by⎧⎪⎪⎨⎪⎪⎩

z′′ = Δz + uχω on (0,∞) ×Ω,

z = 0 on (0,∞) × Γ,

z(0, x) = z0(x), ∂z

∂t(0, x) = z1(x) on Ω.

(3.36)

This system has the form (3.25) if we set H = L2(Ω), V = H10 (Ω), U = L2(Ω) and A = −Δ. Moreover, as

appropriate control operator B : L2(Ω) → L2(Ω), we can set

Bf = χωf. (3.37)

On the other hand, we suppose that we have at hand the following observability condition

T0∫0

∫ω

∣∣ϕ′(t, x)∣∣2 dx dt � c

∫Ω

{∣∣∇ϕ(T0, x)∣∣2 +

∣∣ϕ′(T0, x)∣∣2} dx, (3.38)

for some positive constant c, where ϕ denotes the solution of the uncontrolled wave equation{ϕ′′ = Δϕ on (0,∞) ×Ω,

ϕ = 0 on (0,∞) × Γ.(3.39)

Inequality (3.38) has been established for T0 large enough and ω satisfying the following condition: thereexists x0 ∈ R

N such that ω is a neighborhood of the closure of the set

Γ (x0) ={x ∈ Γ

∣∣ (x− x0).ν(x) > 0}, (3.40)

where ν(x) denotes the unit outward normal at x ∈ Γ [10]. For the one-dimensional case Ω = (0, l), (3.38)can be obtained for T0 � 2l and ω arbitrary [4,7]. Moreover, it is well known that when the domain Ω

satisfies the following “geometric control condition” (GCC): every ray of geometric optics propagating in Ω

and being reflected on its boundary, enters the control region ω in a time less than T0 [1]. Then by applyingTheorem 3.3 and Theorem 3.4, we get either local Uω-exact controllability or global Uω-exact controllabilityin finite time. In both cases, the appropriate control steering the system (3.36) from {z0, z1} to a desiredstate {z1

d, z2d} in time T has the form


u(t, x) = ρ(∥∥χω(.)ϕ′(t, .)

∥∥L2(Ω)

)χω(x)ϕ′(t, x), (3.41)

where ϕ is the solution of (3.39) with terminal data {ϕ(T, .) = ϕ0, ϕ′(T, .) = ϕ1} ∈ H1

0 (Ω)×L2(Ω) satisfying

J1ρ

({ϕ0, ϕ1}

)= min

{ϕ0,ϕ1}J1ρ

({ϕ0, ϕ1}

), (3.42)

with

J1ρ

({ϕ0, ϕ1}

)=

T∫0

Ψρ

(χω(.)ϕ′(t, .)

)dt−

∫Ω

{∇zd0(x).∇ϕ0(x) + zd1(x)ϕ1(x)

}dx

+∫Ω

{∇z0(x).∇ϕ(0, x) + z1(x)ϕ′(0, x)

}dx, (3.43)

where {ϕ(T, .) = ϕ0, ϕ′(T, .) = ϕ1} ∈ H1

0 (Ω) × L2(Ω) are terminal data for the system (3.39). Moreover,the convex functional Ψρ : L2(Ω) → R is defined by

Ψρ(u) =

‖u‖L2(Ω)∫0

sρ(s) ds. (3.44)

Example 2 (Petrowsky system with internal distributed control). As a generalized plate equation, we considerthe Petrowsky system defined by

⎧⎪⎪⎨⎪⎪⎩

z′′ + Δ2z = uχω on (0,∞) ×Ω,

z = Δz = 0 on (0,∞) × Γ,

z(0, x) = z0(x), ∂z

∂t(0, x) = z1(x) on Ω.

(3.45)

This system has the form (3.25) if we set H = L2(Ω), V = H2(Ω)∩H10 (Ω), U = L2(Ω) and A = Δ2. Here,

the space H2(Ω) ∩H10 (Ω) is normed by

‖w‖H2(Ω)∩H10 (Ω) = ‖Δw‖L2(Ω). (3.46)

Moreover, the appropriate control operator B : L2(Ω) → L2(Ω) is defined by (3.37). On the other hand, weconsider the following observability condition

T0∫0

∫ω

∣∣ϕ′(t, x)∣∣2 dx dt � c

∫Ω

{∣∣Δϕ(T0, x)∣∣2 +

∣∣ϕ′(T0, x)∣∣2} dx, (3.47)

for some positive constants c, where ϕ denotes the solution of the uncontrolled system

{ϕ′′ + Δ2ϕ = 0 on (0,∞) ×Ω,

ϕ = Δϕ = 0 on (0,∞) × Γ.(3.48)

In contrast to the wave equation, here ω and T0 can be chosen arbitrarily. Indeed, (3.47) has been establishedwhen Ω is a parallelepipedic domain [6]. On the other hand, regardless of the boundary conditions, wemention the following facts. The inequality (3.47) is valid when ω is a neighborhood of a subset of the


boundary of the form Γ (x0) [5]. Moreover, when Ω is of class C∞, the GCC is sufficient for (3.47) tohold [8].

Then by applying Theorem 3.3 and Theorem 3.4, we get either local Uω-exact controllability or globalUω-exact controllability in finite time. In both cases, the appropriate control steering the system (3.45) from{z0, z1} to the desired state {z1

d, z2d} in time T has the form (3.41), where ϕ is the solution of (3.48) with

terminal data {ϕ(T, .) = ϕ0, ϕ′(T, .) = ϕ1} ∈ H2(Ω) ∩H1

0 (Ω) × L2(Ω) satisfying

J2ρ

({ϕ0, ϕ1}

)= min

{ϕ0,ϕ1}J2ρ

({ϕ0, ϕ1}

), (3.49)

with

J2ρ

({ϕ0, ϕ1}

)=

T∫0

Ψρ

(χω(.)ϕ′(t, .)

)dt−

∫Ω

{Δzd0(x)Δϕ0(x) + zd1(x)ϕ1(x)

}dx

+∫Ω

{Δz0(x)Δϕ(0, x) + z1(x)ϕ′(0, x)

}dx, (3.50)

where {ϕ(T, .) = ϕ0, ϕ′(T, .) = ϕ1} ∈ H2(Ω)∩H1

0 (Ω)×L2(Ω) are terminal data for the system (3.48) andthe convex functional Ψρ : L2(Ω) → R is defined by (3.44).

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A variational approach to constrained controllability for distributed systems