ABC · Stability Analysis of Markovian Jump Systems CONTRIBUTION A L’ETUDE DES SYSTEMES DYNAMIQUES HYBRIDES A COMMUTATIONS ALEATOIRES Realis´ e par: Chafai Imzegouan´ Sous l’encadr

Stability Analysis of Markovian Jump Systems

UNIVERSITE IBN ZOHR

Ecole Nationale des Sciences Appliquees

Formation Doctorale : Sciences et Techniques de l’Ingenieur

Descipline : Mathematiques Appliquees

Specialite : Calcul Stochastique et Systemes Dynamiques

Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 1 / 67


CONTRIBUTION A L’ETUDE DES SYSTEMESDYNAMIQUES HYBRIDES A COMMUTATIONS

ALEATOIRES

Realise par: Chafai ImzegouanSous l’encadrement des Professeurs:Hassane Bouzahir et Brahim Benaid

UNIVERSITE IBN ZOHREcole Nationale des Sciences Appliquees

Laboratoire d’Ingenierie des Systemes et Technologies de l’Information

7 aout 2017




ALEATOIRES




7 aout 2017




ALEATOIRES




7 aout 2017



Plan

1 Introduction2 Mean Exponential Stability of Markovian Jump Linear Systems3 Asymptotic Almost Sure Stability of Markovian Jump Linear

Systems Associated with a Transfer Matrix4 Existence and Uniqueness of Solution for Stochastic Differential

Equations with Infinite Delay5 Stability Analysis for Stochastic Neural Networks with Infinite Delay

and Markovian Switching6 Conclusion and Perspectives



Plan







Plan







Plan







Plan







Plan







Introduction

Switched Systems

1 Introduction

What is a Hybrid Systems ?

A hybrid system is a two-level system with the lower level governed by a set ofmodes described by differential equations and the upper level a coordinatorthat orchestrates the switching among the modes.Clearly, the system admits continuous state that take values from a vectorspace and discrete states that take values from a discrete index set.The interaction between the continuous and discrete states makes switchingdynamical systems widely representative and complicatedly behaved.



Introduction

Switched Systems

Forced-free switched/jump dynamical system

dx(t) = fr(x(t))dt, (1)

where x ∈ Rn is the continuous state, r is the discrete state taking values in afinite state spaceM = 1, 2, ...,N, and fk, k ∈M, are vector fields.



Introduction

Switched Systems

Forced-free switched/jump dynamical system

dx(t) = fr(x(t))dt, (1)

where x ∈ Rn is the continuous state, r is the discrete state taking values in afinite state spaceM = 1, 2, ...,N, and fk, k ∈M, are vector fields.



Introduction

Random Switching

Random switching

We worked on random switching dynamic systems, these hybrid systems witha time-driven switching signal that fluctuates irregularly but obeys adistribution stochastically.A well-known feasible set of random switching signals is the Markov jumpwhere switches between different subsystems are governed by a finite-stateMarkov process/chain.When the subsystems are linear and the switching is a Markov jump, theswitched system is known to be a Markovian jump linear system.Even when all the subsystems are deterministic, a random switching signalmake the switched system random in nature, and the stability notions have tobe defined in a stochastic manner.



Introduction

Random Switching

Random switching




Introduction

Random Switching

Random switching




Introduction

Random Switching

Random switching




Introduction

Motivation

MotivationStability of solutions is important in applications such as communicationnetworks, motor control, economic systems,..., and an important problem is toensure stability.



Introduction

Stability Problems

A typical problem for switched systems goes as follows. It can be that allsub-systems of the switched system are stable but the switched system canbe unstable.



Introduction

Example

Example 1.1

Consider the following randomly switched linear system :

dx(t) = Ar(t)x(t)dt (2)

where r(t) is a continuous-time Markov chain taking values in a finite state

spaceM = 1, 2 with generator Q =

(−1 12 −2

), and

A1 =

(−1 05 −1

), A2 =

(−1 150 −2

)We can check that dx(t) = A1x(t)dt and dx(t) = A2x(t)dt are both stable.However, System (2) is unstable.



Introduction

Example

FIGURE: Jump process r(t) with initialcondition r(0) = 1

FIGURE: Trajectory of x as a function oftime for System (2).



Mean Exponential Stability of Markovian Jump Linear System

Formalisation

1

2 Mean Exponential Stability of Markovian Jump Linear System

Consider the hybrid dynamical system with random switching as following :

dx(t) = Ar(t)x(t)dt, x(0) = x0 ∈ Rn, r(0) = r0 ∈M (3)

where x(t) is the continuous state and r(t) is a Markov process taking valuesin a finite state spaceM = 1, 2, ...,N with generator Q = (qij), qij ≥ 0 fori 6= j and

∑j∈M

qij = 0 for all i ∈M.

Its evolution is governed by the following probability transition :

Pr(t + ∆t) = j/r(t) = i =

qij∆t + o(∆t) i 6= j1 + qii∆t + o(∆t) i = j

(4)




Formalisation

1





∑j∈M



Pr(t + ∆t) = j/r(t) = i =


(4)




Formalisation

1





∑j∈M



Pr(t + ∆t) = j/r(t) = i =


(4)




Formalisation

1





∑j∈M



Pr(t + ∆t) = j/r(t) = i =


(4)




Formalisation

Assume that the Markov chain r(t) is irreducible in the sense that the systemof equations

πQ = 0π1 = 1

(5)

has a unique positive solution termed stationary distribution.The process (x(t), r(t)) is associated with an infinitesimal operator L definedby :For each i ∈M and any g(x, i) ∈ C1(Rn)

Lg(x, i) = 〈Aix,∇g(x, i)〉+Qg(x, .)(i) (6)

where 〈., .〉 is the usual inner product in Rn.∇g(x, i) denotes the gradient (with respect to the variable x) of g(x, i).and Qg(x, .)(i) =

∑j∈M

qijg(x, j).




Formalisation

Definition

For any initial condition (x0, r0), the equilibrium point x = 0 is said to bestochastically stable if there exists a positive constant C(x0, r0) such that

E[ ∫ ∞

0|x(t, x0, r0)|2dt

]≤ C(x0, r0), (7)

mean exponentially stable if there exist positive constants α and β suchthat

E[|x(t, x0, r0)|2

]≤ α|x0|e−βt. (8)




Formalisation

Definition

For any initial condition (x0, r0), the equilibrium point x = 0 is said to bestochastically stable if there exists a positive constant C(x0, r0) such that

E[ ∫ ∞

0|x(t, x0, r0)|2dt

]≤ C(x0, r0), (7)

mean exponentially stable if there exist positive constants α and β suchthat

E[|x(t, x0, r0)|2

]≤ α|x0|e−βt. (8)




Formalisation

The symmetric and skew-symmetric part of a matrix A ∈ Rn×n are expressed,respectively, as

As =12

(A + AT) and Au =12

(A− AT)

Definition

The matrix A ∈ Rn×n is called generalized negative definite if As is negativedefinite.




Formalisation

The symmetric and skew-symmetric part of a matrix A ∈ Rn×n are expressed,respectively, as

As =12

(A + AT) and Au =12

(A− AT)

Definition

The matrix A ∈ Rn×n is called generalized negative definite if As is negativedefinite.




Result

Theorem 2.1

Assume that for any i ∈M, each matrix Ai is generalized definite negative,then, System (3) is mean exponentially stable.




Result

Sketch of proof

Consider the following Lyapunov function

V(x(t), r(t)) = |x(t)|2.

The infinitesimal operator acting on V(x(t), i) is given by :

LV(x(t), i) = 〈Aix(t),∇V(x(t), i)〉+QV(x(t), .)(i)

.

.

≤ −β|x(t)|2 = −βV(x(t), i) (9)

with β =∑

i∈M βi and βi = −λmax(Ai + ATi ) ≥ 0.

By Dynkin’s formula and Gronwall’s inequality, we infer

E[|x(t)|2

]≤ α|x0|e−βt. (10)




Result

Sketch of proof


V(x(t), r(t)) = |x(t)|2.



.

.

≤ −β|x(t)|2 = −βV(x(t), i) (9)

with β =∑



E[|x(t)|2

]≤ α|x0|e−βt. (10)




Result

Sketch of proof


V(x(t), r(t)) = |x(t)|2.



.

.

≤ −β|x(t)|2 = −βV(x(t), i) (9)

with β =∑



E[|x(t)|2

]≤ α|x0|e−βt. (10)




Example

Example 2.1

Consider System (3) with the following specifications. The Markov chain r(t)

has four states, with generator Q =

−3 1 1 10 −2 1 11 1 −3 11 0 1 −2

. The stationary

distribution of the Markov chain r(t) is π = (0.25, 0.25, 0.25, 0.25), which isobtained by solving Equation (5). The matrices are given by :

A1 =

(−2 −1−2 −3

), A2 =

(−1 −22 −5

), A3 =

(−3 30 −1

), A4 =

(−2 0−1 −1

)The eigenvalues of M1 = A1 + AT

1 , M2 = A2 + AT2 , M3 = A3 + AT

3 andM4 = A4 + AT

4 are respectively (−8.1623 − 1.8377), (−10 − 2),(−7.6056 − 0.3944) and (−4.4142 − 1.5858).Then, System (3) is mean exponentially stable.




Example

FIGURE: Markov jump r(t) with initialcondition r0 = 2

FIGURE: Solution curves of System (3)with initial condition x0 = [−7, 9]T



Asymptotic Almost Sure Stability of Markovian Jump Linear System Associated with a Transfer Matrix

Formalisation

1

2

3 Asymptotic Almost Sure Stability of Markovian Jump Linear SystemAssociated with a Transfer Matrix

Consider the system x(t) = Ax(t) + Bu(t)y(t) = Cx(t) + Du(t)

(11)

associated with transfer matrix G(s) = C(sI − A)−1B + D, with state feedbackof the form

ur(t) =1N

(1− r(t))D−1Cx(t)

where r(t) ∈ 1, 2, ...,N.




Formalisation

1

2

3 Asymptotic Almost Sure Stability of Markovian Jump Linear SystemAssociated with a Transfer Matrix

Consider the system x(t) = Ax(t) + Bu(t)y(t) = Cx(t) + Du(t)

(11)

associated with transfer matrix G(s) = C(sI − A)−1B + D, with state feedbackof the form

ur(t) =1N

(1− r(t))D−1Cx(t)

where r(t) ∈ 1, 2, ...,N.




Formalisation

Schematic representation

FIGURE: Dynamical system with Markovian switched controller




Formalisation

We rewrite (11) in following form :

x(t) = Ar(t)x(t) x(0) = x0 ∈ Rn, r(0) = r0 ∈M (12)

with Ar(t) = A + 1N (1− r(t))BD−1C




Definition

DefinitionA jump linear System (12) is said to be

stochastically mean square stable if for any initial state x0 and initialdistribution ρ, we have∫ +∞

0Eρ|x(t; x0, r0)|2dt < +∞. (13)

asymptotically almost surely stable if for any initial state x0 and initialdistribution ρ, we have

P limt→+∞

|x(t; x0, r0)| = 0 = 1. (14)

Lemma 3.1

Any mean square stable jump linear system is asymptotic almost surelystable.




Definition



0Eρ|x(t; x0, r0)|2dt < +∞. (13)


P limt→+∞

|x(t; x0, r0)| = 0 = 1. (14)

Lemma 3.1





Definition



0Eρ|x(t; x0, r0)|2dt < +∞. (13)


P limt→+∞

|x(t; x0, r0)| = 0 = 1. (14)

Lemma 3.1





Definition

lemma (Shorten et al. 2014)

Given a Hurwitz matrix A, the symmetric transfer function matrixG(s) = C(sI − A)−1B + D with D = DT > 0 is strictly positive real (SPR) if andonly if A(A− BD−1C) has no real negative eigenvalue.

Lemma 3.2 (KYP)

Let A be Hurwitz, (A,B) be controllable, and (A,C) be observable. ThenG(s) = C(sI − A)−1B + D is SPR if and only if there exist matrices P = PT > 0,L and W, and a number α > 0 satisfyingATP + PA + αP = −LTLBTP + WTL = CD + DT = WTW.




Definition

lemma (Shorten et al. 2014)

Given a Hurwitz matrix A, the symmetric transfer function matrixG(s) = C(sI − A)−1B + D with D = DT > 0 is strictly positive real (SPR) if andonly if A(A− BD−1C) has no real negative eigenvalue.

Lemma 3.2 (KYP)

Let A be Hurwitz, (A,B) be controllable, and (A,C) be observable. ThenG(s) = C(sI − A)−1B + D is SPR if and only if there exist matrices P = PT > 0,L and W, and a number α > 0 satisfyingATP + PA + αP = −LTLBTP + WTL = CD + DT = WTW.




Result

Theorem 3.3

Assume that the transfer matrix G(s) is SPR, with (A,B) controllable and (A,C)observable. Then the random switching System (12) is asymptotically almostsurely stable.




Result

Sketch of proof

We define a Lyapunov function by the following expression :

V(x(t), r(t)) = xT(t)Pr(t)x(t) (15)

with Pi = Pj = P = PT > 0.Next, by using KYP lemma, we show that for i = 1

LV(x, 1) = −xT(αP1 + LTL)x < 0 (16)

For i = 2, ...,N, by using KYP lemma, we show that

LV(x, i) ≤ −xT[αP +i− 1

N(L−WD−1C)T(L−WD−1C)

]x ≤ 0 (17)




Result

Sketch of proof


V(x(t), r(t)) = xT(t)Pr(t)x(t) (15)


LV(x, 1) = −xT(αP1 + LTL)x < 0 (16)


LV(x, i) ≤ −xT[αP +i− 1


]x ≤ 0 (17)




Result

Sketch of proof


V(x(t), r(t)) = xT(t)Pr(t)x(t) (15)


LV(x, 1) = −xT(αP1 + LTL)x < 0 (16)


LV(x, i) ≤ −xT[αP +i− 1


]x ≤ 0 (17)




Result

Sketch of proof

Using Dynkin’s formula, we infer∫ +∞

0E[|x(s)|2

]ds ≤ C(x0, r0). (18)

This means that the trivial solution of System (12) is stochastically meansquare stable. By Lemma 3.1, System (12) is asymptotically almostsurely stable.




Example

Example 3.1

In this example, we consider G(s) symmetric in order to verify easily that it isSPR.Consider System (12) associated to the symmetric transfer matrixG(s) = C(sI − A)−1B + D with

A =

−2 −1

1 0

, B =

2 1

1 −1

, C =

1 2

−0.3 −0.3

and

D =

1 0

0 2

.




Example

The Markov jump process r(t) takes values inM = 1, 2, ..., 5 with generator

Q =

−4 0 1 1 21 −2 0 1 06 1 −8 0 10 1 1 −3 12 1 1 1 −5

The stationary distribution of irreducible Markov process r(t) isπ = (0.27, 0.24, 0.09, 0.23, 0.17), which is obtained by solving Equation (5).Note that the five Hurwitz matrices associated to System (12) are given by

Ai = A +(1− i)

5BD−1C for i ∈ 1, 2, ..., 5.

Then we have

A1 =

−2.0000 −1.0000

1.0000 0.0000

, A2 =

−2.2350 −1.6310

0.8330 −0.3650

,




Example

A3 =

−2.4700 −2.2620

0.6660 −0.7300

, A4 =

−2.7050 −2.8930

0.4990 −1.0950

and

A5 =

−2.9400 −3.5240

0.3320 −1.4600

.

Note that (A,B) and (A,C) are controllable and observable respectively(rank([B,AB]) = 2 and rank([CT ,ATCT ]) = 2

).

The transfer function G(s) = C(sI − A)−1B + D is symmetric

G(s) =1

(s2 + 2.9s + 2.425)

s2 + 6.9s + 9.535 −s− 1

−s− 1 2s2 + 5.8s + 4.7

and A(A− BD−1C) has no real negative eigenvalue, that means that G(s) isSPR. Then, by Theorem 3.3 the hybrid System (12) is asymptotically almostsurely stable.




Example

A3 =

−2.4700 −2.2620

0.6660 −0.7300

, A4 =

−2.7050 −2.8930

0.4990 −1.0950

and

A5 =

−2.9400 −3.5240

0.3320 −1.4600

.


).


G(s) =1

(s2 + 2.9s + 2.425)

s2 + 6.9s + 9.535 −s− 1

−s− 1 2s2 + 5.8s + 4.7





Example

A3 =

−2.4700 −2.2620

0.6660 −0.7300

, A4 =

−2.7050 −2.8930

0.4990 −1.0950

and

A5 =

−2.9400 −3.5240

0.3320 −1.4600

.


).


G(s) =1

(s2 + 2.9s + 2.425)

s2 + 6.9s + 9.535 −s− 1

−s− 1 2s2 + 5.8s + 4.7





Example

FIGURE: Markov jump r(t) with initialcondition r(0) = 5

FIGURE: Trajectory solution of System (12)with initial condition x(0) = [1, 5]T



Existence and Uniqueness of Solutions of Stochastic Differential Equations with Infinite Delay

1

2

3

4 Existence and Uniqueness of Solutions of Stochastic DifferentialEquations with infinite delay

Let (Ω,F ,P) be a complete probability space with a filtration Ftt≥0 satisfyingthe usual conditions, i.e., it is right continuous and Ft0 contains all P-null sets.M2

((−∞,T];Rn

)denotes the family of all Ft-measurable Rn valued

processes x(t), t ∈ (−∞,T] such that E( T∫−∞|x(t)|2dt

)<∞.

Let Cµ = ϕ ∈ C(−∞; 0];Rn : limθ→−∞

eµθϕ(θ) exists in Rn denote the family of

continuous functions ϕ defined on (−∞, 0] with norm |ϕ|µ = supθ≤0

eµθ|ϕ(θ)|.




1

2

3

4 Existence and Uniqueness of Solutions of Stochastic DifferentialEquations with infinite delay

Let (Ω,F ,P) be a complete probability space with a filtration Ftt≥0 satisfyingthe usual conditions, i.e., it is right continuous and Ft0 contains all P-null sets.M2

((−∞,T];Rn

)denotes the family of all Ft-measurable Rn valued

processes x(t), t ∈ (−∞,T] such that E( T∫−∞|x(t)|2dt

)<∞.

Let Cµ = ϕ ∈ C(−∞; 0];Rn : limθ→−∞

eµθϕ(θ) exists in Rn denote the family of

continuous functions ϕ defined on (−∞, 0] with norm |ϕ|µ = supθ≤0

eµθ|ϕ(θ)|.




Consider the n-dimensional stochastic functional differential equation

dx(t) = f (xt, t)dt + g(xt, t)dW(t), t0 ≤ t ≤ T, (19)

where xt : (−∞, 0] −→ Rn; θ 7−→ xt(θ) = x(t + θ);−∞ < θ ≤ 0 can be regardedas a Cµ-value stochastic processf : Cµ × [t0,T]→ Rn and g : Cµ × [t0,T]→ Rn×m are Borel measurable.Assume that W(t) is an m-dimensional Brownian motion which is defined on(Ω,F ,P).The initial data of the stochastic process is defined on (−∞, t0], withxt0 = ξ = ξ(θ) : −∞ < θ ≤ 0 Ft0 -measurable and ξ ∈M2

(Cµ).







(Cµ).







(Cµ).







(Cµ).




Definition 4.1

The Rn-value stochastic process x(t) defined on −∞ < t ≤ T is called asolution of (19) with initial data xt0 , if x(t) has the following properties :

x(t) is continuous and x(t)t0≤t≤T is Ft-adapted,f (xt, t) ∈ L1

([t0,T];Rn

)and g(xt, t) ∈ L2

([t0,T];Rn×m

),

xt0 = ξ, for each t0 ≤ t ≤ T,

x(t) = ξ(0) +

∫ t

t0f (xs, s)ds +

∫ t

t0g(xs, s)dW(s) almost surely (a.s.) .

x(t) is called a unique solution, if any other solution x(t) is distinguishable withx(t), that is

Px(t) = x(t), for any 0 ≤ t ≤ T = 1. (20)




Definition 4.1



([t0,T];Rn


([t0,T];Rn×m

),


x(t) = ξ(0) +

∫ t

t0f (xs, s)ds +

∫ t



Px(t) = x(t), for any 0 ≤ t ≤ T = 1. (20)




Definition 4.1



([t0,T];Rn


([t0,T];Rn×m

),


x(t) = ξ(0) +

∫ t

t0f (xs, s)ds +

∫ t



Px(t) = x(t), for any 0 ≤ t ≤ T = 1. (20)




Lemma 4.1

If p ≥ 2, g ∈M2([t0,T];Rn×m

)such that E

T∫t0|g(s)|pds <∞, then

E∣∣ T∫

t0

g(s)dW(s)∣∣p ≤ (p(p− 1)

2) p

2 Tp−2

2 E

T∫t0

|g(s)|pds.




Now, we establish existence and uniqueness of solutions for (19) with initialdata xt0 .

Theorem 4.2

Assume that there exist two positive numbers K and K such that(i) For any ϕ,ψ ∈ Cµ and t ∈ [t0,T], it follows that

|f (ϕ, t)− f (ψ, t)|2 ∨ |g(ϕ, t)− g(ψ, t)|2 ≤ K|ϕ− ψ|2µ (21)

(ii) For any t ∈ [t0,T], it follows that f (0, t), g(0, t) ∈ L2(Cµ) such that

|f (0, t)|2 ∨ |g(0, t)|2 ≤ K. (22)

Then, System (19) with initial data xt0 = ξ ∈M2((−∞, 0];Rn), has a uniquesolution x(t). Moreover, x(t) ∈M2

((−∞,T];Rn

).




To show this theorem, we need the following lemma

Lemma 4.3

Let (21) and (22) hold. If x(t) is the solution of (19) with initial data xt0 = ξ, then

E(

supt0≤t≤T

|x(t)|2)≤ Ce6K(T−t0+1)(T−t0) (23)

where C = 3E|ξ|2µ + 6K(T − t0 + 1)(T − t0) + 6K(T − t0 + 1)(T − t0)E|ξ|2µ.Moreover, if ξ ∈M2((−∞, 0];Rn), then x(t) ∈M2((−∞,T];Rn).




Sketch of proof

For each number q ≥ 1, define the stoping time

τq = T ∧ inft ∈ [t0,T] : |xt|µ ≥ q. (24)

Obviously, as q→∞, τq T a.s.Let xq(t) = x(t ∧ τq), t ∈ [t0,T], then xq(t) satisfy the following equation

xq(t) = ξ(0) +

∫ t

t0f (xq

s , s)I[t0,τq](s)ds +

∫ t

t0g(xq

s , s)I[t0,τq](s)dW(s) (25)

By using the Holder inequality, Lemma 4.1, (21) and (22), we show that

E|xq(t)|2 ≤ 3E|ξ|2µ + 6K(t − t0 + 1)(t − t0) + 6K(t − t0 + 1)E∫ t

t0|xq

s |2µds.




Sketch of proof


τq = T ∧ inft ∈ [t0,T] : |xt|µ ≥ q. (24)


xq(t) = ξ(0) +

∫ t

t0f (xq


∫ t

t0g(xq

s , s)I[t0,τq](s)dW(s) (25)



t0|xq

s |2µds.




Sketch of proof


τq = T ∧ inft ∈ [t0,T] : |xt|µ ≥ q. (24)


xq(t) = ξ(0) +

∫ t

t0f (xq


∫ t

t0g(xq

s , s)I[t0,τq](s)dW(s) (25)



t0|xq

s |2µds.




Sketch of proof

Using some properties of the norm, we get

E(

supt0≤s≤T

|xq(s)|2)≤ C + 6K(T − t0 + 1)

∫ T

t0E(

supt0≤r≤T

|xq(r)|2)dr

where C = 3E|ξ|2µ + 6K(T − t0 + 1)(T − t0) + 6K(T − t0 + 1)(T − t0)E|ξ|2µ.By the Gronwall inequality, we infer

E(

supt0≤s≤T

|x(s ∧ τq)|2)≤ Ce6K(T−t0+1)(T−t0).

Letting q −→∞, that implies the following inequality

E(

supt0≤s≤T

|x(s)|2)≤ Ce6K(T−t0+1)(T−t0).




Sketch of proof


E(

supt0≤s≤T

|xq(s)|2)≤ C + 6K(T − t0 + 1)

∫ T

t0E(

supt0≤r≤T

|xq(r)|2)dr


E(

supt0≤s≤T

|x(s ∧ τq)|2)≤ Ce6K(T−t0+1)(T−t0).


E(

supt0≤s≤T

|x(s)|2)≤ Ce6K(T−t0+1)(T−t0).




Sketch of proof


E(

supt0≤s≤T

|xq(s)|2)≤ C + 6K(T − t0 + 1)

∫ T

t0E(

supt0≤r≤T

|xq(r)|2)dr


E(

supt0≤s≤T

|x(s ∧ τq)|2)≤ Ce6K(T−t0+1)(T−t0).


E(

supt0≤s≤T

|x(s)|2)≤ Ce6K(T−t0+1)(T−t0).




Next, to prove the second part or the lemma, suppose thatξ ∈M2((−∞, 0];Rn). Then

E(

sup−∞≤t≤T

|x(t)|2)

= E(

sup−∞≤t≤t0

|x(t)|2)

+ E(

supt0≤t≤T

|x(t)|2)

≤ E(

sup−∞≤t≤t0

|x(t)|2)

+ Ce6K(T−t0+1)(T−t0)

≤ E(

sup−∞≤t−t0≤0

|x(t − t0 + t0)|2)

+ Ce6K(T−t0+1)(T−t0)

≤ E(

sup−∞≤s≤0

|xt0(s)|2)

+ Ce6K(T−t0+1)(T−t0)

≤ E|ξ|2 + Ce6K(T−t0+1)(T−t0)

<∞.




Proof of theorem 4.2

We begin by check uniqueness of solution.Let x(t) and x(t) be two solutions of (19), by lemma 4.3 x(t) andx(t) ∈M2

((−∞,T];Rn

). Note that

x(t)− x(t) =

∫ t

t0

[f (xs, s)− f (xs, s)

]ds +

∫ t

t0

[g(xs, s)− g(xs, s)

]dW(s)

By Holder inequality, Lemma 4.1, (21), (22) and properties of the norm,we show that

E(

supt0≤s≤t

∣∣x(s)− x(s)∣∣2) ≤ 2K(t − t0 + 1)

∫ t

t0E(

supt0≤s≤t

∣∣x(s)− x(s)∣∣2)ds.

Applying the Gronwall inequality to yield

E(|x(t)− x(t)|2

)= 0, t0 ≤ t ≤ T

That is x(t) = x(t) a.s. for t0 ≤ t ≤ T. Therefore, for all−∞ < t ≤ T, x(t) = x(t) a.s.






((−∞,T];Rn

). Note that

x(t)− x(t) =

∫ t

t0

[f (xs, s)− f (xs, s)

]ds +

∫ t

t0


]dW(s)


E(

supt0≤s≤t

∣∣x(s)− x(s)∣∣2) ≤ 2K(t − t0 + 1)

∫ t

t0E(

supt0≤s≤t

∣∣x(s)− x(s)∣∣2)ds.


E(|x(t)− x(t)|2

)= 0, t0 ≤ t ≤ T







((−∞,T];Rn

). Note that

x(t)− x(t) =

∫ t

t0

[f (xs, s)− f (xs, s)

]ds +

∫ t

t0


]dW(s)


E(

supt0≤s≤t

∣∣x(s)− x(s)∣∣2) ≤ 2K(t − t0 + 1)

∫ t

t0E(

supt0≤s≤t

∣∣x(s)− x(s)∣∣2)ds.


E(|x(t)− x(t)|2

)= 0, t0 ≤ t ≤ T





Next, to check the existence, define x0t0 = ξ and x0(t) = ξ(0) for t0 ≤ t ≤ T.

Let xkt0 = ξ, k = 1, 2, ..., and define Picard sequence

xk(t) = ξ(0) +

∫ t

t0f (xk−1

s , s)ds +

∫ t

t0g(xk−1

s , s)dW(s), t0 ≤ t ≤ T (26)

Obviously x0(t) ∈M2((−∞,T];Rn

). By induction, we can see that

xk(t) ∈M2((−∞,T];Rn

).

By Holder inequality, Lemma 4.1, (21) and (22)

E|xk(t)|2 ≤ C1 + C2 + 6K(t − t0 + 1)E∫ t

t0|xk−1(s)|2ds

where C1 = 3E|ξ|2µ + 6K(t − t0 + 1)(t − t0)

and C2 = 6K(t − t0 + 1)(t − t0)E|ξ|2µ.






xk(t) = ξ(0) +

∫ t

t0f (xk−1

s , s)ds +

∫ t

t0g(xk−1

s , s)dW(s), t0 ≤ t ≤ T (26)



xk(t) ∈M2((−∞,T];Rn

).


E|xk(t)|2 ≤ C1 + C2 + 6K(t − t0 + 1)E∫ t

t0|xk−1(s)|2ds

where C1 = 3E|ξ|2µ + 6K(t − t0 + 1)(t − t0)

and C2 = 6K(t − t0 + 1)(t − t0)E|ξ|2µ.






xk(t) = ξ(0) +

∫ t

t0f (xk−1

s , s)ds +

∫ t

t0g(xk−1

s , s)dW(s), t0 ≤ t ≤ T (26)



xk(t) ∈M2((−∞,T];Rn

).


E|xk(t)|2 ≤ C1 + C2 + 6K(t − t0 + 1)E∫ t

t0|xk−1(s)|2ds

where C1 = 3E|ξ|2µ + 6K(t − t0 + 1)(t − t0)

and C2 = 6K(t − t0 + 1)(t − t0)E|ξ|2µ.




By using Gronwall inequality

E|xk(t)|2 ≤ C3e6K(T−t0+1)(T−t0) t0 ≤ t ≤ T, k ≥ 1.

where C3 = C1 + 2C2.From the Holder inequality, Lemma 4.1, (21) and (22), as in a similarearlier inequality, one then has

E(

supt0≤s≤t

|x1(s)− x0(s)|2)≤ 4(T − t0 + 1)(T − t0)(K + KE|ξ|2µ) := R.

By similar arguments as above, we also have

E(

supt0≤s≤t

|x2(s)− x1(s)|2)≤ 2RK(T − t0 + 1)(T − t0) = RM(T − t0),

where M = 2K(T − t0 + 1).Continuing this process to find that

E(

supt0≤s≤t

|xk+1(s)− xk(s)|2)≤

R[M(T − t0)

]k

k!, t0 ≤ t ≤ T







E(

supt0≤s≤t

|x1(s)− x0(s)|2)≤ 4(T − t0 + 1)(T − t0)(K + KE|ξ|2µ) := R.


E(

supt0≤s≤t

|x2(s)− x1(s)|2)≤ 2RK(T − t0 + 1)(T − t0) = RM(T − t0),


E(

supt0≤s≤t

|xk+1(s)− xk(s)|2)≤

R[M(T − t0)

]k

k!, t0 ≤ t ≤ T







E(

supt0≤s≤t

|x1(s)− x0(s)|2)≤ 4(T − t0 + 1)(T − t0)(K + KE|ξ|2µ) := R.


E(

supt0≤s≤t

|x2(s)− x1(s)|2)≤ 2RK(T − t0 + 1)(T − t0) = RM(T − t0),


E(

supt0≤s≤t

|xk+1(s)− xk(s)|2)≤

R[M(T − t0)

]k

k!, t0 ≤ t ≤ T







E(

supt0≤s≤t

|x1(s)− x0(s)|2)≤ 4(T − t0 + 1)(T − t0)(K + KE|ξ|2µ) := R.


E(

supt0≤s≤t

|x2(s)− x1(s)|2)≤ 2RK(T − t0 + 1)(T − t0) = RM(T − t0),


E(

supt0≤s≤t

|xk+1(s)− xk(s)|2)≤

R[M(T − t0)

]k

k!, t0 ≤ t ≤ T




By Alembert’s rule, Chebyshev inequality and Borel-Cantelli’s Lemma, weshow that xk(t) is also a Cauchy sequence in L2. Hence, xk(t)converges uniformly and let x(t) be its limit for any t ∈ (−∞,T]

Noting that the sequence xk(t) → x(t) means that for any given ε > 0there exist k0 such that when k ≥ k0, for any t ∈ (−∞,T], one has

E|xk(t)− x(t)|2 < ε, and∫ T

t0E|xk(t)− x(t)|2dt < (T − t0)ε.

Which implies that∫ t

t0f (xk

s , s)ds→∫ t

t0f (xs, s)ds and

∫ t

t0g(xk

s , s)dW(s)→∫ t

t0g(xs, s)dW(s) in L2.

Taking limits on both sides of (26), we obtain

x(t) = ξ(0) +

∫ t

t0f (xs, s)ds +

∫ t

t0g(xs, s)dW(s) t0 ≤ t ≤ T.







t0E|xk(t)− x(t)|2dt < (T − t0)ε.


t0f (xk

s , s)ds→∫ t

t0f (xs, s)ds and

∫ t

t0g(xk

s , s)dW(s)→∫ t



x(t) = ξ(0) +

∫ t

t0f (xs, s)ds +

∫ t








t0E|xk(t)− x(t)|2dt < (T − t0)ε.


t0f (xk

s , s)ds→∫ t

t0f (xs, s)ds and

∫ t

t0g(xk

s , s)dW(s)→∫ t



x(t) = ξ(0) +

∫ t

t0f (xs, s)ds +

∫ t








t0E|xk(t)− x(t)|2dt < (T − t0)ε.


t0f (xk

s , s)ds→∫ t

t0f (xs, s)ds and

∫ t

t0g(xk

s , s)dW(s)→∫ t



x(t) = ξ(0) +

∫ t

t0f (xs, s)ds +

∫ t




Stability Analysis of Stochastic Neural Network with Infinite Delay and Markovian jump

1

2

3

4

5 Stability Analysis of Stochastic Neural Network with Infinite Delayand Markovian jump

We define Cµα , φ ∈ Cµ; |φ|µ < α.For any M > 0, define two random variables τ y

M and τMy as follows :

τ yM = inft ≥ t0 : |y(t)| ≥ M, |ξ|µ < M, a.s.

τMy = inft ≥ t0 : |y(t)| ≤ M, |ξ|µ > M, a.s.,

where y : [0,+∞)× Ω −→ R is a continuous stochastic process.




The general neural networks (NNs) with infinite delay can be described by aVolterra integro-differential equation :

u(t) = −Du(t) + Ag(u(t)) +

∫ t

−∞CKT(t − s)g(u(s))ds + J, (27)

where u(t) = (u1(t), u2(t), ..., un(t))T ∈ Rn is the state vector associated with theneurons, D = diag(d1, d2, ..., dn) 0 is the firing rate of the neurons,A = (aij)n×n and C = (cij)n×n are connection weight matrices,g(u) = (g1(u1), g2(u2), ..., gn(un)T is the neuron activation function vector,K = (Kij)n×n such that Kij : [0,+∞) −→ [0,+∞) (i, j = 1, 2, ..., n) are piecewisecontinuous on [0,+∞) satisfying∫ +∞

0Kij(s)eµsds = K. i, j = 1, 2, ..., n. (28)

where K is a positive constant depending on µ.and J = (J1, J2, ..., Jn)T is the constant external input vector




By making a transformation x(t) = u(t)− u∗, System (27) has a uniqueequilibrium point, and it can be rewritten as

x(t) = −Dx(t) + AF(x(t)) +

∫ t

−∞CKT(t − s)F(x(s))ds, (29)

where F(x(t)) =(g1(x1(t) + u∗1), ..., gn(xn(t) + u∗n)

)T,(f1(x1(t)), ..., fn(xn(t))

)T .Consider System (29) disturbed by white noise and Markovian switching,which, naturally, called stochastic neural networks with infinite delay andMarkovian switching as follows :

dx(t) =[− Dx(t) + A(r(t))F(x(t)) +

∫ t

−∞C(r(t))KT(t − s)F(x(s))ds

]dt + B(r(t))Q(x(t))dW(t),

(30)

where B(r(t)) =(bij(r(t))

)n×n and Q(x) =

(q1(x1(t)), q2(x2(t)), ..., qn(xn(t))

)T

represents the disturbance intensity of white noise satisfying Q(0) = 0.We also assume that Markov chain r(t) is independent of Brownian motionW(t), and it is irreducible.




By making a transformation x(t) = u(t)− u∗, System (27) has a uniqueequilibrium point, and it can be rewritten as

x(t) = −Dx(t) + AF(x(t)) +

∫ t

−∞CKT(t − s)F(x(s))ds, (29)

where F(x(t)) =(g1(x1(t) + u∗1), ..., gn(xn(t) + u∗n)

)T,(f1(x1(t)), ..., fn(xn(t))

)T .Consider System (29) disturbed by white noise and Markovian switching,which, naturally, called stochastic neural networks with infinite delay andMarkovian switching as follows :

dx(t) =[− Dx(t) + A(r(t))F(x(t)) +

∫ t

−∞C(r(t))KT(t − s)F(x(s))ds

]dt + B(r(t))Q(x(t))dW(t),

(30)

where B(r(t)) =(bij(r(t))

)n×n and Q(x) =

(q1(x1(t)), q2(x2(t)), ..., qn(xn(t))

)T

represents the disturbance intensity of white noise satisfying Q(0) = 0.We also assume that Markov chain r(t) is independent of Brownian motionW(t), and it is irreducible.




For any (φ, k) ∈ Cµ ×M, we denoteE(φ, k)

= −Dφ(0) + A(k)F(φ(0)) +∫ t−∞ C(k)KT(t − s)F(φ(s− t))ds,

H(φ, k)

= B(k)Q(φ(0)).

If V ∈ C2(Rn×R+×M;R+

), define an operator LV from Rn×R+×M to R by

LV(x, t, k) = Vt(x, t, k) + Vx(x, t, k)E(xt, k)

+12

Trace[HT(xt, k)Vxx(x, t, k)H(xt, k)

]+

N∑`=1

γk`V(x, t, `), (31)

where

Vt(x, t, k) =∂V(x, t, k)

∂t, Vx(x, t, k) =

(∂V(x, t, k)

∂x1, ...,

∂V(x, t, k)

∂xn

)and

Vxx(x, t, k) =(∂2V(x, t, k)

∂xi∂xj

)n×n




For any (φ, k) ∈ Cµ ×M, we denoteE(φ, k)

= −Dφ(0) + A(k)F(φ(0)) +∫ t−∞ C(k)KT(t − s)F(φ(s− t))ds,

H(φ, k)

= B(k)Q(φ(0)).

If V ∈ C2(Rn×R+×M;R+

), define an operator LV from Rn×R+×M to R by

LV(x, t, k) = Vt(x, t, k) + Vx(x, t, k)E(xt, k)

+12

Trace[HT(xt, k)Vxx(x, t, k)H(xt, k)

]+

N∑`=1

γk`V(x, t, `), (31)

where

Vt(x, t, k) =∂V(x, t, k)

∂t, Vx(x, t, k) =

(∂V(x, t, k)

∂x1, ...,

∂V(x, t, k)

∂xn

)and

Vxx(x, t, k) =(∂2V(x, t, k)

∂xi∂xj

)n×n




Assumption 5.1

For each j ∈ 1, 2, ..., n, functions gj : R −→ R and qj : R −→ R satisfy globalLipschitz conditions

|gj(x)− gj(y)| ∨ |qj(x)− qj(y)| ≤ Lj|x− y|, for x, y ∈ R, (32)

that is,|F(x)| ∨ |Q(x)| ≤ L|x| (33)

where L = maxL1,L2, ...,Ln. In addition, the initial data xt0 = ξ satisfies|ξ| := sup

θ≤0|ξ(θ)| <∞.




Theorem 5.1

Suppose that Assumption 5.1 holds. Then System (30) has a unique globalsolution on (−∞,∞) with initial data ξ ∈ Cµ and r(t0) = r0.




Sketch of proof

By definition of the right continuous Markov jump r(.), there is asequence τkk≥0 of stoping times such that r(.) is a random constant onevery interval [τk, τk+1), that is r(t) = r(τk) on τk ≤ t < τk+1, for any k ≥ 0.We consider first System (30) for t ∈ [τ0, τ1), we can rewrite it as

dx(t) = E(xt, r0)dt + H(xt, r0)dW(t) (34)

By calculation, we get

|E(ξ, r0)− E(ζ, r0)| ≤(|D|+ L|A(r0)|+ n2L|C(r0)|K

)|ξ − ζ|µ

and |H(ξ, r0)−H(ζ, r0)| ≤ L|B(r0)||ξ − ζ|µ

Then, by Theorem 4.2 System (30) with initial condition ξ ∈ Cµ andr(t0) = r0 has a unique solution on [τ0, τ1).By the same argument of existence and uniqueness as the first stepabove, System (30) with initial condition ξ ∈ Cµ and r(0) = r0 has aunique solution on (−∞,∞).




Sketch of proof





)|ξ − ζ|µ






Sketch of proof





)|ξ − ζ|µ






Sketch of proof





)|ξ − ζ|µ






Sketch of proof





)|ξ − ζ|µ






Definition 5.1The solution of System (30) with initial data xt0 = ξ is said to be stochasticallystable if for every pair ε ∈ (0, 1) and α > 0, there exists a δ = δ(ε, α) > 0 suchthat

P|x(t, t0, ξ)| < α, t ≥ t0

≥ 1− ε,

whenever (ξ, k) ∈ Cµδ ×M.

Definition 5.2The solution of System (30) with initial data xt0 = ξ is said to be stochasticallyasymptotically stable if it is stochastically stable and, moreover, for everyε ∈ (0, 1), there exist δ0 = δ0(ε) > 0 such that

P

limt→∞

x(t, t0, ξ) = 0≥ 1− ε,

whenever (ξ, k) ∈ Cµδ0×M.




Definition 5.1The solution of System (30) with initial data xt0 = ξ is said to be stochasticallystable if for every pair ε ∈ (0, 1) and α > 0, there exists a δ = δ(ε, α) > 0 suchthat

P|x(t, t0, ξ)| < α, t ≥ t0

≥ 1− ε,

whenever (ξ, k) ∈ Cµδ ×M.

Definition 5.2The solution of System (30) with initial data xt0 = ξ is said to be stochasticallyasymptotically stable if it is stochastically stable and, moreover, for everyε ∈ (0, 1), there exist δ0 = δ0(ε) > 0 such that

P

limt→∞

x(t, t0, ξ) = 0≥ 1− ε,

whenever (ξ, k) ∈ Cµδ0×M.




Definition 5.3The solution of System (30) with initial data xt0 = ξ is said to be globallystochastically asymptotically stable if it is stochastically stable and, moreover,for any (ξ, k) ∈ Cµ ×M,

P

limt→∞

x(t, t0, ξ) = 0

= 1.

Let A := −diag(2β1, 2β2, ..., 2βN)−Q where d = mind1, d2, ..., dn. and

βk := −d + L|A(k)|+ 12

L2|B(k)|2 + n2KL|C(k)|, k ∈M.




Definition 5.3The solution of System (30) with initial data xt0 = ξ is said to be globallystochastically asymptotically stable if it is stochastically stable and, moreover,for any (ξ, k) ∈ Cµ ×M,

P

limt→∞

x(t, t0, ξ) = 0

= 1.

Let A := −diag(2β1, 2β2, ..., 2βN)−Q where d = mind1, d2, ..., dn. and

βk := −d + L|A(k)|+ 12

L2|B(k)|2 + n2KL|C(k)|, k ∈M.




Assumption 5.2

There is a λ = (λ1, λ2, ..., λN)T ≥ 0 in RN such that P = Aλ ≥ 0.

Theorem 5.2

Suppose that Assumptions 5.1 and 5.2 hold. Then the trivial solution toSystem (30) is stochastically stable.




Assumption 5.2

There is a λ = (λ1, λ2, ..., λN)T ≥ 0 in RN such that P = Aλ ≥ 0.

Theorem 5.2

Suppose that Assumptions 5.1 and 5.2 hold. Then the trivial solution toSystem (30) is stochastically stable.




Sketch of proof

For any ε ∈ (0.1), and α > 0, we choose a sufficiently small δ(ε, α), such that for anyξ ∈ Cµ

δ(ε,α),

λk|ξ|2µ + 2n2KL|ξ|µ < λkεα2 for any k ∈M

For t ≥ t0, k = 1, 2, ...,N, let

V(x, t, k) =12λk|x|2 +

∫ +∞

t

n∑i=1

n∑j=1

Kij(s− t)|fj(xj(2t − s))|ds (35)

From Assumptions 5.1, 5.2, using the fact that x(t) = x(t + 0) = xt(0) and the transformationv = t − s, we show that

LV(x(t), t, k) ≤ −12

pk|xt|2µ.

Other by Dynkin formula, Assumption 5.1 and Eq. (28) we inferλk

2α2P(ταx < t) ≤ V(x(t ∧ ταx ), t ∧ ταx , k) ≤ EV(x(t0), t0, k)

≤λk

2E|ξ|2µ + n2KLE|ξ|µ <

λk

2εα2

Letting t −→∞ we have Pταx <∞ < ε, which is equivalent toP|x(t, t0, ξ)| ≤ α, t ≥ t0 ≥ 1− ε. (36)




Sketch of proof


δ(ε,α),


For t ≥ t0, k = 1, 2, ...,N, let

V(x, t, k) =12λk|x|2 +

∫ +∞

t

n∑i=1

n∑j=1



LV(x(t), t, k) ≤ −12

pk|xt|2µ.



≤λk


λk

2εα2





Sketch of proof


δ(ε,α),


For t ≥ t0, k = 1, 2, ...,N, let

V(x, t, k) =12λk|x|2 +

∫ +∞

t

n∑i=1

n∑j=1



LV(x(t), t, k) ≤ −12

pk|xt|2µ.



≤λk


λk

2εα2





Sketch of proof


δ(ε,α),


For t ≥ t0, k = 1, 2, ...,N, let

V(x, t, k) =12λk|x|2 +

∫ +∞

t

n∑i=1

n∑j=1



LV(x(t), t, k) ≤ −12

pk|xt|2µ.



≤λk


λk

2εα2





Assumption 5.3

If A is a nonsingular M-matrix, there is aλ = (λ1, λ2, ..., λN)T 0 in RN such that P = Aλ 0.

Theorem 5.3

Suppose that Assumptions 5.1 and 5.3 hold. Then the solution to System (30)is stochastically asymptotically stable.




Sketch of proof

Lemma 5.4

Suppose Assumptions 5.1 and 5.2 hold. Then for any ε ∈ (0, 1) and α > 0,there exists a R(ε, α) > 0, such that for any t0 ≥ 0 and ξ ∈ Cµα a.s.,

P|x(t, t0, ξ)| ≤ R, t ≥ t0 ≥ 1− ε.

From Theorem 5.2, we can easily see that the trivial solution to System(30) is stochastically stable, that is, for any δ1 > 0 and ε ∈ (0, 1), thereexists a δ(ε, δ1) > 0 such that for any ξ ∈ Cµδ(ε,δ1)

,

P(A) ≥ 1− ε,

in which A , ω : |x(t, t0, ξ)| < δ, t ≥ t0.




Sketch of proof

Lemma 5.4

Suppose Assumptions 5.1 and 5.2 hold. Then for any ε ∈ (0, 1) and α > 0,there exists a R(ε, α) > 0, such that for any t0 ≥ 0 and ξ ∈ Cµα a.s.,

P|x(t, t0, ξ)| ≤ R, t ≥ t0 ≥ 1− ε.

From Theorem 5.2, we can easily see that the trivial solution to System(30) is stochastically stable, that is, for any δ1 > 0 and ε ∈ (0, 1), thereexists a δ(ε, δ1) > 0 such that for any ξ ∈ Cµδ(ε,δ1)

,

P(A) ≥ 1− ε,

in which A , ω : |x(t, t0, ξ)| < δ, t ≥ t0.




Sketch of proof

From Lemma 5.4 it follows immediately that for δ1 and any ε1 ∈ (0, 1),there exists a H(ε1, δ1) sufficiently large such that

P|x(t, θ∗, ξθ∗)| ≤ H, t ≥ θ∗ ≥ 1− ε1

4, and |xθ∗ |µ < H,∀θ∗ ≤ t. (37)

Next, we show that if there exists a k > 0, such that

Pω ∈ A : |x(τk, t0, ξ)| = 0, t ≥ k = P(A) ≥ 1− ε,

then the trivial solution to System (30) is stochastically asymptoticallystable




Sketch of proof

From Lemma 5.4 it follows immediately that for δ1 and any ε1 ∈ (0, 1),there exists a H(ε1, δ1) sufficiently large such that

P|x(t, θ∗, ξθ∗)| ≤ H, t ≥ θ∗ ≥ 1− ε1

4, and |xθ∗ |µ < H,∀θ∗ ≤ t. (37)

Next, we show that if there exists a k > 0, such that

Pω ∈ A : |x(τk, t0, ξ)| = 0, t ≥ k = P(A) ≥ 1− ε,

then the trivial solution to System (30) is stochastically asymptoticallystable




Theorem 5.5

Suppose that Assumptions 5.1 and 5.3 hold. Then the solution to System (30)is globally stochastically asymptotically stable.




Sketch of proof

By Theorem 5.2, the solution of System (30) is stochastically stable. Sowe only need to show that for any ξ ∈ Cµ,

P limt→∞

x(t, t0, ξ) = 0 = 1.

Fix any ε ∈ (0, 1) and ξ ∈ Cµ. Let

V(x, t, k) =λk

2

n∑i=1

x2i +

∫ +∞

t

n∑i=1

n∑j=1

Kij(s− t)|fj(xj(2t − s))|ds.

Let H be sufficiently large such that

infω∈Ω,|x|>H,t≥t0

V(x, t, k) ≥ V(ξ(0), t0, k)

ε.

By the generalized Ito formula, we infer

PτHx < t ≤ εV(x(t ∧ τH

x ), t ∧ τHx , k)

V(ξ(0), t0, k)< ε.




Sketch of proof


P limt→∞

x(t, t0, ξ) = 0 = 1.


V(x, t, k) =λk

2

n∑i=1

x2i +

∫ +∞

t

n∑i=1

n∑j=1




V(x, t, k) ≥ V(ξ(0), t0, k)

ε.



x ), t ∧ τHx , k)

V(ξ(0), t0, k)< ε.




Sketch of proof


P limt→∞

x(t, t0, ξ) = 0 = 1.


V(x, t, k) =λk

2

n∑i=1

x2i +

∫ +∞

t

n∑i=1

n∑j=1




V(x, t, k) ≥ V(ξ(0), t0, k)

ε.



x ), t ∧ τHx , k)

V(ξ(0), t0, k)< ε.




Sketch of proof


P limt→∞

x(t, t0, ξ) = 0 = 1.


V(x, t, k) =λk

2

n∑i=1

x2i +

∫ +∞

t

n∑i=1

n∑j=1




V(x, t, k) ≥ V(ξ(0), t0, k)

ε.



x ), t ∧ τHx , k)

V(ξ(0), t0, k)< ε.




Let t −→∞. ThenPτH

x <∞ < ε

namely,Psup

t≥t0|x(t, t0, ξ)| ≤ H ≥ 1− ε.

From here, following the proof of Theorem 5.3 we can easily find that

P limt→∞

x(t, t0, ξ) = 0 ≥ 1− ε.

Since ε is arbitrary, the trivial solution to system (30) is globallystochastically asymptotically stable.





x <∞ < ε

namely,Psup

t≥t0|x(t, t0, ξ)| ≤ H ≥ 1− ε.


P limt→∞

x(t, t0, ξ) = 0 ≥ 1− ε.






x <∞ < ε

namely,Psup

t≥t0|x(t, t0, ξ)| ≤ H ≥ 1− ε.


P limt→∞

x(t, t0, ξ) = 0 ≥ 1− ε.





Example

Example 5.1

Let r(t) be a right-continuous Markovian process taking values inM = 1, 2, 3 with generator

Q =

−2 1 12 −4 23 2 −5

Consider a two-dimensional System (30) with the following specification

D =

(15 00 15

),A(1) =

(2 11 1.5

),B(1) =

(1 00 1

),C(1) =

(0.5 00√

2,

),

A(2) =

(2 0.5

0.3 0.8

),B(2) =

(√0.2 00

√0.2

),C(2) =

(1 00√

2

),

A(3) =

(2 0.25

0.25 0.5

),B(3) =

(1 00 1

)and C(3) =

(0.3 00 0.5,

).




Example

Example 5.1

Let r(t) be a right-continuous Markovian process taking values inM = 1, 2, 3 with generator

Q =

−2 1 12 −4 23 2 −5

Consider a two-dimensional System (30) with the following specification

D =

(15 00 15

),A(1) =

(2 11 1.5

),B(1) =

(1 00 1

),C(1) =

(0.5 00√

2,

),

A(2) =

(2 0.5

0.3 0.8

),B(2) =

(√0.2 00

√0.2

),C(2) =

(1 00√

2

),

A(3) =

(2 0.25

0.25 0.5

),B(3) =

(1 00 1

)and C(3) =

(0.3 00 0.5,

).




Example

We rewrite System (30) in detailed formdx(t) =

[− 15x(t) + a11(r(t))h(x(t)) + a12(r(t))h(y(t))

+∫ t−∞ c11(r(t))es−t(h(x(s)) + h(y(s)))ds

]dt + b11(r(t))q1(x(t))dW(t),

dy(t) =[− 15x(t) + a21(r(t))h(x(t)) + a22(r(t))h(y(t))


]dt + b22(r(t))q2(x(t))dW(t),

(38)where q1(x) = q2(x) = sin x satisfies global Lipschitz condition with Lipschitzconstant L = 1, h(x) = sin x, this means that Assumption 5.1 is verified.

To validate Assumption 5.3, let µ = 0.4, then

A =

+4.2554 −1.000 −1.0000−2.0000 +6.0428 −2.0000−3.0000 −2.0000 21.0421

.

Hence, it is desired that A is a nonsingular M-matrix.By Theorem 5.5, System (30) is globally stochastically asymptotically stable.




Example


[− 15x(t) + a11(r(t))h(x(t)) + a12(r(t))h(y(t))


]dt + b11(r(t))q1(x(t))dW(t),



]dt + b22(r(t))q2(x(t))dW(t),



A =

+4.2554 −1.000 −1.0000−2.0000 +6.0428 −2.0000−3.0000 −2.0000 21.0421

.





Example


[− 15x(t) + a11(r(t))h(x(t)) + a12(r(t))h(y(t))


]dt + b11(r(t))q1(x(t))dW(t),



]dt + b22(r(t))q2(x(t))dW(t),



A =

+4.2554 −1.000 −1.0000−2.0000 +6.0428 −2.0000−3.0000 −2.0000 21.0421

.





Example

FIGURE: Jump process r(t) with initialcondition r(0)=1

FIGURE: Approximate solution ofSystem (38) with initial conditionsx(0) = sin2(0), y(0) = 0.6



Conclusion and perspectives

1

2

3

4

5

6 Conclusion and Perspectives

Our work concerns stochastic stability analysis of hybrid dynamical systemswith Markovian switching, using Lyapunov method, M-matrix theory andstochastic analysis.




As perspectives :

We intend to collaborate with researchers in electrical engineering for theapplication of hybrid dynamic systems with Markovian switching,especially for studying processes altered by abrupt variations.We project to work on other mathematical aspects of study such as

Stochastic optimal controlInfinite dimension...




Thank you for your Attention

Merci pour votre Attention



Publications

Publications

C. Imzegouan, H. Bouzahir, B. Benaid and F. El Guezar, A Note onExponential Stochastic Stability of Markovian Switching Systems,International Journal of Evolution Equations, Vol 10, Issue 2, (2016), pp.189-198.

C. Imzegouan, Stochastic Stability in terms of an Associated TransferFunction Matrix for Some Hybrid Systems with Markovian Switching.Commun. Fac. Sci. Univ. Ankara, Ser. A1, Math. Stat. Volum 67, Number1, pages 1-0 (2018)

H. Bouzahir, B. Benaid and C. Imzegouan, Some Stochastic FunctionalDifferential Equations with Infinite Delay : A Result on Existence andUniqueness of Solutions in a Concrete Fading Memory Space. Chin. J.Math. (N.Y.) 2017.

B. Benaid, H. Bouzahir, C. Imzegouan and F. El Guezar StochasticStability Analysis for Stochastic Neural Networks with MarkovianSwitching and Infinite Delay in a phase space, (In revision).


Documents

ABC · Stability Analysis of Markovian Jump Systems CONTRIBUTION A L’ETUDE DES SYSTEMES DYNAMIQUES HYBRIDES A COMMUTATIONS ALEATOIRES Realis´ e par: Chafai Imzegouan´ Sous l’encadr