Stability Analysis of Markovian Jump Systems UNIVERSITE IBN ZOHR Ecole Nationale des Sciences Appliqu ´ ees Formation Doctorale : Sciences et Techniques de l’Ing´ enieur Descipline : Math ´ ematiques Appliqu ´ ees Sp´ ecialit ´ e : Calcul Stochastique et Syst ` emes Dynamiques Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 ao ˆ ut 2017 1 / 67
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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
Stability Analysis of Markovian Jump Systems
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
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 2 / 67
Stability Analysis of Markovian Jump Systems
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
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 2 / 67
Stability Analysis of Markovian Jump Systems
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
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 2 / 67
Stability Analysis of Markovian Jump Systems
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
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 3 / 67
Stability Analysis of Markovian Jump Systems
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
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 3 / 67
Stability Analysis of Markovian Jump Systems
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
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 3 / 67
Stability Analysis of Markovian Jump Systems
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
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 3 / 67
Stability Analysis of Markovian Jump Systems
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
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 3 / 67
Stability Analysis of Markovian Jump Systems
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
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 3 / 67
Stability Analysis of Markovian Jump Systems
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 4 / 67
Stability Analysis of Markovian Jump Systems
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 5 / 67
Stability Analysis of Markovian Jump Systems
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 5 / 67
Stability Analysis of Markovian Jump Systems
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 6 / 67
Stability Analysis of Markovian Jump Systems
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 6 / 67
Stability Analysis of Markovian Jump Systems
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 6 / 67
Stability Analysis of Markovian Jump Systems
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 6 / 67
Stability Analysis of Markovian Jump Systems
Introduction
Motivation
MotivationStability of solutions is important in applications such as communicationnetworks, motor control, economic systems,..., and an important problem is toensure stability.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 7 / 67
Stability Analysis of Markovian Jump Systems
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 8 / 67
Stability Analysis of Markovian Jump Systems
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 9 / 67
Stability Analysis of Markovian Jump Systems
Introduction
Example
FIGURE: Jump process r(t) with initialcondition r(0) = 1
FIGURE: Trajectory of x as a function oftime for System (2).
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 10 / 67
Stability Analysis of Markovian Jump Systems
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 :
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)
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 11 / 67
Stability Analysis of Markovian Jump Systems
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 :
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)
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 11 / 67
Stability Analysis of Markovian Jump Systems
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 :
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)
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 11 / 67
Stability Analysis of Markovian Jump Systems
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 :
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)
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 11 / 67
Stability Analysis of Markovian Jump Systems
Mean Exponential Stability of Markovian Jump Linear System
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).
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 12 / 67
Stability Analysis of Markovian Jump Systems
Mean Exponential Stability of Markovian Jump Linear System
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)
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 13 / 67
Stability Analysis of Markovian Jump Systems
Mean Exponential Stability of Markovian Jump Linear System
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)
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 13 / 67
Stability Analysis of Markovian Jump Systems
Mean Exponential Stability of Markovian Jump Linear System
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 14 / 67
Stability Analysis of Markovian Jump Systems
Mean Exponential Stability of Markovian Jump Linear System
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 14 / 67
Stability Analysis of Markovian Jump Systems
Mean Exponential Stability of Markovian Jump Linear System
Result
Theorem 2.1
Assume that for any i ∈M, each matrix Ai is generalized definite negative,then, System (3) is mean exponentially stable.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 15 / 67
Stability Analysis of Markovian Jump Systems
Mean Exponential Stability of Markovian Jump Linear System
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)
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 16 / 67
Stability Analysis of Markovian Jump Systems
Mean Exponential Stability of Markovian Jump Linear System
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)
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 16 / 67
Stability Analysis of Markovian Jump Systems
Mean Exponential Stability of Markovian Jump Linear System
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)
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 16 / 67
Stability Analysis of Markovian Jump Systems
Mean Exponential Stability of Markovian Jump Linear System
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 17 / 67
Stability Analysis of Markovian Jump Systems
Mean Exponential Stability of Markovian Jump Linear System
Example
FIGURE: Markov jump r(t) with initialcondition r0 = 2
FIGURE: Solution curves of System (3)with initial condition x0 = [−7, 9]T
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 18 / 67
Stability Analysis of Markovian Jump Systems
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 19 / 67
Stability Analysis of Markovian Jump Systems
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 19 / 67
Stability Analysis of Markovian Jump Systems
Asymptotic Almost Sure Stability of Markovian Jump Linear System Associated with a Transfer Matrix
Formalisation
Schematic representation
FIGURE: Dynamical system with Markovian switched controller
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 20 / 67
Stability Analysis of Markovian Jump Systems
Asymptotic Almost Sure Stability of Markovian Jump Linear System Associated with a Transfer Matrix
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 21 / 67
Stability Analysis of Markovian Jump Systems
Asymptotic Almost Sure Stability of Markovian Jump Linear System Associated with a Transfer Matrix
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 22 / 67
Stability Analysis of Markovian Jump Systems
Asymptotic Almost Sure Stability of Markovian Jump Linear System Associated with a Transfer Matrix
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 22 / 67
Stability Analysis of Markovian Jump Systems
Asymptotic Almost Sure Stability of Markovian Jump Linear System Associated with a Transfer Matrix
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 22 / 67
Stability Analysis of Markovian Jump Systems
Asymptotic Almost Sure Stability of Markovian Jump Linear System Associated with a Transfer Matrix
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 23 / 67
Stability Analysis of Markovian Jump Systems
Asymptotic Almost Sure Stability of Markovian Jump Linear System Associated with a Transfer Matrix
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 23 / 67
Stability Analysis of Markovian Jump Systems
Asymptotic Almost Sure Stability of Markovian Jump Linear System Associated with a Transfer Matrix
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 24 / 67
Stability Analysis of Markovian Jump Systems
Asymptotic Almost Sure Stability of Markovian Jump Linear System Associated with a Transfer Matrix
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)
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 25 / 67
Stability Analysis of Markovian Jump Systems
Asymptotic Almost Sure Stability of Markovian Jump Linear System Associated with a Transfer Matrix
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)
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 25 / 67
Stability Analysis of Markovian Jump Systems
Asymptotic Almost Sure Stability of Markovian Jump Linear System Associated with a Transfer Matrix
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)
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 25 / 67
Stability Analysis of Markovian Jump Systems
Asymptotic Almost Sure Stability of Markovian Jump Linear System Associated with a Transfer Matrix
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 26 / 67
Stability Analysis of Markovian Jump Systems
Asymptotic Almost Sure Stability of Markovian Jump Linear System Associated with a Transfer Matrix
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
.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 27 / 67
Stability Analysis of Markovian Jump Systems
Asymptotic Almost Sure Stability of Markovian Jump Linear System Associated with a Transfer Matrix
Example
The Markov jump process r(t) takes values inM = 1, 2, ..., 5 with generator
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
,
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 28 / 67
Stability Analysis of Markovian Jump Systems
Asymptotic Almost Sure Stability of Markovian Jump Linear System Associated with a Transfer Matrix
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 29 / 67
Stability Analysis of Markovian Jump Systems
Asymptotic Almost Sure Stability of Markovian Jump Linear System Associated with a Transfer Matrix
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 29 / 67
Stability Analysis of Markovian Jump Systems
Asymptotic Almost Sure Stability of Markovian Jump Linear System Associated with a Transfer Matrix
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 29 / 67
Stability Analysis of Markovian Jump Systems
Asymptotic Almost Sure Stability of Markovian Jump Linear System Associated with a Transfer Matrix
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
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 30 / 67
Stability Analysis of Markovian Jump Systems
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)|.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 31 / 67
Stability Analysis of Markovian Jump Systems
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)|.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 31 / 67
Stability Analysis of Markovian Jump Systems
Existence and Uniqueness of Solutions of Stochastic Differential Equations with Infinite Delay
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).
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 32 / 67
Stability Analysis of Markovian Jump Systems
Existence and Uniqueness of Solutions of Stochastic Differential Equations with Infinite Delay
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).
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 32 / 67
Stability Analysis of Markovian Jump Systems
Existence and Uniqueness of Solutions of Stochastic Differential Equations with Infinite Delay
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).
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 32 / 67
Stability Analysis of Markovian Jump Systems
Existence and Uniqueness of Solutions of Stochastic Differential Equations with Infinite Delay
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).
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 32 / 67
Stability Analysis of Markovian Jump Systems
Existence and Uniqueness of Solutions of Stochastic Differential Equations with Infinite Delay
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)
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 33 / 67
Stability Analysis of Markovian Jump Systems
Existence and Uniqueness of Solutions of Stochastic Differential Equations with Infinite Delay
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)
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 33 / 67
Stability Analysis of Markovian Jump Systems
Existence and Uniqueness of Solutions of Stochastic Differential Equations with Infinite Delay
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)
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 33 / 67
Stability Analysis of Markovian Jump Systems
Existence and Uniqueness of Solutions of Stochastic Differential Equations with Infinite Delay
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 34 / 67
Stability Analysis of Markovian Jump Systems
Existence and Uniqueness of Solutions of Stochastic Differential Equations with Infinite Delay
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
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
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 42 / 67
Stability Analysis of Markovian Jump Systems
Existence and Uniqueness of Solutions of Stochastic Differential Equations with Infinite Delay
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 43 / 67
Stability Analysis of Markovian Jump Systems
Existence and Uniqueness of Solutions of Stochastic Differential Equations with Infinite Delay
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 43 / 67
Stability Analysis of Markovian Jump Systems
Existence and Uniqueness of Solutions of Stochastic Differential Equations with Infinite Delay
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 43 / 67
Stability Analysis of Markovian Jump Systems
Existence and Uniqueness of Solutions of Stochastic Differential Equations with Infinite Delay
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 43 / 67
Stability Analysis of Markovian Jump Systems
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 44 / 67
Stability Analysis of Markovian Jump Systems
Stability Analysis of Stochastic Neural Network with Infinite Delay and Markovian jump
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
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Stability Analysis of Markovian Jump Systems
Stability Analysis of Stochastic Neural Network with Infinite Delay and Markovian jump
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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 46 / 67
Stability Analysis of Markovian Jump Systems
Stability Analysis of Stochastic Neural Network with Infinite Delay and Markovian jump
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.
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Stability Analysis of Markovian Jump Systems
Stability Analysis of Stochastic Neural Network with Infinite Delay and Markovian jump
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Stability Analysis of Markovian Jump Systems
Stability Analysis of Stochastic Neural Network with Infinite Delay and Markovian jump
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)| <∞.
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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.
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Stability Analysis of Markovian Jump Systems
Stability Analysis of Stochastic Neural Network with Infinite Delay and Markovian jump
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 (−∞,∞).
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Stability Analysis of Markovian Jump Systems
Stability Analysis of Stochastic Neural Network with Infinite Delay and Markovian jump
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 (−∞,∞).
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Stability Analysis of Markovian Jump Systems
Stability Analysis of Stochastic Neural Network with Infinite Delay and Markovian jump
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 (−∞,∞).
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Stability Analysis of Markovian Jump Systems
Stability Analysis of Stochastic Neural Network with Infinite Delay and Markovian jump
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 (−∞,∞).
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Stability Analysis of Markovian Jump Systems
Stability Analysis of Stochastic Neural Network with Infinite Delay and Markovian jump
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 (−∞,∞).
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Stability Analysis of Markovian Jump Systems
Stability Analysis of Stochastic Neural Network with Infinite Delay and Markovian jump
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.
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Stability Analysis of Markovian Jump Systems
Stability Analysis of Stochastic Neural Network with Infinite Delay and Markovian jump
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.
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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.
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Stability Analysis of Markovian Jump Systems
Stability Analysis of Stochastic Neural Network with Infinite Delay and Markovian jump
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.
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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.
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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.
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Stability Analysis of Markovian Jump Systems
Stability Analysis of Stochastic Neural Network with Infinite Delay and Markovian jump
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
Letting t −→∞ we have Pταx <∞ < ε, which is equivalent toP|x(t, t0, ξ)| ≤ α, t ≥ t0 ≥ 1− ε. (36)
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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.
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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.
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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.
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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
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Stability Analysis of Markovian Jump Systems
Stability Analysis of Stochastic Neural Network with Infinite Delay and Markovian jump
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
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Theorem 5.5
Suppose that Assumptions 5.1 and 5.3 hold. Then the solution to System (30)is globally stochastically asymptotically stable.
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Stability Analysis of Stochastic Neural Network with Infinite Delay and Markovian jump
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.
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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.
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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.
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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.
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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.
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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.
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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.
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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,
).
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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,
).
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 61 / 67
Stability Analysis of Markovian Jump Systems
Stability Analysis of Stochastic Neural Network with Infinite Delay and Markovian jump
(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.
(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.
(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.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 63 / 67
Stability Analysis of Markovian Jump Systems
Conclusion and perspectives
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6 Conclusion and Perspectives
Our work concerns stochastic stability analysis of hybrid dynamical systemswith Markovian switching, using Lyapunov method, M-matrix theory andstochastic analysis.
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 64 / 67
Stability Analysis of Markovian Jump Systems
Conclusion and perspectives
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...
Chafai IMZEGOUAN (ENSA-AGADIR) Stability Analysis of Markovian Jump Systems 7 aout 2017 65 / 67
Stability Analysis of Markovian Jump Systems
Conclusion and perspectives
Thank you for your Attention
Merci pour votre Attention
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Stability Analysis of Markovian Jump Systems
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).
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