EXPONENTIAL STABILITY OF STOCHASTIC HYBRID SYSTEMS WITH NONDIFFERENTIABLE AND INTERVAL TIME-VARYING DELAY

Abstract: This paper addresses exponential stability problem for a class of stochastic hybrid systems with time-varying delay. The time delay is any continuous function belonging to a given interval, but not necessary to be differentiable. By constructing a suitable augmented Lyapunov-Krasovskii functional combined with Leibniz-Newton’s formula, new delay-dependent sufficient conditions for exponential stability of stochastic hybrid systems with time-varying delay are first established in terms of LMIs.


Introduction
As an important class of hybrid systems, switched systems arise in many practical processes that cannot be described by exclusively continuous or exclusively discrete models, such as manufacturing, communication networks, automotive engineering control and chemical processes (see, e.g., [1][2][3][4][5][6][7][8][9] and the references therein).The analysis of stochastic systems with respect to mean square stability of their equilibria has attracted many researchers.Such systems occur in a large number of applications as in Physics, Optics or Mechanical Engineering.Often, these systems can generally be written as systems of stochastic differential equations(SDEs).There stability examinations play an essential role in judgement on qualitative behaviour of natural processes.The concept of mean square stability is one of the most attractive and feasible ones within the large branch of stability analysis.To the best of our knowledge, exponential stability of switched stochastic systems with interval time-varying delay, non-differentiable time-varying delays have not been fully studied yet (see, e.g., [10][11][12][13][14][15][16][17][18] and the references therein).Which are important in both theories and applications.This motivates our research.
This paper gives the improved results for the mean square exponential stability of stochastic hybrid systems with interval time-varying delay.The time delay is assumed to be a time-varying continuous function belonging to a given interval, but not necessary to be differentiable.Specifically, our goal is to develop exponential stability of stochastic hybrid systems with interval timevarying delay.By constructing augmented Lyapunov functional combined with LMI technique, we propose new criteria for the exponential stability of stochastic hybrid systems with interval time-varying delay.The delay-dependent exponential stability conditions are formulated in terms of LMIs.
The paper is organized as follows: Section 2 presents definitions and some well-known technical propositions needed for the proof of the main results.Delay-dependent exponential stability conditions of stochastic hybrid systems with interval time-varying delay is presented in Section 3. The conclusions are drawn in Section 4.

Preliminaries
The following notations will be used in this paper.R + denotes the set of all real non-negative numbers; R n denotes the n−dimensional space with the scalar product ., .and the vector norm .; M n×r denotes the space of all matrices of (n × r)−dimensions; A T denotes the transpose of matrix A; A is symmetric if A = A T ; I denotes the identity matrix; λ(A) denotes the set of all eigenvalues of A; λ min/max (A) = min/max{Reλ; λ ∈ λ(A)}; x t := {x(t + s) : * denotes the symmetric term in a matrix.Consider a switched stochastic system with interval time-varying delay of the form where x(t) ∈ R n is the state; γ(.) : R n → N := {1, 2, . . ., N } is the switching rule, which is a function depending on the state at each time and will be designed.A switching function is a rule which determines a switching sequence for a given switching system.Moreover, γ(x(t)) = i implies that the system realization is chosen as the i th system, i = 1, 2, ..., N. It is seen that the system (1) can be viewed as an autonomous switched system in which the effective subsystem changes when the state x(t) hits predefined boundaries.A i , D i ∈ M n×n , i = 1, 2, ..., N are given constant matrices, and and .., N is the continuous function, and is assumed to satisfy that where ρ 1 > 0 and ρ 2 > 0, are known constant scalars.For simplicity, we denote σ(x(t), x(t − h(t)), t) by σ, respectively.The time-varying delay function h(t) satisfies The mean square stability problem for switched stochastic system (1) is to construct a switching rule that makes the system mean square exponentially stable.
Definition 1.Given α > 0. The switched stochastic system (1) is αexponentially stable in the mean square if there exists a switching rule γ(.) such that every solution x(t, φ) of the system satisfies the following condition We end this section with the following technical well-known propositions, which will be used in the proof of the main results.
It is easy to see that the system {J i } is strictly complete if and only if where We end this section with the following technical well-known propositions, which will be used in the proof of the main results.
Proposition 1. (see [19]) The system {J i }, i = 1, 2, . . ., N, is strictly complete if there exist If N = 2 then the above condition is also necessary for the strict completeness.(see [19]) Let E, H and F be any constant matrices of appropriate dimensions and F T F ≤ I.For any ǫ > 0, we have

Main Results
Let us set The main result of this paper is summarized in the following theorem.
Theorem 1.Given α > 0. The zero solution of the switched stochastic system (1) is α−exponentially stable in the mean square if there exist symmetric positive definite matrices U , and matrices S i , i = 1, 2, such that satisfying the following conditions: The switching rule is chosen as γ(x(t)) = i, whenever x(t) ∈ ᾱi .Moreover, the solution x(t, φ) of the switched stochastic system satisfies Proof.We consider the following Lyapunov-Krasovskii functional for the system (1)

It easy to check that
Taking the derivative of V 1 along the solution of system (1) and taking the mathematical expectation, we obtained

Using Proposition 2 gives
Similarly, we have Therefore, we have where Therefore, we finally obtain from (6) and the condition (ii) that We now apply the condition (i) and Proposition 1, the system J i is strictly complete, and the sets α i and ᾱi by (4) are well defined such that Therefore, for any x(t) ∈ R n , t ∈ R + , there exists i ∈ {1, 2, . . ., N } such that x(t) ∈ ᾱi .By choosing switching rule as γ(x(t)) = i whenever γ(x(t)) ∈ ᾱi , from (6) we have E V (.) + 2αV (.) ≤ E x T (t)J i x(t) < 0, t ∈ R + , and hence E V (t, x t ) ≤ E {−2αV (t, x t )} , ∀t ∈ R + .
Furthermore, taking condition (5) into account, we have By Definition 1, the system (1) is exponentially stable in the mean square.The proof is complete.

Conclusion
In this paper, we have proposed new delay-dependent conditions for exponential stability of stochastic hybrid systems with time-varying delay.Based on the improved Lyapunov-Krasovskii functional and linear matrix inequality technique, a switching rule for exponential stability of stochastic hybrid systems with timevarying delay have been established in terms of LMIs.

Proposition 2 .
(Cauchy Inequality) For any symmetric positive definite marix N ∈ M n×n and a, b ∈ R n we have+a T b ≤ a T N a + b T N −1 b.Proposition 3. (see[19]) For any symmetric positive definite matrix M ∈ M n×n , scalar µ > 0 and vector function ω : [0, µ] → R n such that the integrations concerned are well defined, the following inequality holds µ 0 ω(s) ds T M µ 0 ω(s) ds ≤ µ µ 0 ω T (s)M ω(s) ds .