Almost Sure Stability and Stabilization for Hybrid Stochastic Systems with Time-Varying Delays

The problems of almost sure a.s. stability and a.s. stabilization are investigated for hybrid stochastic systems HSSs with time-varying delays. The different time-varying delays in the drift part and in the diffusion part are considered. Based on nonnegative semimartingale convergence theorem, Hölder’s inequality, Doob’s martingale inequality, and Chebyshev’s inequality, some sufficient conditions are proposed to guarantee that the underlying nonlinear hybrid stochastic delay systems HSDSs are almost surely a.s. stable. With these conditions, a.s. stabilization problem for a class of nonlinear HSDSs is addressed through designing linear state feedback controllers, which are obtained in terms of the solutions to a set of linearmatrix inequalities LMIs . Two numerical simulation examples are given to show the usefulness of the results derived.


Introduction
In the past decades, the problems of stability analysis and stabilization synthesis of stochastic systems have received significant attentions, and many results have been reported; see, for example 1-7 and the references therein.Commonly, the above problems can be solved not only in moment sense 8-10 but also in a.s.sense 11, 12 .However, in recent years, much interest has been focused on a.s.stability problems for stochastic systems; see, for example 8, 13 and the references therein.
It is well known that a lot of dynamical systems have variable structures subject to abrupt changes in their parameters, which are usually caused by abrupt phenomena such as component failures or repairs, changing subsystem interconnections, and abrupt environmental disturbances.The HSSs, which are regarded as the stochastic systems with Notation 1.The notation used here is fairly standard unless otherwise specified.R n and R n×m denote, respectively, the n dimensional Euclidean space and the set of all n × m real matrices, and let R 0, ∞ .Ω, F, {F t } t≥0 , P be a complete probability space with a natural filtration {F t } t≥0 satisfying the usual conditions i.e., it is right continuous, and F 0 contains all P-null sets .If x, y are real numbers, then x ∨ y stands for the maximum of x and y, and x ∧ y the minimum of x and y.M T represents the transpose of the matrix M. λ max M and λ min M denote the largest and smallest eigenvalue of M, respectively.| • | denotes the Euclidean norm in R n .E{•} stands for the mathematical expectation.P{•} means the probability.C −τ, 0 ; R n denotes the family of all continuous R n -valued function ϕ on −τ, 0 with the norm |ϕ| sup{|ϕ θ | : −τ ≤ θ ≤ 0}.C b F 0 −τ, 0 ; R n being the family of all F 0 -measurable bounded C −τ, 0 ; R n -value random variables ξ {ξ θ : −τ ≤ θ ≤ 0}.L 1 R ; R denotes the family of functions λ : R → R such that ∞ 0 λ t dt < ∞.

Problem Formulation
In this paper, let r t , t ≥ 0 be a right-continuous Markov chain on the probability space taking values in a finite state space S {1, 2, . . ., N} with generator Γ γ ij N×N given by where Δ > 0 and γ ij ≥ 0 is the transition rate from mode i to mode j if i / j while γ ii − j / i γ ij .Assume that the Markov chain r • is independent of the Brownian motion B • .It is known that almost all sample paths of r • are right-continuous step functions with a finite number of simple jumps in any finite subinterval of R : 0, ∞ .Let us consider a class of stochastic systems with time-varying delays: R n and r 0 r 0 ∈ S, where τ max{τ 1 , τ 2 }, τ 1 and τ 2 are positive constant and τ 1 t and τ 2 t are nonnegative differential functions which denote the time-varying delays and satisfy

2.3
The nonlinear functions f :

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But it is easy to see that the results in this paper can be applied to the systems 2.5 by the similar assumption in 2.4 .Let C 2,1 R n × R × S; R denote the family of all nonnegative functions V x, t, i on R n × R × S that are twice continuously differentiable in x and once in t.

2.6
Remark 2.2.LV is thought as a single notation and is defined on Definition 2.3.The system 2.2 is said to be a.s.stable if for all ξ ∈ C b F 0 −τ, 0 ; R n and r 0 ∈ S P lim t → ∞ x t; ξ, r 0 0 1. 2.7

Main Results
Theorem 3.1.Assume that there exist nonnegative functions where k 1 , k 2 and k 3 are positive numbers satisfying To prove this theorem, let us present the following lemmas.
That is, all of the three processes X t , A 2 t , and M t converge to finite random variables with probability one.

has a unique global solution.
Proof.Fix any initial data ξ, r 0 , and let β be the bound for ξ.For each integer k ≥ β, define where we set |x| ∧ k/|x| x 0 when x 0. Define g k x, z, t, i similarly.By 2.4 , we can observe that f k and g k satisfy the global Lipschitz condition and the linear growth condition.By the known existence-and-uniqueness theorem, there exists a unique global solution x k t on t ∈ −τ, ∞ to the equation with initial data {x k θ : −τ ≤ θ ≤ 0} ξ and r 0 r 0 .Define the stopping time 3.9 where we set inf ∅ ∞ as usual.It is easy to show that It is clear that x t is a unique solution of 2.2 for t ∈ −τ, σ .To complete the proof, we only need to show P{σ ∞} 1.By Lemma 3.2, we have that for any t > 0, where operator L k V is defined similarly as LV was defined by 2.6 .By the definitions of

3.11
By the conditions of 3.1 and 3.2 , we derive that

3.12
On the other hand,

3.13
This yields Letting k → ∞ and using 3.3 , we obtain P σ ≤ t 0. Since t is arbitrary, we must have P σ ∞ 1.The proof is therefore complete.
Let us now begin to prove our main result.
x for all x ∈ R n .Inequality 3.2 implies ω x > 0 whenever x / 0. Fix any initial value ξ and any initial state r 0 , and for simplicity write x t; ξ, r 0 x t .By Lemma 3.2 and condition 3.1 , we have V x s , s, r 0 l r s , α − V x s , s, r s μ ds, dα where inf ∅ ∞ as usual.Clearly, ρ k → ∞ a.s. as k → ∞.Moreover, for any given ε > 0, there is It is straightforward to see from 3.16 that lim t → ∞ inf ω x t 0 a.s.; then we claim that lim t → ∞ ω x t 0 a.s..

3.20
The rest of the proof is carried out by contradiction.That is, assuming that 3.20 is false, we have Furthermore, there exist ε 0 > 0 and ε > ε 1 > 0 such that where Z is a set of natural numbers and {σ j } j≥1 are a sequence of stopping times defined by

3.23
By the local Lipschitz condition 2.4 , for any given k > 0, there exists L k > 0 such that f x, y, t, i ∨ g x, z, t, i ≤ L k , 3.24 for all |x| ∨ |y| ∨ |z| ≤ k, t ≥ 0 and i ∈ S.

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Meanwhile, we can also choose T T ε, ε 1 , k sufficiently small for And then, 3.27 and 3.28 yield where In the following, we can obtain from 3.16 and 3.29 that Tε 0 ε 1 ∞.

3.30
This is a contradiction.So there is an

3.31
Finally, any fixed ω ∈ Ω, {x t, ω } t≥0 is bounded in R n .By Bolzano-Weierstrass theorem, there is an increasing sequence{t i } i≥1 such that {x t, ω } i≥1 converges to some z ∈ R n with |z| < ∞.Since ω x > 0 whenever x / 0, we must have ω x 0 if and only if x 0. This implies that the solution of 2.2 is a.s.stable, and the proof is therefore completed.
Remark 3.5.The techniques proposed in Theorem 3.1 can be used to deal with the a.s.stability problem for other HSDSs, such as the ones in 25 .In a very special case when τ 1 t τ 2 t τ for all t ≥ 0 and i ∈ S, it is easy to see that τ1 t τ2 t 0, and Theorem 3.1 is exactly Theorem 2.1 in 25 .Similarly, Theorem 2.2 in 25 can be generalized to system 2.2 as a LaSalle-type theorem see 24, 26 for HSSs with multiple time-varying delays.

Almost Sure Stabilization of Nonlinear HSDSs
Consider the following nonlinear HSDSs: where, for each r t j ∈ S, A r t , A d r t are known constant matrices with appropriate dimensions, and F i r t ∈ R n×n , G i r t ∈ R n×n i 1, 2 are positive definite matrices.
In the sequel, we denote the matrix associated with the ith mode by where the matrix Γ could be As the given HSDSs 4.1 is nonlinear, we here consider the resulting systems can be stabilized only by linear state feedback controller which is of the form u t K r t x t , 4.5 where K r t are controller parameters to be designed.Under control law 4.5 , the closed-loop system can be given as follow:

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The stabilization problem is therefore to design matrices K r t for the closed-loop system 4.6 to be a.s.stable.In order to guarantee the solvability of K r t , the following theorem is given.Theorem 4.1.If there exist sequences of scalars ε 1i > 0, ε 2i > 0, δ i > 0, positive definite matrices X i > 0 and matrices Y i such that the following LMIs hold, where 4.9 then the controlled system 4.6 is a.s.stable and the state feedback controller determined by where

4.13
By assumption 1, it is easy to see that we can choose Q 1 and Q 2 such that ω 2i x ≥ 0, ω 3i x ≥ 0 for all x ∈ R n , i ∈ S.
Noting that P i X −1 i and Y i K i X i , we can pre-and postmultiply 4.7 by diag P i , . . ., P i , and using Schur complements, we can obtain where

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Clearly Moreover, by 4.24 we further obtain The required assertion now follows from Theorem 3.1. If hold, where where

4.25
It is easy to see that we can choose Q 1 and Q 2 such that ω 2i x ≥ 0, ω 3i x ≥ 0 for all x ∈ R n , i ∈ S.
Noting that P i X −1 i and Y i K i X i , we can pre-and postmultiply 4.7 by diag P i , . . ., P i , and using Schur complements, we can obtain where

4.27
This implies Let ω 1 x min i∈S ω 1i x , ω 2 x max i∈S ω 2i x , and ω 3 x max i∈S ω 3i x .Clearly Moreover, by 4.24 we further obtain The required assertion now follows from Theorem 3.1.

Examples
In this section we will provide two examples to illustrate our results.In the following examples we assume that B t is a scalar Brownian motion, γ t is a right-continuous Markov chain independent of B t and taking values in S {1, 2}, and the step size Δ 0.0001.By using the YALMIP toolbox, simulations results are shown in Figures 1-3. Figure 1   where  To examine the stability of system 5.2 , we consider a Lyapunov function candidate V : R × S → R as V x, i x 2 for i 1, 2. Then we have LV x, z, t, 1 ≤ −10x 6/5 4z 6/5 , LV x, z, t, 2 ≤ 3x 3 √ 1 t − 6x 6/5 25 8 z 6/5 .

Conclusions
In this paper, we have investigated the a.s.stability analysis and stabilization synthesis problems for nonlinear HSDSs.Some sufficient conditions are given to guarantee the resulting systems to be a.s.stable.Under these conditions, a.s.stabilization problem for a class of nonlinear HSDSs is solved in terms of the solutions to a set of LMIs.Finally, the results of this paper have been demonstrated by two numerical simulation examples.

3 . 25 where
I A is the indicator of set A. Since ω x is continuous in R n , it must be uniformly continuous in the closed ball S k {x ∈ R n : |x| ≤ k}.For any given b > 0, we can choose c b > 0 such that |ω x − ω y | < b whenever x, y ∈ S k and |x − y| < c b .Furthermore, let us choose

Figure 2 :
Figure 2: The state evolution of Example 5.1.

Figure 3 :
Figure 3: The state evolution of Example 5.2.
are known constant matrices with appropriate dimensions and B t represents a scalar Brownian motion Wiener process on Ω, F, {F t } t≥0 , P that is independent of Markov chain r t and satisfies: f and g are both functions from R n × R n × R × S to R n which satisfy local Lipschitz condition and the following assumptions: the systems 4.6 reduces to linear HSDSs of the form If there exist sequences of scalars ε 1i > 0, ε 2i > 0, positive definite matrices X i > 0 and matrices Y i such that the following LMIs gives a portion of state γ t of Example 5.1 for clear display.Figure2simulates the numerical results for Example 5.1.The simulation results have illustrated our theoretical analysis.Following from Theorem 4.1, the simulation results for Example 5.2 can be founded in Figure3, which verify our desired results.The state γ t of Example 5.1.