Stability of Invariant Sets of Itô Stochastic Differential Equations with Markovian Switching

Consider the nonlinear Itô stochastic differential equations with Markovian switching, some sufficient conditions for the invariance, stochastic stability, stochastic asymptotic stability, and instability of invariant sets of the equations are derived. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Introduction
Invariant sets of dynamic systems play an important role in many situations when the dynamic behavior is constrained in some way.Knowing that a set in the state space of a system is invariant means that we have bounds on the behavior.We can verify that pre-specified bounds which originate from, for example, safety restrictions, physical constraints, or state-feedback magnitude bounds are not invalidated.
Recently, much work has been done on stochastic differential equations with Markovian switching [1, 3, 5-14, 19, 20].In particular, we here highlight Mao's significant contribution [6,11,12].However, to the best of the author's knowledge to date, the problem of the invariant sets of equations of this kind, has not been investigated yet.
The aim of the present paper is to study the invariant sets of nonlinear It ô stochastic differential equations with Markovian switching.Similar to the result of [18], which investigates the usual stochastic differential equations, some sufficient conditions for the invariance and stochastic stability of invariant sets of equations of this kind are derived.At the same time, we establish some conditions for stochastic asymptotic stability and instability of the invariant sets, which are not discussed in [18] even in the case of equations without Markovian switching.

Stochastic differential equations with Markovian switching
Let {Ω, Ᏺ,{Ᏺ t } t≥0 ,P} be a complete probability space with a filtration satisfying the usual conditions, that is, the filtration is continuous on the right and Ᏺ 0 contains all P-zero sets.Let w(t) = (w 1 (t),w 2 (t),...,w m (t)) T be an m-dimensional Brownian motion defined on the probability space.
We assume that the Markov chain r(•) is independent of the Brownian motion w(•).It is known that almost every sample path of r(t) is a right-continuous step function with a finite number of simple jumps in any finite subinterval of R + , and r(t) is ergodic.Consider the It ô stochastic differential equations with Markovian switching: where In this paper we always assume that both f and g satisfy the local Lipschitz condition and the linear growth condition.Hence it is known from [6] that (2.2) has a unique continuous bounded solution x(t) = x(t,t 0 ,x 0 ) on t ≥ t 0 .
Denote by C 2,1 (R + × R n × S;R + ) the family of all nonnegative functions V (t,x,i) on R + × R n × S which are continuously twice differentiable with respect to x and once differentiable with respect to t.For any (t,x,i) ∈ R + × R n × S, we define an operator ᏸ by where (2.4) The generalized Itô formula reads as follows: if ᏸV s,x(s),r(s) ds. (2.5) Jiaowan Luo 3

Denote by
where ρ(x,A) denotes the distance between a point x and a set A. A set Q is called unstable if there exist ε 1 > 0 and ε 2 > 0 such that, for any δ > 0, there exist x 0 and t * such that for ρ(x 0 ,Q r0 t0 ) < δ the following holds: Denote by Γ the set of its zeros of the function then the set Γ is a positive invariant set for (2.2).In addition, if for any δ > 0, then the set Γ is stochastically stable.
Remark 3.7.It is obvious that the condition (3.7) is automatically satisfied if the function V is independent on t.
Remark 3.8.If both of the coefficients of (2.2), f and g, are independent of the Markov chain r(t), then Theorem 3.6 in this paper reduces to [18,Theorem 1].

Definition 3 . 4 .
A set Q is called stochastically asymptotically stable if it is stochastically stable