Stability of reaction – diffusion systems with stochastic switching ∗

Lijun Pan, Jinde Cao, Ahmed Alsaedi School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang Guangdong, 524048, China plj1977@126.com School of Mathematics, Southeast University, Nanjing 210096, China jdcao@seu.edu.cn Nonlinear Analysis and Applied Mathematics(NAAM) Research Group, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia aalsaedi@hotmail.com


Introduction
Random disturbance exists in the natural world owing to various environmental noise.Such phenomena can be described by stochastic differential systems, which have been successfully applied to problems in mechanics, engineering, electronics, automation, economics, etc.Therefore, the dynamics of stochastic systems have become a hot topic in recent years [9, 11, 20-23, 25, 29].However, a simple color noise is said to be telegraph L. Pan et al. noise, which is demonstrated as a switching between two or more environmental regimes.If the switching is no memory and the switched times follow an exponential distribution, the switched regime can be modelled by a finite state Markov chain.If the switched times are independent random variables and the time intervals have the same expectations, the switched regime can be modelled by independent and identically distributed switching.Hajnal [8] investigated the behavior of finite nonhomogenous Markov chain having regular transition matrices in 1956.Salehi and Jadbabaie [27] provided consensus algorithms under ergodic stationary graph process.In [12], Kim extended Markov switching model to a more general state space model and proposed a new algorithm.In [3][4][5]10,24], the authors studied the stability of a linear systems with Markov switching or independent and identically distributed process.Gray et al. [7] examined the effects of telegraph noise on the classic SIS model and proved the extinction and persistence for a finite state Markov chain.In [17], the authors investigated the synchronization of complex networks with stochastic switching.In practical application, influence of diffusion is inevitable.Therefore, we must consider the state variables varying with time and space variables.With respect to reaction-diffusion systems, there are many reports of the stability in the literature [6, 14-16, 18, 19, 26, 28, 30].For instance, Luo and Zhang [26] investigated the asymptotical stability in probability and almost sure exponential stability of stochastic reaction-diffusion systems by using the Lyapunov method.In [30], Zhu et al. proved the stability for stochastic bidirectional associative memory neural networks with reaction-diffusion term.In [15], Li et al. investigate the synchronization problem for delayed reaction-diffusion neural networks (RDNNs) with unknown timevarying coupling strengths by an adaptive learning control strategy.However, to the best of our knowledge, none of the authors have considered the stability of reaction-diffusion systems with stochastic switching, which motivates our current research.
The aim of this paper is to study stability for reaction-diffusion systems with stochastic switching of finite state space.Two types of switched models are considered: (i) Markov switching and (ii) independent and identically distributed switching.
If the switched sequence is Markov chain, by means of ergodic property for Markov chain [1] and the Lyapunov method, sufficient condition is obtained to confirm that the zero solution of switched system can achieve almost sure stability.If the switched sequence is an independent and identically distributed process, we also derive the stability by the theorem in Durrent [2].It is interesting that if some subsystems are not stable, but the other subsystems are stable, eventually the overall system will reach stability, which means that Markov switching, as well as independent and identically distributed switching, play an essential role in the stable behavior of reaction-diffusion systems.In addition, an example with simulations is provided to demonstrate the applicability of our results.The rest of this paper is organized as follows.Reaction-diffusion system model with stochastic switching is presented in Section 2 together with some definitions of stability for the zero solution.In Section 3, almost surely exponential stability and exponential stability in the mean square of switching systems are derived.A numerical example is given to demonstrate our results in Section 4.

System description and definitions
Let (Ω, F, {F t }, P) be a probability space related to an increasing right-continuous filtration {F t } t 0 .E[•] denotes mathematical expectation.We present two types of stochastic switchings in probability space (Ω, F, {F t }, P).

Markov switching process
Let r(t) be a right-continuous Markov chain on (Ω, F, {F t }, P) taking values in the state space S = {1, 2, . . ., M } with generator Γ = (δ ij ) M ×M generated by P{r(t + ) = j | r(t) = i} = δ ij + o( ), where > 0, δ ij is the transition rate from i to j satisfying δ ij > 0, i = j, and for all t 0.Moreover, the Markov chain has a unique stationary distribution Π = (π 1 , π 2 , . . ., π M ) T satisfying ΠΓ = 0 and Let X be a compact set with smooth boundary ∂X and measure µ(X) > 0 in R m ; L 2 (R × X) denotes the space of real Lebesgue measurable functions of R × X.It is a Banach space for the 2-norm u(t) 2 = ( X u(t, y) 2 dy) 1/2 , where • is Euclid norm.
Consider the following reaction-diffusion with Markovian switching: where

Independent and identically distributed switching process
Let r(t) be independent and identically distributed sequence on (Ω, F, {F t }, P) taking value in S = {1, 2, . . ., M }.Let {τ k } k 0 be a sequence of finite-valued F t stopping time where ρ l 0 and M l=1 ρ l = 1.Then the reaction diffusion system with independent and identically distributed switching can be written as Throughout this paper, we suppose that there exists a constant L > 0 such that Only a limited number of switching occurs for each finite time interval, which precludes the possibility of infinitely fast switching.In view of the Lipschitz condition of f , we see that there exists a unique solution for (1) or (2).Also we suppose that f (t, y, 0, r(t)) = 0 for any t 0, y ∈ X, which implies that u(t, y) ≡ 0 is a trivial solution.
Definition 1.The zero solution of system (1) or ( 2) is said to be almost surely exponentially stable if there exists λ > 0 such that for any initial value u 0 and t 0, lim sup Definition 2. The zero solution of system (1) or ( 2) is said to be exponentially stable in the mean square if there exist λ > 0, d > 0 such that for any initial value u 0 and t 0, 3 Almost surely exponential stability and exponential stability in the mean square In this section, some criteria on almost surely exponential stability are established.The following lemma is important for almost surely exponential stability and exponential stability in the mean square of switched systems (1) or (2).
Remark 1.The existence and uniqueness of the solution can guarantee that the zero solution of system (1) or ( 2) is almost surely unique equilibrium point, which are the precondition for studying the behavior of solution for the system with stochastic switching.
https://www.mii.vu.lt/NAProof.For any u 0 = 0, it follows from Lemma 1 that X u(t, y) dy = 0 a.s.for t 0. Thus, By (ii), we see that ∂ 2 V (t, u)/∂u i ∂u j = 0, i = j, i, j ∈ {1, 2, . . ., n}.From Neumann value condition and integration by parts, together with (iii), we get By (iv), we have It follows from the ergodic property of Markov chain [6] that Thus, by ( 5), (i) and (v), we have Remark 2. Condition (i) is a general condition for the stability of a reaction-diffusion system.Conditions (ii)-(iv) are required due to stochastic switching of the system.Further, in view of the ergodicity of the Markov process, we can choose condition (v).
Proof.For t 0, we get V t, u(t, y) dy.
Remark 3. The reaction-diffusion system (1) or (2) can be considered as the following subsystems: switching from one to the other by the law of Markov chain or independent and identically distributed process.If some of η i or ξ i are not negative, i.e., the zero solution for some of subsystem achieve stability, but the zero solution for the other subsystems do not achieve stability.However, if the rate of stochastic switching from unstable state to stable state is faster than that from stable state to unstable state, so that M i=1 π i η i < 0 or M i=1 ρ i ξ i < 0, then the overall system will achieve stability, which means that stochastic switching play an essential role in the stability of reaction-diffusion system.Remark 4. Theorems 1-3 establish a general framework for analyzing the stable behaviors of reaction diffusion systems with stochastically switching.Sufficient conditions are derived under which system (1) or (2) can achieve stability in the two kinds of stochastic switchings sense.

Conclusions
In this paper, stability analysis of reaction diffusion systems with stochastically switched parameter has been studied.The switched model includes two kinds of stochastic switchings.Switched process takes values in finite state space.By method of stochastic analysis and Lyapunov function, some new stability criteria have been derived.Finally, a standard numerical package illustrate that the new results are practical.Our future work will focus on the stability of delayed reaction diffusion systems with stochastic switching.

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Figure 2 (
a) shows the trajectory of the state u 1 (t, y) of system(10).