pth Moment Exponential Stability of Nonlinear Hybrid Stochastic Heat Equations
We are concerned with the exponential stability problem of a class of nonlinear hybrid stochastic heat equations (known as stochastic heat equations with Markovian switching) in an infinite state space. The fixed point theory is utilized to discuss the existence, uniqueness, and th moment exponential stability of the mild solution. Moreover, we also acquire the Lyapunov exponents by combining the fixed point theory and the Gronwall inequality. At last, two examples are provided to verify the effectiveness of our obtained results.
There have been enormous theories for both definite and stochastic partial differential equations (SPDEs) since they can describe various phenomena in reality recently. Throughout the previous literature, besides the existence and uniqueness, the stability for SPDEs is a popular topic. Some results on exponential stability of SPDEs were investigated by Caraballo and Liu  and Luo .
At present, the hybrid SDEs and hybrid SPDEs (also known as SDEs and SPDEs with Markovian switching), for example, hybrid stochastic heat equations, have been paid much attention owing to their wide applications in natural science, engineering, biology, finance, and other areas. In addition, one important reason to consider hybrid SDEs or SPDEs is that real situations may exhibit sudden changes or go into different cases in different periods resulting in parameter transition and probably changes in branch structure so that we need Markov chains to characterize such systems. For example, [3, 4] discussed some properties of the solutions to SDEs with Markovian switching. So far, the th moment exponential stability of stochastic heat equations without Markovian switching has been studied extensively, for example, [5, 6], while hybrid stochastic heat equations remain open due to the difficulty arising from Markov chains. As a consequence, it is more challenging and exciting to explore the stabilization of the systems with Markov chains.
Up to now, there have been many methods to study the stability of SPDEs as well as hybrid stochastic heat equations like Lyapunov’s function method, successive approximation approach, large deviation technique, and so on. For example, Deng et al.  obtained some sufficient conditions on the stability of hybrid stochastic differential equations by using the Lyapunov’s function method and stochastic feedback controls. In , Ubøe and Zhang constructed sequence solutions to approximate the exact solution successively. Gong and Qian studied the large deviation rate function of the Markov chains more completely in . In , Bao et al. discussed the th moment exponential stability of linear hybrid stochastic heat equations by employing the explicit formulae and large deviation technique. However, we cannot use the method of the linear case to obtain the stability property of a class of nonlinear hybrid stochastic heat equations. Generally speaking, it is very difficult to calculate the explicit formulae of solutions to nonlinear hybrid stochastic heat equations. Furthermore, the Markov chain has no unique stationary probability distribution when it is not irreducible in an infinite state space, which makes the analysis of Lyapunov exponents more difficult.
To overcome difficulties raised by what was mentioned above, we utilize a new method—the fixed point theory—to deal with the stability of a class of nonlinear hybrid stochastic heat equations in this paper. It is well known that the fixed point theory first put forward by Burton is usually applied to study the almost sure exponential stability and the th moment exponential stability of both SDEs and SPDEs such as second-order stochastic evolution equations, for example, , nonlinear neutral SDEs, for example, , and stochastic Volterra-Levin equations, for example, [13–15]. Different from the method in , we use the fixed point theory to analyze the stability of nonlinear hybrid stochastic heat equations. Firstly, we prove the existence and uniqueness of the solution, as well as the th moment exponential stability. Then, we obtain the th moment Lyapunov exponents by using the Gronwall inequality instead of seeking the proper rate function which was complicated to compute in . Moreover, we provide two examples and some comparisons to show that our results extend and improve those given in the previous literature.
The rest of this paper is organized as follows. In Section 2, we introduce the notations and the model of nonlinear hybrid stochastic heat equations along with some necessary assumptions. In Section 3, by applying the fixed point theorem, we prove the existence, uniqueness, and th moment exponential stability of hybrid stochastic heat equations. Besides, based on the Gronwall inequality, we further obtain the Lyapunov exponents of the solutions and make some comparisons with the previous results. In Section 4, we provide two simple examples to verify the effectiveness of the obtained results. In the last section, we conclude the paper with some general remarks.
2. Preliminaries, Model, and Assumptions
In this section, we introduce some preliminaries and common notations for a more detailed description, and then give the model that we will deal with.
Throughout this paper, the following notations will be used. Let be a real-valued Brownian motion defined on the complete probability space which has a filtration satisfying the usual conditions. We denote a bounded domain , equipped with boundary . In this paper, is defined as usual, that is, the family of all-real valued square integrable functions with their inner product , and norm , . Also, we can define , , by and by . Then, we can denote the Laplace operator , with domain , which generates a strongly continuous semigroup . Furthermore, let be a right continuous Markov chain which takes values in a listed state space , where is some positive integer or arrives at . Moreover, we assume that the Markov chain is independent of the Brownian motion .
Model Definition. In this paper, we consider the following hybrid stochastic heat equation: where are mappings from and we take , simply in this paper. Letting us fix an interval , then the operators are -measurable while the initial value is a -valued, -measurable random variable, which is independent of and , and, for any , .
Next, we will give the definitions of the mild solution to (1) and th moment exponential stability as well as inequality. For the convenience of writing, we will take and such that is a -valued stochastic process since is a -valued random variable.
Definition 1. A -valued stochastic process is called a mild solution to (1) if the following conditions are satisfied: (i) and for any , is adapted to with (ii)stochastic integral equation holds a.s. for any and .
Definition 2. Equation (1) is called exponentially stable in the th moment if there exists a pair of constants and such that , .
Lemma 3 ( inequality). Let be an arbitrary family of random variables; then one has where , ; , .
Finally, we present some necessary assumptions to close this section.
Assumptions. Consider the following:(A1) for some constants and ;(A2)the operators satisfy the following properties: for any and , there exist positive constants such that (A3), .
Remark 4. It is obvious to see that (1) has a trivial solution under Assumptions when the initial value is equal to zero.
3. th Moment Exponential Stability
In this section, we will use the fixed point principle to discuss the existence and uniqueness of the mild solution to (1) and prove the exponential stability result.
Theorem 5. Let and suppose that Assumptions hold. Then, (1) is exponentially stable in the th moment if the following holds: where , , .
Proof. Let be the Banach space of all -adapted continuous processes consisting of functions such that , , where , . Below we denote the norm in by .
Then, we derive an operator as follows: It is easy to prove that the following holds by inequality yields: Next, we will divide the proof into three steps.
Claim 1. is continuous in the th moment on .
Proof of Claim 1. Let , and let be sufficiently small,
Letting , it is easy to see that
Then, by Burkhölder-Davis-Gundy inequality, the following holds when : Hence, from (10)–(12), we see that is th continuous on .
Claim 2. is contained in .
Proof of Claim 2. It follows from (8) that By Assumptions and the Hölder inequality, we have
From (13)-(14), it is easy to see that , where , which implies .
Claim 3. is contractive for arbitrary with .
Proof of Claim 3. Consider the following: where .
Recalling condition (6) and noting that , we see that is a contraction mapping. By the fixed point theory, we derive that has a unique fixed point in , which is also exponentially stable in the th moment from the proof of three claims. Therefore, the desired assertion in Theorem 5 is completed.
Remark 6. If , then it is obvious that (1) is mean square exponentially stable.
Remark 7. In Theorem 5, we apply the fixed point theory to obtain the existence and uniqueness of the solution for a class of nonlinear hybrid stochastic heat equations. Obviously, our results extend and improve those given in [3, 5, 10].
Remark 8. According to the proof of Theorem 5 and based on the Gronwall inequality, we can easily obtain the Lyapunov exponents of the solution as follows:
In particular, we derive that when in Theorem 5 such that the solution of (1) is exponentially stable in the th moment.
Remark 9. In , Bao et al. discussed the Lyapunov exponents of linear hybrid stochastic heat equations by employing the explicit formula of solution. However, we consider a class of nonlinear hybrid stochastic heat equations in this paper. Obviously, the method in  fails in our result since the explicit formula of solution cannot be obtained in the nonlinear case. So we apply succesfully the fixed point principle to study the th moment exponential stability of a class of nonlinear hybrid stochastic heat equations, which obviously generalizes the linear case.
4. Two Examples
In this section, we provide two examples of heat equations with Markovian switching as the applications of our main results and give some remarks compared to the previous articles.
Example 1. Consider the following stochastic heat equation with Markovian switching:
where is a right continuous Markov chain which takes values in an infinite state space . Take , , . Below we recall that is an infinitesimal generator with a strongly continuous semigroup so that the eigenfunctions of are , , and the relevant eigenvalues of are . Observing that , , and , , we have , .
Since we can choose and similarly . It is easy to calculate that with , . Therefore, by Theorem 5 we see that (17) is th moment exponentially stable.
Remark 10. The state space in (17) is infinite in which Markov chain takes values while generally we assume the state space to be finite to gain the th moment exponential stability of the solution, so our results generalize those in the previous literature (e.g., see [4, 10]).
Remark 11. If , then (17) is mean square exponentially stable.
Example 2. We compare the following linear stochastic heat equation with that in : where takes values in with the generator : which implies that the Markov chain has a unique invariant measure . Take , , , . Then, it is easy to compute . So we can conclude that the system of (19) is th moment exponentially stable if .
Remark 12. Example 2 is also considered by Bao et al.  based on the large deviation technique, in which a proper should be found to calculate the rate function to testify whether the solution of (19) is th moment exponentially stable or not. However, it is very difficult and complicated in comparison with our conditions, especially in the nonlinear hybrid stochastic heat equations. So our results generalize and imporove those given in .
In this paper, we have studied the stability problem of a class of nonlinear hybrid stochastic heat equations. Based on the fixed point theory and Burkhölder-Davis-Gundy inequality, we not only establish the existence and uniqueness of the equation in an infinite state space, but also prove the th moment exponential stability of the system. Moreover, we give two simple examples to verify all our conditions at the end of this paper. In the future work, we will focus on the stability of more complicity models such as neutral stochastic differential equations with Markov chains and variable time delay.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was jointly supported by the National Natural Science Foundation of China (61374080), the Natural Science Foundation of Zhejiang Province (LY12F03010), the Natural Science Foundation of Ningbo (2012A610032), and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
J. W. Luo, “Exponentially stable stationary solutions for delay stochastic evolution equations,” Progress in Probability, vol. 65, pp. 169–178, 2011.View at: Google Scholar
T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover, Mineola, NY, USA, 2006.View at: MathSciNet