Stability and Boundedness of Stochastic Volterra Integrodifferential Equations with Infinite Delay
We make the first attempt to discuss stability and boundedness of solutions to stochastic Volterra integrodifferential equations with infinite delay (IDSVIDEs). By the Lyapunov-Krasovskii functional approach, we get kinds of sufficient criteria for stability and boundedness of solutions to IDSVIDEs. The main innovation here is that stochastic systems with infinite delay can retain stability and boundedness of corresponding deterministic systems under some conditions.
Recently, stochastic functional differential equations with infinite delay (IDSFDEs) have attracted broad attention of many researchers. In the literature, there are two main lines of research on the IDSFDEs. On one hand, existence and uniqueness of solution are basic properties for equations. So a great number of authors have devoted themselves to this research, and thus many excellent results on the existence and uniqueness of the solutions to IDSFDEs and neutral IDSFDEs can be found in [1–6] and references cited therein. On the other hand, the study of stability and boundedness of solutions is one of the most attracting topics in the qualitative theory of differential equations because of its various applications in many areas such as physics and control theory [7, 8]. Hence, more and more researchers study them and especially focus on the stability of solutions. An important issue in stochastic analysis is whether or not random disturbance can change the qualitative properties of system, which is particularly important in control field. In most cases, people are interested in the performance of antidisturbance of system. So it is vital to seek some antidisturbance systems or present the intensity of stochastic perturbation that stable system can tolerate without losing the property of stability . In recent years, many meaningful works on this topic have come out; see, for example, [10–21].
Volterra integrodifferential equations (VIDEs) are widely applied in biology, ecology, medicine, physics, among other scientific areas and thus have been encountered by many researchers in numerical and theoretic analysis; see [7, 22–25]. It is well known that concrete systems are inevitably affected by external perturbations usually modeled by stochastic noise. So a great deal of attention has been paid to the research of stochastic VIDEs [9, 26, 27]. Additionally, time delay is always ubiquitous and infinite delay systems have wide applications in many fields. Hence, there is naturally an important kind of IDSFDEs, that is, stochastic Volterra integrodifferential equations with infinite delay (IDSVIDEs). In practice, many applications of IDSVIDEs are greatly dependent on the stability and boundedness of their solutions. However, to the best of the authors' knowledge, few research results mentioned above focus on the stability and boundedness of IDSVIDEs, which motivates the present study. Precisely, this paper investigates in detail the problem of stability and boundedness of solutions for the following IDSVIDE: where , and are continuous functions. and are also continuous. Therein, , and denote the intensity of disturbance to , and , respectively.
Compared with the existing results in the literature, contributions of this paper are mainly as follows.(1)Both stochastic perturbation and infinite delay are considered in the IDSVIDEs.(2)A new Lyapunov function is constructed to derive stability and boundedness criteria for IDSVIDEs efficiently. (3)The problem of how much the stochastic noise VIDEs with infinite delay can tolerate without losing the properties of stability and boundedness has been solved.
Let be a complete probability space with a filtration satisfying the usual conditions. As usual, denotes a scalar Brownian motion defined on the space and is the mathematical expectation with respect to . Write as the family of bounded continuous real-valued functions defined on with the norm .
In this paper, we suppose there exists a constant such that . Then by Theorem of , there exists a unique global solution to (1) if and are bounded. For more details, readers can see [29, 30]. Here and in the rest of the paper, write the solutions with the initial condition as . Throughout this paper, unless otherwise specified, we use the following Lyapunov function: For any , define two stopping times:
In order to study the stability and boundedness of solutions to (1), we state the following assumptions:(K1) There exist nonnegative continuous functions and , such that (K2) For any , there exists such that where .(K3) For any , there exist and such that where is defined in (K2).
Remark 1. Conditions (K2) and (K3) are the conditions about the intensity of perturbations. Note that constant in equality (2) is the same as the one which appeared in conditions (K2) and (K3).
3. Stability of Solutions of IDSVIDEs
In order to consider the stability of (1), without loss of generality, we suppose that and hold in this section. Hence, (1) has the trivial solution . First, we introduce three kinds of definitions about stability of solutions to (1). It is easy to see that these definitions are a strict generalization of deterministic cases.
Definition 2 (see ). The trivial solution of (1) is said to be stochastically stable if for every pair and , there exists a such that
whenever . Additionally, it is said to be stochastically uniformly stable if is independent of .
The trivial solution of (1) is said to be stochastically asymptotically stable if it is stochastically stable and, moreover, for any , there is such that whenever .
The trivial solution of (1) is said to be stochastically globally asymptotically stable if it is stochastically stable and, moreover, for all , it follows that
In the following, we will apply the Lyapunov-Krasovskii functional approach to delve into some sufficient criteria, under which the trivial solution to (1) is stochastically stable, stochastically asymptotically stable, and stochastically globally asymptotically stable, respectively. The Itô formula used in this paper can be seen in .
Theorem 3. Suppose that (K1) and (K2) hold. Then the trivial solution of (1) is stochastically uniformly stable.
Proof. For any and , choose sufficiently small for . For any given satisfying , we can get from (2) that Then it follows that, for any , Then by a straightforward computation we obtain that From the definition of , it follows that So Let . We deduce that that is, For any , we can have which implies that This completes the proof.
Remark 4. If in (1), then we can get the corresponding disturbance free system. Theorem 3 really tells us that stochastic perturbation cannot disturb the stability of original deterministic system if the noisy intensity satisfies (K2).
Lemma 5. Assume that (K1) and (K2) hold. Then for any , there is such that for any , a.s.
Proof. For any given , and satisfying , choose sufficiently large such that Denote . By using a similar argument as in Theorem 3, we have that, for any , Making use of the definition of yields Combining this and (21), we have Let . Then which can imply our desired result
Theorem 6. Suppose that (K1) and (K3) hold. Then the trivial solution is stochastically asymptotically stable and stochastically globally asymptotically stable.
Proof. (I) The proof of stochastic asymptotic stability.
For any , is -adapted -valued random variable such that . Let denote the solution with the initial value . From Theorem 3, the trivial solution is stochastically uniformly stable. Hence, for any given , there exists , such that whenever . Fix such that . For any given , define So Lemma 5 shows that, for above and any , there is such that If there exists such that then the trivial solution of system (1) is stochastically asymptotically stable. If not, there are a sequence and increasing sequence such that and Define It is not difficult to see that Hence, there are and positive sequence such that If for any , the following holds: which can show the trivial solution of system (1) is stochastically asymptotically stable. If not, there is such that Obviously, , , and for any , For any given such that . Choose sufficiently large such that, for any , For any given , define So (28) can tell us that From here, we let , and let be the same as (2).
By The Itô formula and (K3), for any , where is the same as (10). So, Letting can yield Clearly, it follows from (39) that . Therefore, Hence, Choose sufficiently large such that Then Define two stopping times: So for any , Note that if , then Consequently, Noting , it yields that In view of the definition of and condition (K1), which, in conjunction with (51) and (52), yields that Letting , it yields that Hence, This implies immediately that At last, from the arbitrariness of , it must be Since is arbitrary, we then obtain that Hence, from (40), there is such that But this is in contradiction with (36), and thus (34) must hold and this completes the proof.
(II) The proof of stochastic global asymptotic stability.
Give any , . Let be sufficiently large such that Then we can easily have From the assumption of , So Let . we could get namely, Arguing as part (I), we obtain that From the arbitrariness of , the trivial solution of (1) is stochastically globally asymptotically stable, which ends the proof.
In the last part of this section, we give two examples for better understanding the stability theorems above.
Example 7. Consider the following IDSVIDE: where is a constant.
Obviously, , so . Let , . Then we have and For any , let . Easily to get Therefore, Hence all the conditions of Theorem 3 have been verified, and it follows that the trivial solution of (1) is stochastically uniformly stable.
Example 8. Consider an IDSVIDE as follows: where are constants.
Obviously, , so we can let . Let , . Then we have and Additionally, for any , let and . Then it is not difficult to check that Consider function . Clearly, is differentiable and So , that is, . Hence, Hence, all of the conditions of Theorem 6 are satisfied and we can assert that the trivial solution of (1) is stochastically asymptotically stable and stochastically globally asymptotically stable.
4. Boundedness of Solutions of IDSVIDEs
In the section, we begin with three types of definitions about boundedness for solutions of (1).
Definition 9. A solution of (1) is stochastically bounded, if, for any , , there exists , such that
The solutions of (1) are stochastically equibounded, if, for any , , and , there exists , such that
Fix ; the solutions of (1) are stochastically ultimately bounded for bound , if, for any , , there exists a , such that
Now we apply the Lyapunov-Krasovskii method to give the sufficient conditions, under which the solutions of (1) are stochastically bounded, stochastically equibounded, and stochastically ultimately bounded, respectively.
Theorem 10. Suppose that (K1) and (K2) hold. If , then the solutions to (1) are stochastically bounded and stochastically equibounded.
Proof. (I) Proof of stochastic boundedness.
Let , be arbitrary. Choose sufficiently large such that . By the Itô formula, for any , we can obtain that Taking expectation on both sides of (80), it follows by using (10) that From the assumption , we derive that We therefore must have Letting , we can obtain that That is, The proof of stochastic boundedness is complete.
(II) Proof of stochastic equiboundedness.
Let and be arbitrary. Give any , such that . Choose such that . By applying the Itô formula, for any , Taking the expectation for both sides of (86), we have In view of Consequently, Letting , it yields that that is, This completes the proof.
Remark 11. Notice that, when conditions (K1) and (K2) hold, the solutions of stochastic system (1) are stable and bounded. That is, environmental noise (in the sense of Itô) cannot disturb stability and boundedness of solutions for some systems if noisy intensity satisfies (K2). Hence, we can construct some anti-interference systems in practice.
Theorem 12. Suppose that (K1) and (K3) hold. If , then the solutions to (1) are stochastically ultimately bounded.
Proof. Give . For any , . From Theorem 10, there exists , such that Choose , and define four stopping times as follows: From here, we can show in the same way as in the proof of Theorem 6 that we could choose sufficiently small , such that and Then it follows that After letting , we have Let . Then the above inequality is equivalent to The proof is complete.
Remark 13. Obviously, we can verify that the solutions of IDSVIDE (68) are stochastically bounded and stochastically equibounded. And the solutions of IDSVIDE (72) are stochastically ultimately bounded.
Throughout this paper, by combining the Lyapunov-Krasovskii method, we have obtained various kinds of sufficient stability and boundedness criteria, where the ranges of noisy intensity that stable and bounded systems can tolerate without losing the properties of stability and boundedness are presented, respectively. These sufficient conditions are very necessary for us to verify stability and boundedness of stochastic Volterra integrodifferential equations with infinite delays. Moreover, the conditions obtained in this paper can also help us to construct some antidisturbance systems in the applications. In addition, two examples have been given to illustrate our theoretical results.
The authors really appreciate the reviewers' valuable comments and the reviewers’ helpful suggestions to improve the paper. This work was supported by the NNSF of China (nos. 11171081, 11171056, and 11271065), the NNSF of Shandong Province (no. ZR2010AQ021), the Key Project of Science and Technology of Weihai (no. 2011dxgj06), and the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (no. HIT.NSRIF.2011104).
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