Abstract
We investigate LaSalle-type theorems for general nonlinear stochastic functional differential equations. With some preliminaries on lemmas and the derivation techniques, we establish three LaSalle-type theorems for the general nonlinear stochastic functional differential equations via multiple Lyapunov functions. For the typical special case with estimations involving for the derivatives of the Lyapunov functions, a theorem is established as the corollary of the main theorem. At the end of the paper, an example is given to illustrate the usage of the method proposed and show the advantage of the results obtained.
1. Introduction
As it is well known, the Lyapunov function method is the most widely used tool to establish criteria for stability or other asymptotic properties of dynamic systems governed by differential equations or difference equations. With this method, the derivatives of the Lyapunov functions or their upper bounds are often desired to be negative definite. For some complex equations, for example, the functional differential equations, this point may be somehow difficult for us at some times. Thus a spontaneous question will arise, that is, may we weaken the negative definiteness conditions for the derivatives of the Lyapunov functions or their upper bounds? In fact, some investigations have been made in the past years in this aspect. For example, LaSalle established a very important theorem named LaSalle invariance principle or LaSalle’s theorem [1], which weakened the condition of the Lyapunov function method on the negative definiteness of the derivatives of the Lyapunov functions along the solutions of the equations, and it has been widely used in the theory of ordinary differential equations. In the recent years, LaSalle’s theorem has been generalized directly to the stochastic differential delay equations by Mao [2–5], and a kind of LaSalle-type theorems had been established. Then the LaSalle-type theorems for stochastic differential delay equations have also been generalized to a kind of stochastic functional differential equations with distributed delays by Shen et al. [6–11]. Limited by the derivation techniques, this kind of theorems has not been generalized to the most general nonlinear stochastic functional differential equations so far.
In this paper, we generalize the investigation by Xuerong Mao, Yi Shen, and other authors to the general nonlinear stochastic functional differential equations. With some preliminaries on lemmas and the derivation techniques, we establish three LaSalle-type theorems for the general nonlinear stochastic functional differential equations via multiple Lyapunov functions. For the typical special case with estimations involving for the derivatives of the Lyapunov functions, a theorem is established as the corollary of the main theorem of the paper. The key point of the paper lies in the treatment of the general retarded terms in the estimations for the derivatives of the Lyapunov functions or their upper bounds. At the end of the paper, an example is given to illustrate the usage of the method proposed in the paper.
2. Preliminaries
2.1. Basic Notations
Throughout the paper, unless otherwise specified, we will employ the following notions. is a positive constant which stands for the upper bound for the bounded time delays involved possibly in the involved inequalities or equations, and , . denotes the space of continuous functions from to with norm , where is any kind of norms for vectors. Let , . denotes the family of all -measurable bounded -valued random variables.
For a given function , the associated function is defined as , . In real applications of the results of this paper, the criteria may be used in the function space ; in this case, one can extend the norm of as . For the general theory of functional differential equations, the readers are referred to [12–17].
In the paper, we will replace the function in by when necessary via an operator “”, named freezing operator. In this case, as a function in , will be counted as a constant; that is, for any , ; thus we have . For example, for a functional , if we replace by , then it becomes
In the paper, we also define and .
Let be a complete probability space with a filtration satisfying the usual conditions; that is, it is right continuous and contains all -null sets, and let be an -dimensional Wiener process defined on the probability space .
2.2. Equation Description and Basic Assumptions
Given the stochastic functional differential equation where the state . and , . The given initial data is given by . is a stochastic process in . When the criteria are used in the function space , the initial data can extend to by for .
As a standing condition, we impose the Lipschitz condition for the coefficients.
Both and satisfy the Lipschitz condition. That is, there is a constant such that for all and , , which may be directly useful in our estimations when necessary.
In this case, and also satisfy the linear growth condition. In fact, by , we have , for . Thus, under the Lipschitz condition, there exists a unique global solution, which is denoted by in this paper, to the equation for each initial data ; see [14].
It is also known that, under the Lipschitz condition , the solution with initial data to (2) is continuous, satisfying for and arbitrary ; see [14].
In the paper, some coefficients and functions will be involved. Assume that, for each , functions , , , , , , , .
For , we further assume the following.
Along the solution of the equation, for each , we have estimation where , are nonnegative continuous functions on and with . , , and, especially, where where will be defined in Lemma 4.
Remark 1. The assumption on can also be described in the following form theoretically if is differentiable. Denote
Assume that
Let ; that is, is once differentiable in and twice continuously differentiable in . Define a differential operator , associated with (2), acting on by
where
The assumptions for the involved Lyapunov functions will, respectively, be as follows:
2.3. Lemmas
By the nonnegative semimartingale convergence theorem [18], with a simple variable substitution for time , we directly have the following.
Lemma 2. Let and be two continuous adapted increasing processes on with , a.s., a real-valued continuous local martingale with , a.s., and a nonnegative -measurable random variable such that . Define If is nonnegative, then where , a.s. means . In particular, if , a.s., then for almost all , That is, both and a.s. converge to finite random variables as .
Lemma 3. Under the assumption , along the solution of (2), one has, for each , where
Proof . Firstly, we directly have and thus we have Combining with the assumption , this yields The proof is complete.
Lemma 4. One has the following estimation: where and
Proof . We directly have and this completes the proof.
3. Main Results
Theorem 5. Assume that holds. If there are Lyapunov functions satisfying , and for each , , then for all , the solution of (2) satisfies
Proof . Denote . Firstly, by Lemma 3 we have
and, combining with Lemma 4, we have
Secondly, by the first assumption of the theorem, we have
and thus we have
due to . With this we know that
and this implies that converges, so is , a.s.
Third, denote
For every integer , define a stopping time
Define sets ; then one easily verifies that ; thus we have ; that is, the sequence is increasing for .
Denote ; then we have
By the above derivations, we have
and thus we have , and then , a.s.
By the Itô's rule, we have
By Lemma 2, we obtain , a.s., and , a.s. Thus for sufficient large , the probability for is obviously ; thus there is a subset of with such that for every there is an such that
On the other hand, we have, for any ,
Letting yields
which implies that
with probability 1. Hence there is another subset of with such that, if , holds for every . Therefore, for any , we have
Since , we must have
By Itô’s rule again, together with the above derivation, we get
Notice that all the terms in the left side are a.s. bounded; then we have
It follows that
since . In fact, we have
If not, there must be some with such that, for any ,
Hence, for any , one can find a pair of and such that
Consequently
This is a contradiction, so (45) holds. It now follows from (45) that
that is, we have the conclusion of the theorem. The proof is complete.
Theorem 6. Assume that holds. If there are Lyapunov functions satisfying , , and for some , implies ; , , imply ; then ; the solution of (2) satisfies
Similarly, we have the following theorem.
Theorem 7. Assume that holds. If there are Lyapunov functions satisfying and , then ; the solution of (2) satisfies
4. Special Case:
Consider the special case with , where is nonincreasing, and , , .
Theorem 8. Assume that and hold, one of and holds, and for , with parameters , , . If there are Lyapunov functions satisfying or and , then ; the solution of (2) satisfies or respectively.
Proof . We prove the first case, that is, the case with assumption .
Define , by the definition of ; there exists a such that . By the definition of freezing operator , we have ; thus
Denote the th column of by ; then we can rewrite (2) as
Then by Itô’s rule, we have
where
Thus we have
By the linear growth condition, we have
With these, we then have
Denote
and, for each , denote the -algebra generalized by by ; then we have
By Theorem 5, we get the conclusion of the theorem.
Remark 9. Theorem 8 is considered for the case . The homogeneous case for , , and is , .
5. Remarks for Special Coefficients
Remark 10. For the case , , we have where thus the condition becomes , .
Remark 11. If , , and , then we have
Remark 12. If , , , and for , , then , where due to , . In this case, the condition for is . A sufficient condition for this is which simplifies the condition of the theorem greatly.
6. Example
To illustrate the usage of our results, we construct an example.
Consider a 2-dimensional stochastic functional differential equation with time-varying delay where is a scalar standard Wiener process defined on the complete probability space .
For this equation, the Lipschitz condition is satisfied and is the trivial solution. Besides, we have time-varying delay with .
It is obvious that the results in the previous literature are not adapted to this kind of equations with time-varying delay. We now give a conclusion for the asymptotic property of the solution of the equation by our theorems.
Define two Lyapunov functions , , and denote the states , , and , which is the retarded state determined by . Define , , , , , , , , and . By a simple computation, we have , , and as well as Computations show that where .
By the above computations, we have , , , , , and , and then , , , , , , , , , , , , , and ; thus, by Theorem 8 and Remark 10, we have , a.s. Because , , then it follows that , a.s.
Taking initial data , , Figure 1 verifies our result about the asymptotic property of the solution of the equation.
7. Conclusion
In this paper, we have generalized a kind of LaSalle-type theorems established by Xuerong Mao, Yi Shen, and other authors to the general nonlinear stochastic functional differential equations. With some preliminaries on lemmas and the derivation techniques, we establish three LaSalle-type theorems for the general nonlinear stochastic functional differential equations by the multiple Lyapunov functions. It should be pointed out that the models investigated in this paper are not desired in any kind of special form. The key point of the paper lies in the treatment of the general retarded terms in the estimations for the derivatives of the Lyapunov functions or their upper bounds. Of course, for the typical special case with estimation involving for the derivatives of the Lyapunov functions, a theorem is established as the corollary of the main theorem. At the end of the paper, an example is given to illustrate the usage of the method proposed in the paper. In fact, the asymptotic properties of the solutions of this kind of equations cannot be judged by the results in the previous literature mentioned above.
It is obvious that the results of this paper can be rewritten in another form by mathematical expectation with only minor changes for the assumptions on the stochastic derivatives of the Lyapunov functions.
At the end, we point out that one may determine the asymptotic behavior of the solutions of the equations by virtue of the integrability of the terms as directly if suitable assumptions are added to the related coefficients.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the editor and reviewers for their insightful comments and help as well as the National Natural Science Foundation of China under Grant 61273126, the Natural Science Foundation of Guangdong Province under Grants 10251064101000008 and S2012010009675, and the Research Fund for the Doctoral Program of Higher Education of China under Grant 20130172110027, for their financial support.