Abstract
A novel and effective approach to stability of the solutions of nonlinear systems with impulsive effect is considered. The investigations are carried out by means of a class of vector Lyapunov functions and differential inequalities for piecewise continuous functions. Simulation examples are given to illustrate the presented results.
1. Introduction
There has been a significant development in the theory of the impulsive differential systems within the past 20 years (see [1–17]). The efficient applications of impulsive differential systems require the finding of criteria for stability of their solutions.
It is well known that the more complex the considered dynamical system is, the more difficult it is to find a Lyapunov function. This leads to employing several Lyapunov functions, which is called to be a vector Lyapunov function, for which each component provides information about a part of the dynamics. Then some of the complex aspects can be reduced to the connected dynamical systems and less rigid requirements can be satisfied. Hence, the corresponding theory, namely, the method of vector Lyapunov functions, offers a very flexible process (see [1–5, 11, 13, 18]).
In this paper, without assuming Lipschitz conditions on these nonlinear functions, which play important roles in the considered systems, as needed in most other papers, we will use a class of vector Lyapunov functions, by means of the comparison principle (see [1, 2, 13]), to study global exponential stability of the solution of nonlinear impulsive differential systems.
2. Statement of the Problem
Let be the -dimensional Euclidean space with norm , , let be a bounded domain in containing the origin, and , where , , and .
Let . Consider the impulsive functional differential system where , , , , and .
Definition 1 (see [12, 14, 15, 19]). System (1) is called to be exponentially stable on a neighborhood of the equilibrium point, if there exist constants , such that where is any solution of (1) initiated from .
We introduce a kind of partial ordering defined in the following natural way. For , if and only if for any .
Definition 2 (see [1, 2, 13]). The function is called monotonically increasing function in if for and for , .
Definition 3 (see [1, 2, 13]). The function is called quasi-monotonically increasing function in if, for any two points and of and for any , the inequality holds always when and , that is, if, for any fixed and any , the function is nondecreasing with respect to .
In the further considerations we will use the class of piecewise continuous auxiliary functions which are analogues of Lyapunov functions (see [20]).
We denote by the class of all continuous and strictly increasing functions such that .
We are interested in focusing on the possibility of applying the comparison method to the stability of the zero solution of system (1). Together with the system (1), we will also consider the following comparison system: where , , , , , and is an open subset of .
Lemma 4 (see [1, 2, 13]). Let the following conditions be fulfilled:(1),
where is continuous, quasi-monotonically increasing in and for and ;(2)there exists a which satisfies when , are monotonically increasing in and ;(3)there exist the functions which satisfy for all , where .
If the zero solution of system (3) is stable, then the zero solution of system (1) is stable with respect to .
3. Theory Results of Impulsive Stability
3.1. Basic Results
Consider the following system where , , and .
We will say that condition (A) is satisfied if the following conditions hold.
(A) The functions , are continuous in . There exists a function , and the condition holds for any .
We will say that condition (B) is satisfied if the following conditions hold.
(B) The functions , are continuous in . There exist functions , and the condition holds for any .
We will say that condition (C) is satisfied if the following conditions hold.
(C) The functions , are continuous in . There exist functions , and the condition holds for any .
Theorem 5. If the operators in system (5) satisfy condition (A) for any , then and initiated from , satisfy
Proof. For all and , we have
The derivative with respect to time is
That is,
It is easy to prove that is monotonically decreasing on ; thus the limit
exists. From the norm and condition (A), we obtain
where and .
Namely,
Corollary 6. When , , and , let and holds for any , where . Then one obtains a similar result with Theorem 5:
Corollary 7. When , , and , let and holds for any , where , . Then one obtains a similar result with Theorem 5:
Remark 8. When , , and , the constant is called the nonlinear measure [21–26]. But the condition is weaker than the nonlinear measure.
When system (5) is written as where , , and , , we have the following conclusion.
Corollary 9. If the operators in system (22) satisfy condition (B) for any , then and initiated from , satisfy
When system (5) is written as where , , and , , we have the following conclusion.
Corollary 10. If the operators , , in system (24) satisfy condition (B) for any , then and initiated from , satisfy
Corollary 11. If the operators , , in system (24) satisfy condition (B) for any , then and initiated from , and the constant satisfy
When system (5) is where , , , , and is the time delay, we have the following conclusion.
Corollary 12. If the operators in system (27) satisfy conditions (B) and (C) for any , then and initiated from , satisfy
When system (5) is written as where , , , and , we have the following conclusion.
Corollary 13. If the operators in system (29) satisfy the condition for any and , then , and initiated from , , and satisfy
3.2. Impulsive Stability
Consider the impulsive differential system where , , , and .
Theorem 14. If the operators in system (32) and the constants , , satisfy condition (B) for any and the matrix is invertible, is continuous, quasi-monotonically increasing in , are monotonically increasing, , and . If the zero solution of system (3) is stable, , , and , , then the zero solution of system (32) is also stable.
Proof. From Corollary 10, we can obtain It is easy to verify that all the conditions in the lemma are satisfied when . So it is true that the zero solution of system (32) is stable.
Remark 15. Imitating Theorem 14 and combining the above 7 corollaries, respectively, we can obtain similar results with Theorem 14, respectively.
4. Example
Example 1. We consider the system [3, 13]
where , , , and . Consider
So , .
Similarly, , .
When for all , we can obtain . That is, and , respectively. Then we get the comparison system
When for all , we can obtain the comparison system
We know that when the constants satisfy the condition in Theorem 14, in the systems (38) and (39) are stable by stability theory of impulsive differential systems. So it is true that the zero solution of system (36) is stable. Taking , , and initial conditions , the stability simulations of the states of system (36) are as shown in Figures 1 and 2, respectively.
Example 2. We consider the system [3]
Combining Corollary 11 and imitating Example 1, we can obtain the comparison system
where , , and . When the constants satisfy the condition in Theorem 14, it is true that the zero solution of system (40) is stable. Taking initial conditions , the stability simulations of the states of system (40) are as shown in Figures 3 and 4, respectively.
Remark 16. Comparing Examples 1 and 2 in this paper with Examples 2 and 4 in the literature [3, 13] or [3], the vector Lyapunov functions , in this paper are simpler than the vector Lyapunov functions , in [3, 13] or [3], respectively, and the approach is easy to operate.
Example 3. We consider the system
where , , , and .
Combining Corollary 12 and imitating Example 1, we can obtain the comparison system
where , . When the constants satisfy the condition in Theorem 14, it is true that the zero solution of system (42) is stable. Taking , , , and initial conditions , the stability simulations of the states of system (42) are as shown in Figures 5 and 6, respectively.
Example 4. We consider the system
where , , , and .
Combining Corollary 6 or Corollary 13 and imitating Example 1, we can obtain the comparison system
where , , or the comparison system
where , , and . When the constants , and satisfy the conditions in Theorem 14 or Remark 15, it is true that the zero solution of system (44) is stable. Taking , , and initial conditions , the stability simulations of the states , and of system (44) are as shown in Figures 7, 8, and 9, respectively.
5. Conclusion
This letter, which uses a class of vector Lyapunov functions, has studied the issue on impulsive stability of nonlinear systems and improves the existing results. Numerical simulations have verified the effectiveness of the method.
Conflict of Interests
The author declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author was supported financially by the National Natural Science Foundation of China (11071141), the Natural Science Foundation of Shandong Province of China (ZR2011AL018, ZR2011AQ008), the Project of Shandong Province Higher Educational Science and Technology Program (J11LA06, J13LI02), and the Research Fund Project of Heze University (XY10KZ01).