Practical Stability and Integral Stability for Singular Differential Systems with Maxima

Bao, Junyan; Wang, Peiguang; Li, Yanjun

doi:https://doi.org/10.1155/2020/4792183

Mathematical Problems in Engineering

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Abstract Introduction and Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

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Control Problems of Nonlinear Systems with Applications 2020

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Research Article | Open Access

Volume 2020 | Article ID 4792183 | https://doi.org/10.1155/2020/4792183

Practical Stability and Integral Stability for Singular Differential Systems with Maxima

Junyan Bao,¹Peiguang Wang,¹and Yanjun Li¹

Guest Editor: Rongwei Guo

Received17 Feb 2020

Accepted14 Mar 2020

Published14 Apr 2020

Abstract

In this paper, we introduce various definitions of practical stability and integral stability for nonlinear singular differential systems with maxima and give criteria of stability for such systems via the Lyapunov method and comparison principle.

1. Introduction and Preliminaries

Differential equations with maxima are a special type of differential equations that contain the maximum of the unknown function over a previous interval, of which many examples are found in the fields of application such as automatic control, population dynamics, disease control, and so on. Recently, the research interest in differential equations with maxima has increased exponentially. Some stability results for such equations can be found in the monographs [1, 2], the papers [3–9], and references cited therein.

In practical applications, many problems can be described by singular system models, such as optimal control problems and constrained control problems, which can be found in the monographs of Campbell [10] and Dai [11]. Singular system is a type of dynamic system which is more complicated than the ordinary one. Owing to its complicated structure and many other factors, the study of stability for singular systems involves greater difficulty than that of nonsingular systems. Till now, various types of stability for singular systems have been investigated via Lyapunov functions. However, most previous studies focused on the singular systems described by ordinary differential equations [10–13], difference equations [14–17], and delay differential equations [18–20], and there are a few results for singular differential systems with maxima. In addition, differential equations with maxima have some different properties from the well-known differential equations and delay differential equations.

The purpose of this paper is to integrate these two areas and analyze the practical stability and integral stability of nonlinear singular systems with maxima. To extend Lyapunov’s stability and support the specific needs of singular systems, we introduce the function and obtain some different types of stability criteria by using the Lyapunov function method and comparison principle.

2. Practical Stability

The practical stability, being quite different from the stability in the sense of Lyapunov, is neither weaker nor stronger than the usual stability. It is significant from the perspective of engineering application (see [21–25]). In this section, by using Lyapunov functions and the comparison principle, we study some practical stability for the following singular differential systems.

Consider the singular differential systems with maximawhere with rank is a singular constant matrix, , , , is a constant, and .

Firstly, we introduce the following notations and sets for convenience.

Let , where ; , , . is a set of all consistent initial functions at initial time . Then, for any , there exists at least one continuous solution of systems (1) in through (see [20]). means that implies

We denote by the solution of the initial value problems (1).

Definition 1. Let and we define the derivative of the function V (t; x) along the trajectory of solution of the singular systems (1) as follows:

Definition 2. Let . The singular systems (1) is said to be stable with respect to if for any , and some , there exists , such that

Definition 3. Let The singular systems (1) is said to be practically stable for given with and some , such that uniformly practically stable if holds for all practically quasistable for given with , and some , we have uniformly practically quasistable if holds for all strongly practically stable if and hold simultaneously strongly uniformly practically stable if and hold simultaneously

Remark 1. then Definitions 2 and 3 reduce to the concepts of classic Lyapunov stability.
It is well known that the comparison principle plays an important role in the development of stability theory. By the comparison principle, we can reduce the study of a given complicated differential system to that of a relatively simpler differential equation. For this purpose, we give the following lemma and definition.

Lemma 1 (See [1]). Assume that the following conditions hold and for any such that for , the inequality holds, where , the maximal solution of the scalar equation exists, on . Then, , provided .

Definition 4. Comparison equation (7) is said to be practically stable if for given with and some , we have implies , for uniformly practically stable if holds for all practically quasistable if for given with , , , and some , we have that implies , for uniformly practically quasistable if holds for all

Theorem 1. Assume that the following conditions hold. with are given there exists a function and such that(i)for any , , , the inequality holds, where and (ii), where , and Then, equation (7) is (uniformly) practically stable with respect to implies that system (1) is (uniformly) practically stable with respect to

Proof. Assume that is a solution of the equation (7), and is practically stable with respect to for given . Let , where is a solution of the systems (1). From the condition (i) of , it follows that , for . LetBy Lemma 1, we know that the inequality , holds, where is the maximal solution of comparison equation (7) existing on . Assume that , then, we haveFurthermore, from the condition (ii) of and Lemma 1, we get . Thus, implies , , that is, system (1) is practically stable with respect to .
Similarly, we can prove that equation (7) is uniformly practically stable with respect to implies that the systems (1) is uniformly practically stable with respect to . The proof is completed.
By Theorem 1, we can obtain the following corollaries.

Corollary 1. Assume that the conditions and (ii) of (A₄) hold in Theorem 1, and there exists a function and such that for any , , , the inequality holds. Then, system (1) is uniformly practically stable with respect to .

The conclusion of Corollary 1 can be obtained by considering the case of and is uniformly practically stable with respect to for given .

Corollary 2. Assume that the conditions and (ii) of hold in Theorem 1, and there exists a function and such that for any , , , the inequality holds, where and the inequalities hold Then, system (1) is uniformly practically stable with respect to

Proof. By Theorem 1, we only prove that the system is uniformly practically stable with respect to . In fact, let . Assume that , it follows from the condition (ii) of thatFurthermore, by condition , the inequalityholds. Then, system (1) is uniformly practically stable with respect to . The proof is completed.

Corollary 3. Assume that the conditions (A₃) and (ii) of (A₄) hold in Theorem 1, and there exists a function and such that for any , for , the inequality holds, in which and are positive constants, the inequalities hold Then, system (1) is uniformly practically stable with respect to

Proof. In fact, we only need to prove that the system is uniformly practically stable with respect to . Let ; then, we haveFurthermore, by condition , the inequalityholds. Then, system (1) is uniformly practically stable with respect to .

Theorem 2. Assume that the conditions (A₃) and (ii) of (A₄) hold in Theorem 1, and (iii) , where .

Then, equation (7) is (uniformly) practically stable with respect to implies that the systems (1) is (uniformly) practically stable with respect to .

Proof. In fact, by the condition (iii) of , we haveThen, we can get the result by using a method similar to Theorem 1. We omit its details.

Theorem 3. Assume that the following conditions hold with , and are given there exists a function and such that(i)for any , , , the inequality holds, where , (ii), where , and Then, equation (7) is (uniformly) practically quasistable with respect to implies that system (1) is (uniformly) practically quasistable with respect to .

Proof. Assume that is a solution of equation (7) and is practically quasistable with respect to for given , and . Let , where is a solution of system (1). It follows from the condition thatLet . Then, by Lemma 1, we know that the inequalityholds, where is the maximal solution of comparison equation (7) existing on . Assume that and . Then, we obtainFurthermore, by comparison equation (7) which is practically quasistable with respect to , the condition , and Lemma 1, we getThus, implies , , that is, system (1) is practically quasistable with respect to .
Similarly, we can prove that equation (7) is uniformly practically stable with respect to implies that the systems (1) is uniformly practically stable with respect to .

Theorem 4. Assume that the conditions (A₁₀) and (i) of (A₄) hold in Theorem 3, and the condition (ii) of (A₄) is replaced by(iv), where Then, equation (7) is (uniformly) practically quasistable with respect to implies that the systems (1) is (uniformly) practically quasistable with respect to .

The proof of Theorem 4 is similar to that of Theorem 3, so we omit its details.

3. Integral Stability

The concept of integral stability, which was introduced for ordinary differential equations by Vrhoc in 1959 [26] and Lakshmikantham in 1969 [27], enlarges the collection of dynamical properties of solutions which can be investigated by the direct Lyapunov method. The integral stability theory has been rapidly developed recently. For example, Martynyuk [28], Salvadori and Visentin [29], Soliman and Abdalla [30] obtained the integral stability criteria for nonlinear differential equations, respectively; Hristova [31] obtained the integral stability in terms of two measures for impulsive differential equations; and Sood and Srivastava [32] gave the -integral stability criteria for impulsive differential equations. The main purpose of this section is to discuss the integral stability of singular differential systems with maxima and its perturbed systems.

Consider singular differential system (1) and its perturbed systemswhere , .

Let be a set of all consistent initial functions of (1) and (26) in through . For any , assume that there exists a continuous solution of (1) and (26) in through at least.

Definition 5. Let . Singular system (1) is said to be equi-integrally stable on , if for given and , there exists a positive function , which is continuous in for each and , such that, for every solution of the perturbed systems (26), holds, provided that and uniformly integrally stable on , if the in is independent of equiasymptotically integrally stable on , if holds and for every , and , there exist positive functions and , which are continuous in for each and , and for every solution of the perturbed systems (26), holds, provided that , and uniformly asymptotically integrally stable on if the and in are independent of and holdsNow, we consider comparison scalar differential equation (7) and its perturbed equationwhere , , .

Definition 6. Equation (7) is said to be equi-integrally stable, if for given , there exists a positive function that is continuous in for each and , such that, for every solution of the perturbed differential equation (32), the inequalityholds, provided that and for every .

Remark 2. Similar to Definition 5, we can give the corresponding concepts of stability of equation (7).
Next, we investigate the integral stability of system (1) via the Lyapunov function method and comparison principle.

Theorem 5. Assume that the condition (i) of (A₄) holds in Theorem 1, and condition (ii) of (A₄) is replaced by

Then, equation (7) which is equi-integrally stable implies that system (1) is equi-integrally stable on .

Proof. Let for every , . Since is Lipschitzian in , we haveLet be any solution of (26). Thus, by the condition (i) of and (29), we getDefining and choosing , by Lemma 1, we havewhere is the maximal solution of (32).
If equation (7) is equi-integrally stable, then for and , there exists a , which is continuous in for each and , such that, for every solution of (32), the inequalityholds, whenever and for any .
By the condition , it is possible to choose a satisfyingIt is easily shown that is continuous in for each and for each . Moreover, we claim that system (1) is equi-integrally stable on . In fact, if this is not true, there exists a such thatFrom (37)–(40), we havewhich is a contradiction. Thus, system (1) is equi-integrally stable on .

Corollary 4. Assume that the conditions of Theorem 5 hold. Then, equation (7) which is uniformly integrally stable implies that system (1) is uniformly integrally stable on {q(t; x); T_k}.

The detailed proof of Corollary 4 is similar to the proof in Theorem 5, so we omit it.

Theorem 6. Assume that the conditions of Theorem 5 hold and t_k = + Then, equation (7) which is equiasymptotically integrally stable implies that system (1) is equiasymptotically integrally stable on {q(t. x), R₊}.

Proof. It can be known from the proof of Theorem 5 that system (1) is equi-integrally stable on . Let and , for given and . For given .., and , there exist and , such that for every solution of (32),holds, whenever and .
Choosing a positive number such that , for given and , system (1) satisfies of Definition 5. In fact, suppose that the conclusion is not true, then , cannot be satisfied whenLet be a sequence such that and . Suppose that there is a solution of system (26), such that for every ,By the condition and (44), we obtainFurthermore, by the equiasymptotical integral stability of equations (7), (37), and (42)–(59), we can getwhich is a contradiction. Thus, system (1) is equiasymptotically integrally stable on .

Corollary 5. Assume that the conditions of Theorem 6 hold. Then, equation (7) which is uniformly asymptotically integrally stable implies that system (1) is uniformly asymptotically integrally stable on {q(t; x); T_k}.

In fact, we can show that the positive numbers and in proof of Theorem 6 are independent of ; therefore, (44) implies that is independent of . The rest of the proof is similar to that of Theorem 6, so we omit the details here.

For the comparison equation (7), if we suppose that the function is nonincreasing in for ; then, we can get the uniform asymptotic integral stability of system (1) by the uniform asymptotic stability of the comparison equation (7). Therefore, we firstly give the following definition and Lemmas, which can be found in [27].

Definition 7. Comparison equation (7) is said to be equistable, if for each , , there exists a positive function , such that uniformly stable if in is independent of equiasymptotically stable, if it is equistable and for each , , there exists a positive function , , such that uniformly asymptotically stable if holds and the numbers and in are independent of

Lemma 2 (see [27]). Equation (7) is uniformly stable if and only if there exists a function a(r) K, such that

Lemma 3 (see [27]). Equation (7) is uniformly asymptotically stable if and only if there exist functions a(r) K and , such thatwhere is monotone decreasing in , and as .

Theorem 7. Assume that the conditions of Theorem 6 hold, the function be nonincreasing in for any , and

Then, equation (7) which is uniformly asymptotically stable implies that system (1) is uniformly asymptotically integrally stable on .

Proof. Firstly, we prove system (1) is uniformly integrally stable. Because equation (7) is uniformly stable, by Lemma 2, there exists a function , such thatFor given and , where , let . Since is Lipschitzian in , we haveLet be any solution of (26) with , andwhere . According to the condition (i) of , we getBy (51) and Lemma 1, we havewhere is the maximal solution of (7) with .
We choosing , such thatIn view of the condition as , the choice of is reasonable. It is obvious that and . At the same time, we can claim that system (1) is uniformly integrally stable. In other words, the solution of system (26) satisfies , , whenever , andSuppose that this is not true; there exists a such thatFrom the condition and (54)–(58), we getThis is a contradiction, and then system (1) is uniformly integrally stable.
Secondly, we prove that system (1) is uniformly asymptotically integrally stable. By the uniform asymptotic stability of equation (7) and Lemma 3, we havewhere and . For given , and , let , andwhere . For any solution of (26) and (56), holds whenever . By (54), (56), and (61), together with , we can obtain the inequalitySince , then there exists a , such thatFurthermore, we havewhich implies thatprovided , and (62) is satisfied. Therefore, system (1) is uniformly asymptotically integrally stable.

4. Conclusion

This paper discussed a class of nonlinear singular differential systems with maxima. Some notions of practical stability and integral stability for such systems were introduced, and various stability criteria were obtained by using the Lyapunov method and comparison principle.

Data Availability

Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors read and approved the final manuscript.

Acknowledgments

This paper was supported by the National Natural Science Foundation of China (11771115 and 11271106).

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Copyright © 2020 Junyan Bao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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