Abstract

We investigate the qualitative behavior of a perturbed causal differential equation that differs in initial position and initial time with respect to the unperturbed causal differential equations. We compare the classical notion of stability of the causal differential systems to the notion of initial time difference stability of causal differential systems and present a comparison result in terms of Lyapunov functions. We have utilized Lyapunov functions and Lyapunov functional in the study of stability theory of causal differential systems when establishing initial time difference stability of the perturbed causal differential system with respect to the unperturbed causal differential system.

1. Introduction

Lyapunov’s second method is a standard technique used in the study of the qualitative behavior of causal differential systems along with a comparison result [1] that allows the prediction of behavior of a causal differential system when the behavior of the null solution of a comparison system is known. However, there has been difficulty with this approach when trying to apply it to unperturbed causal differential systems and associated perturbed causal differential systems with an initial time difference. The difficulty arises because there is a significant difference between initial time difference (ITD) stability and the classical notion of stability. The classical notions of stability are with respect to the null solution, but ITD stability is with respect to unperturbed causal differential system where the perturbed causal differential system and the unperturbed causal differential system differ in both initial position and initial time [2–4].

In this paper, we have resolved this difficulty and have a new comparison result which again gives the null solution a central role in the comparison differential system. This result creates many paths for continuing research by direct application and generalization.

In Section 2, we present definitions and necessary background material. In Section 3, we discuss and compare the differences between the classical notion of stability and the recent notion of initial time difference (ITD) stability. In Section 4, we have comparison theorems via Lyapunov functions with initial time difference. In Section 5, we have our main results of initial time difference stability criteria and asymptotic stability via Lyapunov functions and we finally have our main result of initial time difference uniformly asymptotic stability via Lyapunov functionals.

2. Preliminaries

In order to investigate the theory of stability for causal differential equations, we need comparison results in terms of Lyapunov-like functions.

Consider the causal differential systems and the perturbed system of (2) where is a causal operator and . A special case of (3) is where and is the perturbation term. We assume the existence and uniqueness of solutions and of (1) and (3) through and , respectively. We need to have some classes of functions to utilize Lyapunov-like functions for the generalized derivative of Lyapunov functions which has to satisfy suitable conditions as follows: where(i) is continuously differentiable on and ;(ii) is a Lyapunov function.

Before we can establish our comparison theorem and Lyapunov stability criteria for initial time difference we need to introduce the following definitions.

Definition 1. The solution of system (3) through is said to be initial time difference stable with respect to the solution , where is any solution of system (1) for , , and if and only if given any there exist and such that If , are independent of , then the solution of system (3) is initial time difference uniformly stable with respect to the solution . If the solution of system (3) through is initial time difference stable and there exist and such that for all and with and for then it is said to be initial time difference asymptotically stable with respect to the solution . It is initial time difference uniformly asymptotically stable with respect to the solution if and are independent of .

Definition 2. A function is said to belong to the class if , , and is strictly monotone increasing in .

Definition 3. For a real-valued function one defines the Dini derivatives as follows:(a) for .(b)One defines the generalized derivatives (Dini-like derivatives) as follows: for , where is the solution of system (3) and , where is any solution of system (1) for , , and .(c)Let be any Lyapunov functional. One defines its generalized derivatives as follows: where is the solution of IVP (3) through and is the solution of IVP through , , where is any solution of the IVP (1), for .

3. Causal Stability and New Notion of ITD Causal Stability

3.1. Causal Stability

Let be any solution of the causal differential system (1) with where and is the set .

Assume that for so that is a null solution of (1) through . Now, we can state the well-known definitions concerning the stability of the null solution.

Definition 4. The null solution of (1) is said to be stable if and only if, for each and for all , there exists a positive function that is continuous in for each such that If is independent of , then the null solution of (1) is said to be uniformly stable.

Definition 5. The solution of   (10) through is said to be stable with respect to the solution of (1) for if and only if given any there exists a positive function that is continuous in for each such that If is independent of , then the solution of system (10) is uniformly stable with respect to the solution .
We remark that, for the purpose of studying the classical stability of a given solution of system (10), it is convenient to make a change of variable. Let and be the unique solutions of (1) and (10), respectively, and set for . Then It is easy to observe that is a solution of the transformed system if which implies for . Since and is the null solution, the solution of   (1) corresponds to the identically null solution of where . Hence, we can assume, without loss of generality, that is the null solution of system (10) and we can limit our study of stability to that of the null solution [5]. However, it is not possible to do a similar transformation for initial time difference stability analysis.

3.2. New Notion of ITD Causal Stability

Let be a solution of (2) and where is any solution of system (1) for . Let us make a transformation similar to that in (15). Set for . Then One can observe that even if , is not zero and is not the null solution of the transformed system and the solution does not correspond to the identically zero solution of . Consequently, stability properties of null solution cannot be used in order to find ITD stability properties using this approach.

4. Comparison Theorems via Lyapunov Functions with Initial Time Difference

In our earlier work and in the work of others [1, 2, 5], the differences between the classical notion of stability and ITD stability did not allow the use of the behavior of the null solution in our ITD stability analysis. The main result presented in this section resolves those difficulties with a new approach that allows the use of the stability of the null solution of the comparison system to predict the stability properties of the solution of (3) with respect to where is any solution of system (1).

Theorem 6. Assume that(i) and is locally Lipschitzian in ;(ii)for and , where with and ;(iii) is the maximal solution of the scalar differential equation existing on .
Then, if , , where is any solution of the causal differential system (1) and is any solution of causal differential system (3) existing on such that implies

Proof. Let , , and let be any solution of the causal differential system (1) and let be any solution of causal differential system (3) for . Define so that . For some sufficiently small , consider the differential equation whose solutions exist as far as . In order to prove the conclusion of the theorem, we need to show that If this is not true, then there exists a such that It then follows that By using the assumption on and , the solutions are nondecreasing in . Since for , we get Consequently, .
The standard computation for small enough implies Since is locally Lipschitzian in and is the Lipschitzian constant, we obtain where and are error terms. This shows that since as . Hence, at , we have which contradicts (27). Hence , which yields the desired estimate as : Therefore these complete the proof.

Corollary 7. Let satisfy the conditions of Theorem 6 with and . Then where and are any solutions of the initial value problems (1) and (2), respectively. Equivalently we have