Approximate Controllability and Ulam Stability for Second-Order Impulsive Integrodifferential Evolution Equations with State-Dependent Delay

Bensalem, Abdelhamid; Salim, Abdelkrim; Benchohra, Mouffak; N’Guérékata, Gaston

doi:https://doi.org/10.1155/2024/8567425

International Journal of Differential Equations

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Research Article | Open Access

Volume 2024 | Article ID 8567425 | https://doi.org/10.1155/2024/8567425

Approximate Controllability and Ulam Stability for Second-Order Impulsive Integrodifferential Evolution Equations with State-Dependent Delay

Abdelhamid Bensalem,¹Abdelkrim Salim,^1,2Mouffak Benchohra,¹and Gaston N’Guérékata³

Academic Editor: Amar Nath Chatterjee

Received18 Apr 2023

Revised31 May 2023

Accepted29 Jan 2024

Published29 Mar 2024

Abstract

In this paper, we shall establish sufficient conditions for the existence, approximate controllability, and Ulam–Hyers–Rassias stability of solutions for impulsive integrodifferential equations of second order with state-dependent delay using the resolvent operator theory, the approximating technique, Picard operators, and the theory of fixed point with measures of noncompactness. An example is presented to illustrate the efficiency of the result obtained.

1. Introduction

In applied mathematics, control theory is crucial; it involves building and evaluating the control framework. Controllability analysis is used to solve a variety of real-world issues, such as issues with rocket launchers for satellite and aircraft control, issues with missiles and antimissile defense, and issues with managing the economy’s inflation rate. Over the last twenty years, a lot of work has been done for controllability of evolution equations [1–13].

In addition, a key aspect of the field of mathematical analysis study is stability analysis. The concept of Ulam stability is applicable in various branches of mathematical analysis and is used in the cases where finding the exact solution is very difficult. A number of researchers have been working on the study of Ulam-type stabilities of differential and integrodifferential equations recently, and they have produced some remarkable findings, see [14–16], and the references therein.

During the past ten years, impulsive differential equations have attracted a lot of interest. Dynamic systems that contain jumps or discontinuities are represented using impulsive differential equations. In contrast, integrodifferential equations are found in many scientific fields where it is important to include aftereffect or delay (for example, in control theory, biology, ecology, and medicine). In fact, one always uses integrodifferential equations to describe a model that has heritable characteristics. As a result, these equations have attracted a lot of attention (see for instance, [17–23]). In [24], the authors studied some local and global existence and uniqueness results for abstract differential equations with state-dependent argument.

Second-order nonautonomous differential systems have received a lot of interest. There is no need to transform a second-order differential system into a first-order system in order to solve it. Various second-order nonautonomous differential systems existence results are presented in [5, 20, 25–29] and references therein.

In [30], Balachandran and Sakthivel considered the following integrodifferential system:where takes values in a Banach space with the norm and the control function is given in , a Banach space of admissible control functions, with being a Banach space. are given functions, and is a bounded linear operator from into . Here, .

In [8], the authors investigated the controllability of the functional differential equation with a random effect:where is a complete probability space with being the event space and being the probability function (see [31], for more information), is a given function, are given measurable functions, and is a real Banach space. is the control function defined in , a Banach space of admissible control functions with being a Banach space, and is a bounded linear operator from into . The main result is based upon a generalization of the classical Darbo fixed-point theorem and the concept of measure of noncompactness combined with the family of cosine operators.

Arthi and Balachandran et al. [32] considered the following abstract control system:where a Banach space of admissible control functions with being a Banach space and being a bounded linear operator; the function , is the phase space; are prefixed numbers; are appropriate functions.

Motivated by the abovementioned works, we derive some sufficient conditions for the existence, approximate controllability, and Ulam-type stability for impulsive integrodifferential equations of second order with state-dependent delay described in the form:where , , and , with . , are closed linear operators on , with dense domain , which is independent of , and ; the operator is defined by

The nonlinear term , and are given functions. The jumps at the points are given by and , in the states and , respectively, where stand for left and right limits of at . Similarly, stand for right and left limits of at . The jumps at the points are determined by the nonlinear functions , where . The control function is a given function in the Banach space of admissible control , where is also a Banach space. is a bounded linear operator from into , and is a Banach space.

The work is organized as follows: In section two, we recall some definitions and facts about the resolvent operator, Picard operator, and measure of noncompactness. In section three, we give the existence of mild solutions to the problem (4). Section four is devoted to approximate controllability of mild solution and section five to the generalized Ulam–Hyers–Rassias (U-H-R) stability. In the last section, we present an example to illustrate our main result.

2. Preliminaries

Let be the Banach space of all continuous functions mapping into . Let for . We define the space of piecewise continuous functions:with the norm

Next, we consider the second-order integrodifferential system [26]:for . We denote . Let: (B1) For each is a bounded linear operator, for every is continuous and for , . (B2) There exists where . (B3) There exists such that

Under these conditions, it has been established that there exists a resolvent operator associated with systems (2).

Definition 1 (see [26]). A family of bounded linear operators on is a resolvent operator for (2) if it verifies the following:(a)The map is strongly continuous; is continuously differentiable for all and (b)Assume . The function is a solution for systems (6) and (7). Thus, for all .

By (a), there are and , such that

Moreover,can be extended to where

Then, there exists where

Let the state space be a seminorm linear space of functions mapping into , and verifying (see [33]): If and , then for : There exists where There exist and with continuous and bounded and locally bounded where For the function in is a -valued continuous function on . The space is complete. We denote

For , we define the spaceand the space

In the following, consider .

Lemma 2 (see [34]). Let the following inequality holds:where is nondecreasing, . Then, for , the following inequality is valid:

Definition 3 (see [35]). Let be a metric space. is a Picard operator if there exists , such that(i) where is the fixed point set of (ii) converges to for all

Lemma 4 (see [35]). Let be an ordered metric space and . We assume the following:(i) is a Picard operator (ii) is an increasing operatorThen, we have(a)(b)

Definition 5 (see [36]). Let be a Banach space and be the bounded subsets of . The Kuratowski measure of noncompactness is the map given bywhere

Lemma 6 ([37]). If is a bounded subset of a Banach space , then for each , there is a sequence such that

Lemma 7 (see [38]). If is uniformly integrable, then the function is measurable and

Lemma 8 (see [36]). (i)If is bounded, then for any where .(ii)If is piecewise equicontinuous on , then is piecewise continuous for , and(iii)If is bounded and piecewise equicontinuous, then is piecewise continuous for and where denotes the Kuratowski measure of noncompactness in the space .

Theorem 9 (see [39]). Let be a nonempty, bounded, closed, and convex subset of a Banach space and let be a continuous mapping. Assume that there exists a constant , such thatfor any nonempty subset of . Then, has a fixed point in set .

Theorem 10 (see [40]). Let be a nonempty complete metric space with a contraction mapping . Then, admits a unique fixed point in .

3. Existence of Mild Solutions

Definition 11. A function is called a mild solution of problem (1) if it satisfies

The following assumption will be needed throughout the paper: is a Carathéodory function, and there exist positive constants and continuous nondecreasing functions such that for . There exists a positive constant , such that for any bounded set and , and each , we have with The function is continuous, and there exists , such that Let Assume that hold, and there exist , , and , such that The functions are continuous, and there exist positive constants , such that and where Set . We assume that is continuous. Moreover, we assume the following assumption:(i) The function is continuous from into , and there exists a continuous and bounded function such that

Remark 12 (see [41]). The condition is verified by functions continuous and bounded.

Lemma 13 (see [42]). If is a function such that , thenwhere .

Now, we define a measure of noncompactness in the space . Let us fix a nonempty bounded subset of the space and . Then, for , , such that , we denote the modulus of continuity of the function on , namely,

Consider the function defined on the family of subset of bywith and .

The function is a sublinear measure of noncompactness on the space . For details on the definition and properties of the measure of noncompactness on the space of piecewise continuous functions , the reader is referred to [43].

Theorem 14. Suppose that and are verified. Then, (1) has at least one mild solution.

Proof. We transform problem (1) into a fixed-point problem and define the operator byLet be the function defined byThen, , and for each , with , we denote the function byIf satisfies (3), we can decompose it as , which implies , and the function satisfiesSetLet the operator be defined byThe operator has a fixed point which is equivalent to say that has one, so it turns to prove that has a fixed point. We shall check that the operator satisfies all conditions of Darbo’s theorem.
Let }, withsuch that are constants, they will be specific later.
The set is bounded, closed, and convex. Step 1: . For , and by , we have and Then, Thus, Therefore, , which implies that is bounded. Step 2: is continuous. Let be a sequence such that in At the first, we study the convergence of the sequences . If is such that , then we have which proves that in , as , for where . If , we get which also shows that in , as , for every such that . Then, for , we have Since and are continuous, we obtain and By the Lebesgue-dominated convergence theorem, Then, by , we get Since are continuous, by the Lebesgue-dominated convergence theorem, we obtain Thus, is continuous. Step 3: is -contraction. Let be a bounded equicontinuous subset of , , and , with , we haveBy the strong continuity of , we getThus, is equicontinuous; then, .
Now, for , and for any , there exists a sequence such that for , we haveSince is arbitrary, we getThus,By Theorem 9, it follows that there exists at least one fixed point within . Consequently, the point is a fixed point for the operator , which is a mild solution to (1).

4. Controllability Results

Definition 15. The reachable set of system (1) is given by

In case , system (1) reduces to the corresponding linear system. The reachable set in this case is denoted by .

Definition 16. If , then the semilinear control system is approximately controllable on . Here, represents the closure of . Clearly, if , then the linear system is approximately controllable.

We define the operator as follows:

It is demonstrated that the approximate controllability of the linear system extends from the semilinear system, given certain conditions on the nonlinear component. Let us now consider the ensuing linear system:and the semilinear system

The following hypotheses must be introduced in order to demonstrate the main aim of this section, that is, the approximate controllability of system (5):(1) Linear system (4) is approximately controllable(2) Range of the operator is a subset of the closure of range of , i.e.,

Theorem 17. If hypotheses are verified, then system (1) is approximately controllable.

Proof. The mild solution of system (4) corresponding to the control is given byAssume the following system:Since , there exists a control function such thatNow, assume that is the mild solution of system (1) corresponding to given byThen, if , we get .
And, for , we haveNow, for any , we define the function , and from the definition of the function and Lemma 13, we obtainThen,Therefore, according to Lemma 2, we getBy taking suitable control function , we make arbitrary small. Therefore, the reachable set of (4) is dense in the reachable set of (70), which is dense in due to . Hence, the approximate controllability of (70) implies that of the semilinear control system (4).

5. Ulam–Hyers–Rassias Stability Results

Let and be nondecreasing and consider the following inequalities:

Let the space be

The following concepts are inspired by papers [14, 15] and references therein.

Definition 18. Equation (1) is generalized U-H-R stable with respect to , if there exists , such that for each solution of inequality (6), there exists a mild solution of equation (1), with

Remark 19. A function is a solution of inequality (6) if and only if there exist and , such that , and , , , , .

Remark 20. If is a solution of inequality (6) then is a solution of the following integral inequality:

We also need the following additional assumption to discuss about stability:. We assume that for a nondecreasing function , there exists , such that

Theorem 21. If , and are satisfied, withthen, equation (1) is generalized U-H-R stable with respect to .

Proof. Let be a solution of (6) and be the mild solution of (1) with and .
Then, we getOn the other hand, we getHence, for , we haveLet , andFor , letNow, we will prove that is a Picard operator. For that, let and , if , we get , and if , we haveTherefore, is a contraction; hence, from Theorem 10, there exists a unique in , and from Definition 3, we deduce that is a Picard operator.
Furthermore, we haveWe can see that is an increasing function and is nonnegative.
So, for , we haveThen,From Lemma 2, we getIn particular, if , then we have , and applying the abstract Gronwall lemma, we obtain . It follows thatNow, if , we getThen, if we putThus, we have for all which implies that (4) is generalized U-H-R stable with respect to .

6. An Example

Consider the following class of partial integrodifferential system:where , , .

Letbe the Hilbert space with the scalar product , and the normand the phase space be , the space of bounded uniformly continuous functions endowed with the following norm: . It is well known that satisfies the axioms and with and (see [41]). We define induced on as

Then, is the infinitesimal generator of a cosine function of operators on associated with sine function . In addition, has discrete spectrum which consists of eigenvalues for , with corresponding eigenvectors . The set is an orthonormal basis of . Applying this idea, we can write

The cosine family associated with is given by :and the sine function is given by

Thus, and is compact for all . We define on . Clearly, is a closed linear operator. Therefore, generates such that is compact and self-adjoint for all (see [26]).

We define the operators as follows:

The assumption holds under more suitable conditions on the operator . Furthermore, are fulfilled. Then, there exists a resolvent compact operator [26, 44].

Now, let be defined by , where is linearly continuous, and for , we put , such that holds, and let be continuous on .

We put , for , and define

These definitions allow us to depict system (7) in the abstract form (4).

Now, for , we have

So, are continuous nondecreasing functions, and we have

And for any bounded set , and , we get

Now, about , we obtain

Now, similar reasoning as in [28], if the corresponding linear system is approximately controllable, then system (7) is approximately controllable.

Furthermore, we have

Thus, all the assumptions of Theorem 21 are fulfilled. Consequently, the mild solution of problem (101) is generalized U-H-R stable.

Data Availability

Data sharing is not applicable to this paper as no datasets were generated or analyzed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The study was carried out in collaboration of all authors. All the authors read and approved the final manuscript.

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