International Journal of Differential Equations
Volume 2024, Issue 1 8567425
Research Article
Open Access

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

Abdelhamid Bensalem, 

Abdelhamid Bensalem

Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbès, P.O. Box 89, Sidi Bel-Abbès 22000, Algeria univ-sba.dz

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Abdelkrim Salim, 

Abdelkrim Salim

Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbès, P.O. Box 89, Sidi Bel-Abbès 22000, Algeria univ-sba.dz

Faculty of Technology, Hassiba Benbouali University of Chlef, P.O. Box 151, Chlef 02000, Algeria

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Mouffak Benchohra, 

Mouffak Benchohra

Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbès, P.O. Box 89, Sidi Bel-Abbès 22000, Algeria univ-sba.dz

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Gaston N’Guérékata, 

Corresponding Author

Gaston N’Guérékata

NEERLab Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21252, USA morgan.edu

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First published: 29 March 2024
https://doi.org/10.1155/2024/8567425
Academic Editor: Amar Nath Chatterjee

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 ϑ(⋅) takes values in a Banach space with the norm ‖⋅‖ and the control function u(⋅) is given in L2(Θ, U), a Banach space of admissible control functions, with U being a Banach space. are given functions, and B is a bounded linear operator from U into . Here, ∇ = {(ς, ε) : 0 ⩽ ε ⩽ ς ⩽ b}.
In [8], the authors investigated the controllability of the functional differential equation with a random effect:
()
where (Ϝ, F, P) is a complete probability space with F being the event space and P being the probability function (see [31], for more information), is a given function, ȷ1, ȷ2 : Ϝ⟶Ξ are given measurable functions, and (Ξ, |⋅|) is a real Banach space. is the control function defined in , a Banach space of admissible control functions with Λ being a Banach space, and B is a bounded linear operator from Λ into Ξ. 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 L2(I, U) a Banach space of admissible control functions with U being a Banach space and being a bounded linear operator; the function , is the phase space; 0 < ς1 < ⋯<ςn < a 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 0 = <ς1 < ς2 < …<ςk < ςm+1 = T. , Υ(ς, ε) are closed linear operators on , with dense domain , which is independent of ς, and ; the operator Ϝ is defined by
()

The nonlinear term , and are given functions. The jumps at the points ςk ∈ (0, T) are given by and , in the states ϑ and ϑ′, respectively, where stand for left and right limits of ϑ at . Similarly, stand for right and left limits of ϑ′ at . The jumps at the points are determined by the nonlinear functions , where k = 1, 2, 3, …, m. The control function u is a given function in the Banach space of admissible control L2(Θ, U), where U is also a Banach space. is a bounded linear operator from U 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 ϑ mapping Θ into . Let for . We define the space of piecewise continuous functions:
()
with the norm
()
Next, we consider the second-order integrodifferential system [26]:
()
for 0 ≤ ε ≤ T. We denote ∇ = {(ς, ε) : 0 ≤ ε ≤ ς ≤ T}. Let:
  •  

    (B1) For each is a bounded linear operator, for every is continuous and

    ()

  •  

    for ι > 0, ε, ς ∈ ∇.

  •  

    (B2) There exists LΥ > 0 where

    ()

  •  

    .

  •  

    (B3) There exists b1 > 0 such that

    ()

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

Definition 1 (see [26].)A family of bounded linear operators (ℸ(ς, ε))ς≥ε on is a resolvent operator for (2) if it verifies the following:

  • (a)

    The map is strongly continuous; ℸ(ς, ⋅)γ is continuously differentiable for all and ∂/∂ς ℸ(ς, ε)|ε=ς = −I

  • (b)

    Assume . The function is a solution for systems (6) and (7). Thus,

    ()

  •  

    for all 0 ≤ ε ≤ ς ≤ T.

By (a), there are Mℸ > 0 and , such that
()
Moreover,
()
can be extended to where
()
Then, there exists Lℸ > 0 where
()
Let the state space be a seminorm linear space of functions mapping (−∞, 0] into , and verifying (see [33]):
  •  

    (A1) If ϑ ∈ C and , then for ς ∈ Θ:

    •  

      (i)

    •  

      (ii) There exists H > 0 where

    •  

      (iii) There exist Φ1(⋅) and with Φ1 continuous and bounded and Φ2 locally bounded where

      ()

  •  

    (A2) For the function ϑ in (A1), ϑς is a -valued continuous function on .

  •  

    (A3) The space is complete. We denote

For , we define the space
()
and the space
()

In the following, consider .

Lemma 2 (see [34].)Let the following inequality holds:

()
where
()
a 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 ϑ∗ 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 operator

Then, 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 by

()
where
()

Lemma 6 ([37]). If is a bounded subset of a Banach space , then for each ϵ > 0, 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 ς ∈ Θ where .

  • (ii)

    If is piecewise equicontinuous on Θ, then is piecewise continuous for ς ∈ Θ, and

    ()

  • (iii)

    If is bounded and piecewise equicontinuous, then is piecewise continuous for ς ∈ Θ and

    ()

  •  

    where αPC denotes the Kuratowski measure of noncompactness in the space .

Theorem 9 (see [39].)Let Δ be a nonempty, bounded, closed, and convex subset of a Banach space and let be a continuous mapping. Assume that there exists a constant k ∈ [0, 1), such that

()
for any nonempty subset M of Δ. Then, has a fixed point in set Δ.

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

    (C1) is a Carathéodory function, and there exist positive constants ξ1, ξ2 and continuous nondecreasing functions such that

    ()

  •  

    for . There exists a positive constant lΨ, such that for any bounded set and , and each , we have

    ()

  •  

    with

    ()

  •  

    (C2) The function is continuous, and there exists , such that

    ()

  •  

    Let

    ()

  •  

    (C3) Assume that (B1) − (B3) hold, and there exist , ζ ≥ 0, and , such that

    ()

  •  

    (C4) The functions are continuous, and there exist positive constants , such that

    ()

  •  

    and

    ()

  •  

    where

    ()

  •  

    (CH) Set . We assume that is continuous. Moreover, we assume the following assumption:

    • (i)

      The function is continuous from ℸ(ρ−) 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 , then

()
where .

Now, we define a measure of noncompactness in the space . Let us fix a nonempty bounded subset S of the space and . Then, for v ∈ S, ϵ > 0, κ1, κ2 ∈ ×, such that |κ1 − κ2| ≤ ϵ, we denote ωT(v, ϵ) the modulus of continuity of the function v on ×, namely,
()
Consider the function χPC defined on the family of subset of by
()
with and .

The function χPC 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 PC, the reader is referred to [43].

Theorem 14. Suppose that (C1) − (C4) and (CH) are verified. Then, (1) has at least one mild solution.

Proof. We transform problem (1) into a fixed-point problem and define the operator by

()

Let be the function defined by

()

Then, , and for each , with w(0) = 0, we denote the function by

()

If ϑ satisfies (3), we can decompose it as ϑ(ς) = w(ς) + x(ς), which implies ϑς = wς + xς, and the function w(⋅) satisfies

()

Set

()

Let the operator be defined by

()

The 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 }, with

()
such that are constants, they will be specific later.

The set is bounded, closed, and convex.

  •  

    Step 1: .

  •  

    For , ς ∈ Θ and by (C1) − (C3), we have

    ()

  •  

    and

    ()

  •  

    Then,

    ()

  •  

    Thus,

    ()

  •  

    Therefore, , which implies that is bounded.

  •  

    Step 2: is continuous.

  •  

    Let be a sequence such that wm⟶ℷ∗ in At the first, we study the convergence of the sequences . If ε ∈ Θ is such that ρ(ε, wε) > 0, then we have

    ()

  •  

    which proves that in , as m⟶∞, for ε ∈ Θ where ρ(ε, wε) > 0. If ρ(ε, wε) < 0, we get

    ()

  •  

    which also shows that in , as m⟶∞, for every ε ∈ Θ such that ρ(ε, wε) < 0. Then, for ς ∈ Θ, we have

    ()

  •  

    Since and Ψ are continuous, we obtain

    ()

  •  

    and

    ()

  •  

    By the Lebesgue-dominated convergence theorem,

    ()

  •  

    Then, by (C1), we get

    ()

  •  

    Since are continuous, by the Lebesgue-dominated convergence theorem, we obtain

    ()

  •  

    Thus, is continuous.

  •  

    Step 3: is ζC-contraction.

  •  

    Let Π be a bounded equicontinuous subset of , w ∈ Π, and κ1, κ2 ∈ Θ, with κ2 > κ1, we have

    ()

By the strong continuity of ℸ, we get

()

Thus, is equicontinuous; then, .

Now, for w ∈ Π, and for any ϱ > 0, there exists a sequence such that for ς ∈ Θ, we have

()

Since ϱ is arbitrary, we get

()

Thus,

()

By Theorem 9, it follows that there exists at least one fixed point ℷ∗ within . Consequently, the point ℷ∗ + x 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 Ψ ≡ 0, 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 [0, T]. 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)

    [(C5)] Linear system (4) is approximately controllable

  • (2)

    [(C6)] Range of the operator ℵ is a subset of the closure of range of , i.e.,

()

Theorem 17. If hypotheses (C1) − (C6) are verified, then system (1) is approximately controllable.

Proof. The mild solution of system (4) corresponding to the control v is given by

()

Assume the following system:

()

Since , there exists a control function u ∈ L2(Θ, U) such that

()

Now, assume that ϑ is the mild solution of system (1) corresponding to (v − u) given by

()

Then, if , we get ‖ȷ(ς) − ϑ(ς)‖ = 0.

And, for ς ∈ Θ, we have

()

Now, for any , we define the function Ϝ(ς) = supε∈[0, ς]‖ȷ(ε) − ϑ(ε)‖, and from the definition of the function ρ and Lemma 13, we obtain

()

Then,

()

Therefore, according to Lemma 2, we get

()

By taking suitable control function u, we make ‖ȷ(ς) − ϑ(ς)‖ arbitrary small. Therefore, the reachable set of (4) is dense in the reachable set of (70), which is dense in due to (C5). Hence, the approximate controllability of (70) implies that of the semilinear control system (4).

5. Ulam–Hyers–Rassias Stability Results

Let ∇1, ∇2, ∇3 ≥ 0 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 (ν, ∇1, ∇2, ∇3), if there exists θΨ,∇,ν > 0, 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

  •  

    (a1), and ,

  •  

    (a2),

  •  

    (a3),

  •  

    (a4),

  •  

    (a5).

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

()

We also need the following additional assumption to discuss about stability:(Cν). We assume that for a nondecreasing function , there exists cν > 0, such that
()

Theorem 21. If (C1) − (C4), (CH), and (Cν) are satisfied, with

()
then, equation (1) is generalized U-H-R stable with respect to (ν, ∇1, ∇2, ∇3).

Proof. Let ȷ be a solution of (6) and be the mild solution of (1) with and ȷ′(0) = ȷ′(0) = ν0.

Then, we get

()

On the other hand, we get

()

Hence, for , we have

()

Let , and

()

For , let

()

Now, we will prove that is a Picard operator. For that, let and , if , we get , and if , we have

()

Therefore, is a contraction; hence, from Theorem 10, there exists a unique ℷ∗ in , and from Definition 3, we deduce that is a Picard operator.

Furthermore, we have

()

We can see that ℷ∗ is an increasing function and is nonnegative.

So, for , we have

()

Then,

()

From Lemma 2, we get

()

In particular, if , then we have , and applying the abstract Gronwall lemma, we obtain ℷ(ς) ≤ ℷ∗(ς). It follows that

()

Now, if , we get

()

Then, if we put

()

Thus, we have for all

()
which implies that (4) is generalized U-H-R stable with respect to (ν, ∇1, ∇2, ∇3).

6. An Example

Consider the following class of partial integrodifferential system:
()
where , , αk, βk ∈ (0, e−π/17).
Let
()
be the Hilbert space with the scalar product , and the norm
()
and the phase space be , the space of bounded uniformly continuous functions endowed with the following norm: . It is well known that satisfies the axioms (A1) and (A2) with K = 1 and Φ1(ς) = Φ2(ς) = 1 (see [41]). We define induced on as
()
Then, is the infinitesimal generator of a cosine function of operators on H associated with sine function . In addition, has discrete spectrum which consists of eigenvalues −n2 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, ‖C0(ς)‖ ≤ 1 and S0(ς) is compact for all . We define on . Clearly, is a closed linear operator. Therefore, generates (S(ς, ε))(ς, ε)∈∇ such that S(ς, ε) is compact and self-adjoint for all (ς, ε) ∈ ∇ = {(ς, ε) : 0 ≤ ε ≤ ς ≤ 1} (see [26]).

We define the operators as follows:
()

The assumption (C4) holds under more suitable conditions on the operator Γ. Furthermore, (B1) − (B3) 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 ℸ(ρ−).

We put ν(ς)(x) = ν(ς, x), for ς ∈ [0, 1], and define
()

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

Now, for ς ∈ [0, 1], we have
()
So, ψi+1(ς) = ς/1 + i;  i = 0, 1 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.

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.

Data Availability

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

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