Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 54, pp. 1–18. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.54 OPTIMAL CONTROL AND APPROXIMATE CONTROLLABILITY FOR SECOND-ORDER INTEGRO-DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAY AND NON-INSTANTANEOUS IMPULSES ABDELHAMID BENSALEM, ABDELKRIM SALIM, MOUFFAK BENCHOHRA, GASTON M. N’GUÉRÉKATA Abstract. This article concerns the optimal control and the approximate controllability for second order integro-differential equation with state-dependent delay and non-instantaneous impulses. We first establish the existence of mild solution for the control system. Then, based on these results, we investigate the approximate controllability and show the existence of optimal controls for Bolza problems by using the resolvent family of linear operators, Mönch’s fixed point theorem, and the resolvent condition. Finally, we give an example to illustrate the effectiveness of the results. 1. Introduction Numerous physical phenomena, such as shocks and natural disasters, exhibit dynamics that are susceptible to sudden alterations. These phenomena involve brief disturbances that are negligible in magnitude when contrasted with the overall course of the evolution. Occasionally, these impul- sive effects persist for extended periods, and they are referred to as non-instantaneous impulses. The publications [1, 6, 7, 8, 13, 22, 36, 39] and their associated references contain the latest findings on evolution equations with impulses. A multitude of natural phenomena spanning diverse fields, such as electronics, fluid dynamics, biological models, and chemical kinetics, can be mathematically modeled using integro-differential equations. Conventional differential equations are typically inadequate for explaining the behavior of the majority of these phenomena, thereby piquing the interest of a large number of mathemati- cians, physicists, and engineers, as evidenced in [11, 12, 17, 18, 22, 32]. Numerous publications have investigated integro-differential systems with Υ(θ, ε) = 0 using semigroup methods. Fur- ther details can be found in the aforementioned references. Benchohra et al. [9] have shown existence of solution for second order semilinear volterra-type integro-differential equations with non-instantaneous impulses: ϑ′′(θ) = A(θ)ϑ(θ) +K(θ, ϑ, (Ψϑ)(θ)), if θ ∈ Ik, k ∈ Nm 0 , ϑ(θ) = Υk(θ, ϑ(θ − k )), if θ ∈ Jk, k ∈ Nm 1 , ϑ′(θ) = Θk(θ, ϑ(θ − k )), if θ ∈ Jk, k ∈ Nm 1 , ϑ′(0) = ζ0 ∈ H, ϑ′(0) = ζ1 ∈ H. The concept of controllability has long been recognized as having a significant role in engi- neering and mathematical control theory. In recent years, numerous authors have investigated the controllability of various nonlinear systems. Interested readers may refer to the papers [5, 10, 12, 31, 33, 37] for further study on this topic. When a system is deemed controllable, 2020 Mathematics Subject Classification. 93B05, 49N25, 45J05, 47H08, 34K45. Key words and phrases. Approximate controllability; optimal control; fixed point theorem; infinite delay; second-order integrodifferential equation; impulsive; measures of noncompactness; mild solution; resolvent operator. ©2025. This work is licensed under a CC BY 4.0 license. Submitted March 3, 2025. Published May 26, 2025. 1 2 A. BENSALEM, A. SALIM, M. BENCHOHRA, G. N’GUÉRÉKATA EJDE-2025/54 the question of how to obtain a control function that produces more, faster, better, and more cost-effective outcomes naturally arises. To address this, an optimal control problem is proposed. Optimal control problems play a crucial role in the design and analysis of control systems, and they have numerous applications in diverse fields, including robotics, chemical process control, power plants, and space technology. For additional information on optimal control problems, please refer to [20, 29, 30] and their cited references. This manuscript is devoted to the study of the existence and the approximate controllability of mild solutions, as well as the existence of optimal controls for second-order integro-differential equations with state-dependent delay and non-instantaneous impulses of the form ϑ′′(θ) = Z(θ)ϑ(θ) + ∫ θ 0 Υ(θ, ε)ϑ(ε)dε+K(θ, ϑℑ(θ,ϑθ), (Ψϑ)(θ)) + Pu(θ), if θ ∈ Ik, k ∈ Nm 0 , ϑ(θ) = Υk(θ, ϑ(θ − k )), if θ ∈ Jk, k ∈ Nm 1 , ϑ′(θ) = Θk(θ, ϑ(θ − k )), if θ ∈ Jk, k ∈ Nm 1 , ϑ′(0) = ζ0 ∈ H, ϑ(θ) = ℘(θ), if θ ∈ R−, (1.1) where I0 = [0, θ1], Ik = (εk, θk+1] and Jk = (θk, εk], N m 1 = {1, . . . ,m}, and Nm 0 = Nm 1 ∪ {0} with 0 = ε0 < θ1 ≤ ε1 ≤ θ2 < . . . < εm−1 ≤ θm ≤ εm ≤ θm+1 = T , ∇ = [0, T ], ∇̃ = (−∞, T ], Z(θ) : D(Z(θ)) ⊂ H → H, Υ(θ, ε) are closed linear operators on H, with dense domain D(Z(θ)), which is independent of θ, and D(Z(ε)) ⊂ D(Υ(θ, ε)), the operator Ψ is defined by (Ψϑ)(θ) = ∫ T 0 g(θ, ε, ϑ(ε))dε. The nonlinear termK : ∇×G×H → H, ℘ : R− → H, ℑ : ∇×G → (−∞,∞), Υk,Θk : Jk×H → H, k ∈ Nm 1 , are a given functions, the control function u is give function in L2(∇,Y) Banach space of admissible control with Y as a Banach space. P is a bounded linear operator from Y into H, and (H, ∥ · ∥) is a Banach space. This article is organized as follows: in Section 2, we recall the notation, some concepts, hy- potheses, and basic results about resolvent operator theory, phase space, and measure of noncom- pactness. In Section 3, we prove the existence of a mild solution for problem (1.1). In section 4, we investigate the approximate controllability result. We show the existence of optimal controls in section 5. Finally, an example is provided to show the applications of the obtained results. 2. Preliminaries Let C(∇,H) be the Banach space of continuous functions ϑ mapping ∇ := [0, T ] into H, with ∥ϑ∥∞ = sup θ∈∇ ∥ϑ(θ)∥. A measurable function ϑ : ∇ → H is Bochner integrable if and only if ∥ϑ∥ is Lebesgue integrable [40]. Let L1(∇,H) be the Banach space of measurable functions ϑ : ∇ → H which are Bochner integrable, with the norm ∥ϑ∥L1 = ∫ T 0 ∥ϑ(θ)∥dθ. Now, we consider the second-order integro-differential system [23]: κ′′(θ) = Z(θ)κ(θ) + ∫ θ ε Υ(θ, ν)κ(ν)dν, ε ≤ θ ≤ T, κ(ε) = 0, κ′(ε) = x ∈ H, (2.1) for 0 ≤ ε ≤ T . We denote ∆ = {(θ, ε) : 0 ≤ ε ≤ θ ≤ T}. We now present some properties of Υ: (1) For 0 ≤ ε ≤ θ ≤ T , Υ(θ, ε) : D(Z) → H is a bounded linear operator, for every κ ∈ D(Z),Υ(·, ·)κ is continuous and ∥Υ(θ, ε)κ∥ ≤ β∥κ∥[D(Z)], EJDE-2025/54 SECOND-ORDER IMPULSIVE INTEGRO-DIFFERENTIAL EQUATIONS 3 for β > 0 independent of (ε, θ) ∈ ∆. (2) There exists LΥ > 0 such that ∥Υ(θ2, ε)κ −Υ(θ1, ε)κ∥ ≤ LΥ |θ2 − θ1| ∥κ∥[D(Z)], for all κ ∈ D(Z) and 0 ≤ ε ≤ θ1 ≤ θ2 ≤ T . (3) For 0 ≤ σ ≤ ε ≤ θ ≤ T , there exists b1 > 0 such that ∥ ∫ θ σ S(θ, ε)Υ(ε, σ)κdε∥ ≤ b1∥κ∥, for all κ ∈ D(Z). Under these conditions, it has been established that there exists a resolvent operator (Q(θ, ε))θ≥ε associated with (2.1). Definition 2.1 ([23]). A family of bounded linear operators (Q(θ, ε))θ≥ε on H is a resolvent operator for (2.1) if it satisfies: (a) Q : ∆ → L(H) is strongly continuous, Q(θ, ·)κ is continuously differentiable for all κ ∈ H, Q(ε, ε) = 0, ∂ ∂θ Q(θ, ε) ∣∣ θ=ε = I and ∂ ∂ε Q(θ, ε) ∣∣ ε=θ = −I; (b) For each x ∈ D(Z), the function Q(·, ε)x is a solution for system (2.1). This means ∂2 ∂θ2 Q(θ, ε)x = Z(θ)Q(θ, ε)x+ ∫ θ ε Υ(θ, ν)Q(ν, ε)xdν, for all 0 ≤ ε ≤ θ ≤ T . Thus, there are constatns MQ > 0 and M̃Q > 0 such that ∥Q(θ, ε)∥ ≤MQ, ∥ ∂ ∂ε Q(θ, ε)∥ ≤ M̃Q, (θ, ε) ∈ ∆. Furthermore, P(θ, ν)x = ∫ θ ν Υ(θ, ε)Q(ε, ν)xdε, x ∈ D(Z), 0 ≤ ν ≤ θ ≤ T, can be extended to H. This expansion, denoted by similar notation P(θ, ν),P : ∆ → L(H), is strongly continuous, which is satisfied by Q(θ, ν)x = S(θ, ν)x+ ∫ θ ν S(θ, ε)P(ε, ν)xdε, for all x ∈ H. It follows from this property that Q(·) is uniformly Lipschitz continuous, that is, there exists a constant LQ > 0 such that ∥Q(θ + h, ν)−Q(θ, ν)∥ ≤ LQ|h|, for all θ, θ + h, ν ∈ [0, T ]. Lemma 2.2. Let N1 : Lq(∇,H) → C(∇,H), (q > 1) be defined by (N1f)(θ) = ∫ θ 0 Q(θ, ε)f(ε)dε. If the resolvent operator (Q(θ, ε))θ≥ε is compact then fn w→ f0 in Lq(∇,H) implies N1fn ε→ N1f0 in C(∇,H), and N1 is a strongly continuous operator. The proof of this lemma is similar to that of Lemma 14 in [15]. We omit it. Assume that the phase space (G, ∥ · ∥G) is a seminormed linear space of functions mapping (−∞, 0] into R, and satisfying the following [21]: (A1) If ϑ ∈ PC and ϑ0 ∈ G, then for every θ ∈ ∇: (i) ϑθ ∈ G, (ii) There exists β1 > 0 where |ϑ(θ)| ≤ β1∥ϑθ∥G, (iii) There exist two functions β2(·) and β3(·) : R+ → R+ independent of ϑ with β2 continuous and bounded and β3 locally bounded such that ∥ϑθ∥G ≤ β2(θ) sup{|ϑ(ε)| : 0 ≤ ε ≤ θ}+ β3(θ)∥ϑ0∥G. 4 A. BENSALEM, A. SALIM, M. BENCHOHRA, G. N’GUÉRÉKATA EJDE-2025/54 (A2) For the function ϑ in (A1), ϑθ is a G - valued continuous function on R+ \ Jk. (A3) The space G is complete. We denote β2 ∗ = sup{β2(θ) : θ ∈ ∇}, β3∗ = sup{β3(θ) : θ ∈ ∇}, ℧ = max{β2∗, β3∗}. Now, let p ∈ Nm 0 and (νk)k∈Nm 1 be a sequence defined by νk = { νp+1 − θ, if k = 2p+ 1, θ ∈ R−, εp − θ, if k = 2p, θ ∈ R−. Then, for Iν = R− \ {νk : k ∈ Nm 1 }, we define the space PCν(R−,H) = { ϑ : R− → H : ϑ|Iν is continuous and ϑ(ν−k ), ϑ(ν+k ) exist with ϑ(ν−k ) = ϑ(νk) } , and the space Cν := { ג ∈ PCν(R−,H) : lim ν→−∞ (ν)ג exist in H } , with ν∥ג∥ = sup{|ג(ν)| : ν ≤ 0}. Then, (A1)–(A3) are satisfied in Cν . So in all what follows, we consider the phase space G = Cν . Consider the set PC(∇̃,H) = { ϑ : ∇̃ → H : ϑ|R− ∈ G, ϑ|Jk = Υk; k ∈ Nm 1 , ϑ|Ik ∈ C(Ik,H); k ∈ Nm 0 , ϑ(θ−k ), ϑ(ε − k ), ϑ(ε + k ) and ϑ(θ + k ) exist with ϑ(θ − k ) = ϑ(θk)and ϑ(ε − k ) = ϑ(εk) } , with ∥ϑ∥PC = sup θ∈∇̃ {∥ϑ(θ)∥}. Definition 2.3 ([2]). LetX be a Banach space and ℶX the bounded subsets ofX. The Kuratowski measure of noncompactness is the map χ : ℶX → [0,∞) defined by χ(B) = inf{ϵ > 0 : B ⊆ ∪n i=1Bi and diam(Bi) ≤ ϵ}; where B ∈ ℶX , where diam(Bi) = sup{∥ϑ− v∥X : ϑ, v ∈ Bi}. Lemma 2.4 ([16]). If Y is a bounded subset of a Banach space X, then for each ϵ > 0, there is a sequence {ϑk}∞k=1 ⊂ Y such that χ(Y ) ≤ 2χ({ϑk}∞k=1) + ϵ. Lemma 2.5. ([34]) If {ϑk}∞k=0 ⊂ L1 is uniformly integrable, then the function θ → χ({ϑk(θ)}∞k=0) is measurable and χ ({∫ θ 0 ϑk(ε)dε }∞ k=0 ) ≤ 2 ∫ θ 0 χ({ϑk(ε)}∞k=0)dε. Lemma 2.6 ([2]). If U ⊂ PC(∇;H) is bounded, then χ(U(θ)) ≤ αPC(U), for all θ ∈ ∇; here U(θ) = {ϑ(θ);ϑ ∈ U ⊂ H}. Furthermore if U is equicontinuous on ∇, then χ(U(θ)) is continuous on ∇ and αPC(U) = sup θ∈∇ χ(U(θ)). Theorem 2.7 (Mönch’s fixed point theorem [34]). Let D be a bounded, closed and convex subset of a Banach space X, such that 0 ∈ D, and let U be a continuous mapping of D into itself. If the implication M = convU(M) or M = U(M) ∪ {0} ⇒ χ(M) = 0, holds for every subset M of D, then U has a fixed point. EJDE-2025/54 SECOND-ORDER IMPULSIVE INTEGRO-DIFFERENTIAL EQUATIONS 5 Lemma 2.8 ([4]). Let ϑ(θ) and β(θ) be nonnegative continuous function for θ ≥ α, and let ϑ(θ) ≤ a+ ∫ θ α β(ε)ϑ(ε)dε θ ≥ α, where a ≥ 0 is a constant. Then ϑ(θ) ≤ a exp (∫ θ α β(ε)dε ) , θ ≥ α. 3. Existence of mild solution Let us recall the following special measure of noncompactness on the space X = PC(∇̃,H) which originates from [2], and will be used in our main results. For a nonempty bounded subset S of the space X , and 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 the interval ∇̃, namely, ωT (v, ϵ) = sup{∥e−κ1v(κ1)− e−κ2v(κ2)∥ : κ1, κ2 ∈ ∇̃}, ω0(S) = lim ϵ→0 sup{ωT (v, ϵ) : v ∈ S}. Finally, consider the function χPC defined on the family of subset of X by the formula χPC(S) = ω0(S) + χ(S(θ)), where S(θ) = {ϑ(θ) ∈ H : ϑ ∈ S}. Note that the function χPC is a sublinear measure of noncompactness on the space X . In contrast to the advancements presented in [19, 23, 24], we introduce a new notion of a mild solution for system (1.1). Definition 3.1. A function ϑ ∈ X is called a mild solution of problem (1.1), if the following hold: (i) ϑ′(0) = ζ0 ∈ H and ϑ(θ) = ℘(θ); if θ ∈ R−. (ii) The non-instantaneous conditions ϑ(θ) = Υk(θ, ϑ(θ − k )), if θ ∈ Jk, k ∈ Nm 1 and ϑ′(θ) = Θk(θ, ϑ(θ − k )), if θ ∈ Jk, k ∈ Nm 1 are satsified (iii) ϑ is the solution of the integral equations ϑ(θ) =  −∂Q(θ,ε) ∂ε ∣∣ ε=0 ℘(0) +Q(θ, 0)ζ0 + ∫ θ 0 Q(θ, ε)(K(ε, ϑℑ(ε,ϑε), (Ψϑ)(ε)) + Pu(ε))dε, if θ ∈ I0, −∂Q(θ,ε) ∂ε ∣∣ ε=εk Υk(εk, ϑ(θ − k )) +Q(θ, εk)Θk(εk, ϑ(θ − k )) + ∫ θ εk Q(θ, ε)(K(ε, ϑℑ(ε,ϑε), (Ψϑ)(ε)) + Pu(ε))dε, if θ ∈ Ik, k ∈ Nm 1 . To guarantee the existence of mild solutions, we need the following assumptions: (A4) K : ∇×G×H → H is a Carathéodory function and there exist positive constants ξ1, ξ2 and continuous nondecreasing functions ψ1 K, ψ 2 K : ∇ → (0,+∞) such that ∥K(θ, ϑ1, ϑ2)∥ ≤ ξ1ψ 1 K(∥ϑ1∥G) + ξ2ψ 2 K(∥ϑ2∥), for ϑ1 ∈ G, ϑ2 ∈ H. There exists a positive constant lK, such that for any bounded set B ⊂ H, and Bθ ∈ G and each θ ∈ ∇, we have χ(K(θ,Bθ,Ψ(B(θ)))) ≤ lK ( χ(B(θ)) + sup ν∈(−∞,0] χ(B(ν + θ)) ) . (A5) The function g : Dg ×H → H is continuous and there exists αg > 0, such that ∥g(θ, ε, ϑ1)− g(θ, ε, ϑ2)∥ ≤ αg∥ϑ1 − ϑ2∥, for each (θ, ε) ∈ Dg and ϑ1, ϑ2 ∈ H, with sup Dg ∥g(θ, ε, 0)∥ = g∗0 <∞. 6 A. BENSALEM, A. SALIM, M. BENCHOHRA, G. N’GUÉRÉKATA EJDE-2025/54 (A6) The functions Zi k : Jk × H → H are continuous and there exist LZi k > 0, k ∈ Nm 1 , such that ∥Zi k(θ, ϑ1)− Zi k(θ, ϑ2)∥ ≤ LZi k ∥ϑ1 − ϑ2∥, for all ϑ1, ϑ2 ∈ H, k ∈ Nm 1 , Zi,0 k = ∥Zi k(θ, 0)∥, max k∈Nm 1 {LZi k , k ∈ Nm 1 } = L∗ Zi k < +∞, where Zi k = { Θk, i = 1, Υk, i = 2. (A7) Assume that properties (1)-(2) of Υ hold, and that there exist MQ, M̃Q ≥ 1, µ ≥ 0 and MP > 0, such that ∥Q(θ, ε)∥Υ(H) ≤MQ, ∥∂Q(θ, ε) ∂ε ∥Υ(H) ≤ M̃Q, ∥P∥ =MP , with M̃QLΥk +MQLΘk < 1. (A8) Set R(ℑ−) = {ℑ(ε, φ) : (ε, φ) ∈ ∇ ×G,ℑ(ε, φ) ≤ 0}. We assume that ℑ : ∇×G → R is continuous. (A9) The function θ → ℘θ is continuous from R(ℑ−) into G and there exists a continuous and bounded function L℘ : R(ℑ−) → (0,∞) such that ∥℘θ∥G ≤ L℘(θ)∥℘∥G, for every θ ∈ R(ℑ−). The condition (A9) is frequently satisfied by functions continuous and bounded. For more details, see for instance [26]. Lemma 3.2 ([25]). If ϑ : (−∞,+∞) → H is a function such that ϑ0 = ℘, then ∥ϑε∥G ≤ (β3 ∗ + L℘)∥℘∥G + β2 ∗ sup{|ϑ(θ)| : θ ∈ [0,max{0, ε}]}, ε ∈ R(ℑ−) ∪∇, where L℘ = supς∈R(ℑ−) L℘(ς). Theorem 3.3. Assume that the conditions (A4)–(A9) are satisfied, then the system (1.1) has at least one mild solution. Proof. We transform problem (1.1) into a fixed point problem, by considering the operator Ξ : X → X define by: Ξϑ(θ) =  −∂Q(θ,ε) ∂ε ∣∣ ε=0 ℘(0) +Q(θ, 0)ζ0 + ∫ θ 0 Q(θ, ε) ( K(ε, ϑℑ(ε,ϑε), (Ψϑ)(ε)) + Pu(ε) ) dε, if θ ∈ I0, −∂Q(θ,ε) ∂ε ∣∣ ε=εk Υk(εk, ϑ(θ − k )) +Q(θ, εk)Θk(εk, ϑ(θ − k )) + ∫ θ εk Q(θ, ε) ( K(ε, ϑℑ(ε,ϑε), (Ψϑ)(ε)) + Pu(ε) ) dε if θ ∈ Ik, k ∈ Nm 1 , Υk(θ, ϑ(θ − k )), if θ ∈ Jk, k ∈ Nm 1 , ℘(θ), if θ ∈ R−. (3.1) Let x(·) : (−∞, T ] → H be defined by x(θ) =  −∂Q(θ,ε) ∂ε ∣∣ ε=0 ℘(0) +Q(θ, 0)ζ0, if θ ∈ I0, 0, if θ ∈ (θ1, T ], ℘(θ), if θ ∈ R−. Then x0 = ℘, and for each ϖ ∈ X , with ϖ(0) = 0, we denote by ϖ the function ϖ(θ) = { ϖ(θ), if θ ∈ ∇, 0, if θ ∈ R−. EJDE-2025/54 SECOND-ORDER IMPULSIVE INTEGRO-DIFFERENTIAL EQUATIONS 7 If ϑ satisfies Definition 3.1, then we can decompose it as ϑ(θ) = ϖ(θ) + x(θ), which implies ϑθ = ϖθ + xθ, and the function ϖ(·) satisfies ϖ(θ) =  ∫ θ 0 Q(θ, ε) ( K(ε,ϖℑ(ε,ϖε+xε) + xℑ(ε,ϖε+xε),Ψ(ϖ + x)(ε)) + Pu(ε) ) dε, if θ ∈ I0, −∂Q(θ,ε) ∂ε ∣∣ ε=εk Υk(εk, ϑ(θ − k )) +Q(θ, εk)Θk(εk, ϑ(θ − k )) + ∫ θ εk Q(θ, ε) ( K(ε, ϑℑ(ε,ϑε), (Ψϑ)(ε)) + Pu(ε) ) dε, if θ ∈ Ik, k ∈ Nm 1 , Υk(θ, ϑ(θ − k )), if θ ∈ Jk, k ∈ Nm 1 . (3.2) Set ℶ = {ϖ ∈ X : ϖ(0) = 0}. Let the operator Ξ̂ : ℶ → ℶ defined by Ξ̂ϖ(θ) =  ∫ θ 0 Q(θ, ε) ( K(ε,ϖℑ(ε,ϖε+xε) + xℑ(ε,ϖε+xε),Ψ(ϖ + x)(ε)) + Pu(ε) ) dε, if θ ∈ I0, −∂Q(θ,ε) ∂ε ∣∣ ε=εk Υk(εk, ϑ(θ − k )) +Q(θ, εk)Θk(εk, ϑ(θ − k )) + ∫ θ εk Q(θ, ε) ( K(ε, ϑℑ(ε,ϑε), (Ψϑ)(ε)) + Pu(ε) ) dε, if θ ∈ Ik, k ∈ Nm 1 , Υk(θ, ϑ(θ − k )), if θ ∈ Jk, k ∈ Nm 1 . Obviously, the operator Ξ has a fixed point is equivalent to Ξ̂ having a fixed point, and so we turn to proving that Ξ̂ has a fixed point. We shall use Theorem 2.7 to prove that Ξ̂ has a fixed point. Let ℶℑ = {ϖ ∈ ℶ : ∥ϖ∥ℶ ≤ ℑ}, with 0 < max { ℑ∗ 1,ℑ∗ 2,ℑ∗ 3 } ≤ ℑ, such that ℑ∗ 1 =MQ ( ξ1Tψ 1 K(δ ∗ 1) + ξ2Tψ 2 K(δ ∗ 2) +MPT 1/2∥u∥L2 ) , ℑ∗ 2 = M̃QΥ 0 k +MQ(Θ 0 k + ξ1Tψ 1 K(δ̃ ∗ 1) + ξ2Tψ 2 K(δ̃ ∗ 2) +MPT 1/2∥u∥L2) 1− M̃QL∗ Υk −MQL∗ Θk , ℑ∗ 3 = L∗ Υk ℑ+Υ0 k, δ∗1 = β2 ∗ℑ+ [ β3 ∗ + L℘ + β2 ∗(M̃R∥℘0∥+MR∥ζ0∥ ) β1 ] ∥℘∥B, δ∗2 = T (αg(ℑ++M̃R∥℘0∥+MR∥ζ0∥) + g∗0), δ̃∗2 = (αgℑ+ g∗0)T, δ̃∗1 = ℧(ℑ+ ∥℘∥G). The set ℶℑ is bounded, closed, and convex. Step 1. Ξ̂(ℶℑ) ⊂ ℶℑ. For θ ∈ I0, ϖ ∈ ℶℑ and from (A4)–(A7), it follows that ∥wℑ(ε,wε+xε) + xℑ(ε,wε+xε)∥B ≤ ∥wℑ(ε,wε+xε)∥B + ∥xℑ(ε,wε+xε)∥B ≤ β2(θ) sup [0,ε] |w(θ)|+ ( β3(θ) + L℘ ) ∥℘∥B + β2(θ) sup [0,ε] ∥x(θ)∥ ≤ β2 ∗ℑ+ ( β3 ∗ + L℘ ) ∥℘∥B + β2 ∗(M̃R∥℘0∥+MR∥ζ0∥ ) β1∥℘∥B ≤ β2 ∗ℑ+ [ β3 ∗ + L℘ + β2 ∗(M̃R∥℘0∥+MR∥ζ0∥ ) β1 ] ∥℘∥B = δ∗1 and ∥Ψ(ϖ + x)(ε)∥ ≤ T (αg(ℑ++M̃R∥℘0∥+MR∥ζ0∥) + g∗0) = δ∗2 . Then, we have ∥Ξ̂ϖ(θ)∥ ≤MQ ∫ θ 0 (ξ1ψ 1 K(δ ∗ 1) + ξ2ψ 2 K(δ ∗ 2) + ∥Pu(ε)∥)dε ≤MQ(ξ1Tψ 1 K(δ ∗ 1) + ξ2Tψ 2 K(δ ∗ 2) +MPT 1/2∥u∥L2) ≤ ℑ. 8 A. BENSALEM, A. SALIM, M. BENCHOHRA, G. N’GUÉRÉKATA EJDE-2025/54 Now if θ ∈ Ik and for each ϖ ∈ ℶℑ, by (A4)–(A6), we obtain ∥Υk(θ,ϖ(·))∥ ≤ LΥk (θ)∥ϖ(θ)∥+Υ0 k and ∥Θk(θ,ϖ(.))∥ ≤ LΘk (θ)∥ϖ(θ)∥+Θ0 k. Hence, for δ̃∗2 = (αgℑ+ g∗0)T and δ̃∗1 = ℧(ℑ+ ∥℘∥G), we obtain ∥Ξ̂ϖ(θ)∥ ≤ M̃Q(L ∗ Υk ℑ+Υ0 k) +MQ [ L∗ Θk ℑ+Θ0 k + ξ1Tψ 1 K(δ̃ ∗ 1) + ξ2Tψ 2 K(δ̃ ∗ 2) +MPT 1/2∥u∥L2 ] ≤ ℑ. If θ ∈ Jk and for each ϖ ∈ ℶℑ, from (A6), we obtain ∥Ξ̂ϖ(θ)∥ ≤ L∗ Υk ℑ+Υ0 k ≤ ℑ. Thus, ∥Ξ̂ϖ∥ℶ ≤ ℑ. Consequently, Ξ̂(ℶℑ) ⊂ ℶℑ and Ξ̂(ℶℑ) is bounded. Step 2. Ξ̂ is continuous. Let {ϖn}n∈N be a sequence, such that ϖn → ϖ∗. At the first, we study the convergence of the sequences (ϖn ℑ(ε,ϖn ε ))n∈N, ε ∈ ∇. If ε ∈ ∇ is such that ℑ(ε,ϖε) > 0, then we have ∥ϖn ℑ(ε,ϖn ε ) −ϖ∗ ℑ(ε,ϖ∗ ε ) ∥B ≤ ∥wn ℑ(ε,wn ε ) −ϖ∗ ℑ(ε,ϖn ε )∥B + ∥ϖ∗ ℑ(ε,ϖn ε ) −ϖ∗ ℑ(ε,ϖ∗ ε ) ∥B ≤ β2 ∗∥ϖn −ϖ∗∥+ ∥ϖ∗ ℑ(ε,ϖn ε ) −ϖ∗ ℑ(ε,ϖ∗ ε ) ∥B, which proves that ϖn ℑ(ε,ϖn ε ) → ϖ∗ ℑ(ε,ϖε) in B, as n→ ∞, for every ε ∈ ∇ such that ℑ(ε,ϖε) > 0. Similarly, if ℑ(ε,ϖε) < 0, we obtain ∥ϖn ℑ(ε,ϖn ε ) −ϖ∗ ℑ(ε,ϖε) ∥B = ∥℘n ℑ(ε,ϖn ε ) − ℘ℑ(ε,ϖ∗ ε ) ∥B = 0, which also shows that ϖn ℑ(ε,ϖn ε ) → ϖ∗ ℑ(ε,ϖ∗ ε ) in B, as n→ ∞, for every ε ∈ ∇ such that ℑ(ε,ϖε) < 0. Then for θ ∈ I0, we have ∥(Ξ̂ϖn)(θ)− (Ξ̂ϖ∗)(θ)∥ ≤MQ ∫ θ 0 ∥K(ε,ϖn ℑ(ε,ϖn ε ) + xℑ(ε,ϖn ε +xε),Ψ(ϖn + x)(ε)) −K(ε,ϖ∗ ℑ(ε,ϖ∗ ε ) + xℑ(ε,ϖ∗ ε+xε),Ψ(ϖ∗ + x)(ε))∥dε. By the continuity of g, we obtain g(θ, ε, (ϖn ε + x)(ε)) → g(θ, ε, (ϖ∗ + x)(ε)) as n→ +∞, ∥g(θ, ε, (ϖn + x)(ε))− g(θ, ε, (ϖ∗ + x)(ε))∥ ≤ αg∥ϖn −ϖ∗∥ℶ. By the Lebesgue dominated convergence theorem,∫ T 0 g(θ, ε, (ϖn + x)(ε))dε→ ∫ T 0 g(θ, ε, (ϖ∗ + x)(ε))dε, as n→ +∞. Thus, by the continuity of K, and Lebesgue dominated convergence theorem, ∥(Ξ̂ϖn)(θ)− (Ξ̂ϖ∗)(θ)∥ → 0, as n→ +∞. If θ ∈ Ik, we obtain ∥Ξ̂(ϖn)(θ)− Ξ̂(ϖ∗)(θ)∥ ≤ M̃Q∥Υk(εk, (ϖ n)(θ−k ))−Υk((εk, (ϖ ∗)(θ−k )))∥+MQ∥Θk(εk, (ϖ n)(θ−k ))−Θk((εk, (ϖ ∗)(θ−k )))∥ +MQ ∫ θ εk ∥K(ε,ϖn ℑ(ε,ϖm ε ),Ψ(ϖn)(ε))−K(ε,ϖ∗ ℑ(ε,ϖ∗ ε ) ,Ψ(ϖ∗)(ε))∥dε. Similarly, by the continuity of g, K, Υk and Θk, we obtain ∥(Ξ̂ϖn)(θ)− (Ξ̂ϖ∗)(θ)∥ → 0, as n→ +∞. Now for θ ∈ Jk, we have ∥(Ξ̂(ϖn))(θ)− Ξ̂(ϖ∗)(θ)∥ ≤ ∥Υk(θ, (ϖ n)(θ−k ))−Υk(θ, (ϖ ∗)(θ−k ))∥. By the continuity of Υk, we obtain that ∥(Ξ̂ϖn)(θ) − (Ξ̂ϖ∗)(θ)∥ → 0 as n → +∞. Thus, Ξ̂ is continuous. EJDE-2025/54 SECOND-ORDER IMPULSIVE INTEGRO-DIFFERENTIAL EQUATIONS 9 Step 3. For Π ⊂ ℶℑ, ϖ ∈ Π, and κ1, κ2 ∈ I0, with κ2 > κ1, we have ∥Ξ̂ϖ(κ1)− Ξ̂ϖ(κ2)∥ ≤ ∫ κ1 0 ∥Q(κ1, ε)−Q(κ2, ε)∥(ξ1ψ1 K(δ ∗ 1) + ξ2ψ 2 K(δ ∗ 2) + ∥Pu(ε)∥)dε + ∫ κ2 κ1 ∥Q(κ2, ε)∥(ξ1ψ1 K(δ ∗ 1) + ξ2ψ 2 K(δ ∗ 2) + ∥Pu(ε)∥)dε ≤ ∫ κ1 0 ∥Q(κ1, ε)−Q(κ2, ε)∥(ψ1 K(δ ∗ 1)ξ1 + ψ2 K(δ ∗ 2)ξ2)dε +MP( ∫ θ 0 ∥Q(κ1, ε)−Q(κ2, ε)∥2)1/2dε∥u∥L2 +MQ(ψ 1 K(δ ∗ 1)ξ1 + ψ2 K(δ ∗ 2)ξ2)(κ2 − κ1) +MQMP(κ2 − κ1) 1/2∥u∥L2 . By the strong continuity of Q(·) and assumption (A4), we obtain ∥Ξ̂ϖ(κ1)− Ξ̂ϖ(κ2)∥ → 0, as κ1 → κ2. Now for κ1, κ2 ∈ Ik, we obtain ∥Ξ̂ϖ(κ1)− Ξ̂ϖ(κ2)∥ ≤ ∥Q(κ1, εk)−Q(κ2, εk)∥∥Θk(εk, (ϖ)(θ−k ))∥ + ∥∂Q(κ1, εk) ∂ε − ∂Q(κ2, εk) ∂ε ∥ ∥Υk(εk, (ϖ)(θ−k ))∥ + ∫ κ1 εk ∥Q(κ1, ε)−Q(κ2, ε)∥ ( ξ1ψ 1 K(δ̃ ∗ 1) + ξ2ψ 2 K(δ̃ ∗ 2) + ∥Pu(ε)∥ ) dε + ∫ κ2 κ1 ∥Q(κ2, ε)∥ ( ξ1ψ 1 K(δ̃ ∗ 1) + ξ2ψ 2 K(δ̃ ∗ 2) + ∥Pu(ε)∥ ) dε ≤ ∥Q(κ1, εk)−Q(κ2, εk)∥(L∗ Θk ℑ+Θ0 k) + ∥∂Q(κ1, εk) ∂ε − ∂Q(κ2, εk) ∂ε ∥(L∗ Υk ℑ+Υ0 k) + ( ψ1 K(δ̃ ∗ 1)ξ1 + ψ2 K(δ̃ ∗ 2)ξ2 ) ∫ κ1 εk ∥Q(κ1, ε)−Q(κ2, ε)∥dε +MP (∫ θ 0 ∥Q(κ1, ε)−Q(κ2, ε)∥2 )1/2 dε∥u∥L2 +MQ(κ2 − κ1) ( ψ1 K(δ̃ ∗ 1)ξ1 + ψ2 K(δ̃ ∗ 2)ξ2 ) +MQMP(κ2 − κ1) 1/2∥u∥L2 . By the strong continuity of Q(·) and assumption (A4), we obtain ∥Ξ̂ϖ(κ1)− Ξ̂ϖ(κ2)∥ → 0, as κ1 → κ2. For κ1, κ2 ∈ Jk, we obtain ∥Ξ̂ϖ(κ1)− Ξ̂ϖ(κ2)∥ = ∥Υk(κ1, ϖ(θ−k )−Υk(κ2, ϖ(θ−k )∥. From (A6), we obtain ∥Ξ̂ϖ(κ1)−Ξ̂ϖ(κ2)∥ → 0, as κ1 → κ2. Hence, the set Ξ̂(Π) is equicontinuous, then ω0(Ξ̂(Π)) = 0. Now, let S be a subset of ℶℑ, such that S ⊂ Ξ̂(S) ∪ {0}. S is bounded and equicontinuous; therefore, the function θ → φ(θ) = χ(S(θ)) is continuous. From the properties of the measure χ, we obtain φ(θ) ≤ χ((Ξ̂(S))(θ) ∪ {0}),≤ χ((Ξ̂(S))(θ)). Now for any ϱ > 0, there exists a sequence {ϖk}∞k=0 ⊂ S such that for θ ∈ I0 we have φ(θ) ≤ χ ({∫ θ 0 Q(θ, ε) ( K(ε,ϖℑ(ε,ϖε+xε) + xℑ(ε,ϖε+xε),Ψ(ϖ + x)(ε)) + Pu(ε) ) dε : ϖ ∈ S }) ≤ 2χ ({∫ θ 0 Q(θ, ε)K(ε,ϖk ℑ(ε,ϖk ε+xε) + xℑ(ε,ϖk ε+xε),Ψ(ϖk + x)(ε))dε : ϖ ∈ S }) + ϱ 10 A. BENSALEM, A. SALIM, M. BENCHOHRA, G. N’GUÉRÉKATA EJDE-2025/54 ≤ 4 ∫ θ 0 MQlK ( χ({Π(ε)}) + sup ν∈(−∞,0] χ({Π(ν + ε)}) ) dε+ ϱ ≤ 8 ∫ θ 0 MQlKφ(ε)dε+ ϱ. Since ϱ is arbitrary, we obtain φ(θ) ≤ 8 ∫ θ 0 MQlKφ(ε)dε. From Lemma 2.8, we obtain φ(θ) = χ(S(θ)) = 0. Now if θ ∈ Ik, we have φ(θ) ≤ M̃Qχ( { Υk(εk, ϖ(θ−k )) : ϖ ∈ S } ) +MQχ ({ Θk(εk, ϖ(θ−k )) : ϖ ∈ S }) + χ( {∫ θ εk Q(θ, ε)(K(ε,ϖℑ(ε,ϖε),Ψ(ϖ)(ε)) + Pu(ε))dε : w ∈ S } Big) ≤ ( M̃QLΥk +MQLΘk ) χ(S(θ)) + 4 ∫ θ εk MQlK ( χ({Π(ε)}) + sup ν∈(−∞,0] χ({Π(ν + ε)}) ) dε+ ϱ. Then φ(θ) ≤ 8 1− M̃QLΥk −MQLΘk ∫ θ εk MQlKφ(ε)dε+ ϱ 1− M̃QLΥk −MQLΘk . Since ϱ is arbitrary, we obtain φ(θ) ≤ 8 1− M̃QLΥk −MQLΘk ∫ θ 0 MQlKφ(ε)dε, From Lemma 2.8, we obtain φ(θ) = χ(S(θ)) = 0. If θ ∈ Jk, by (C3) we obtain φ(θ) ≤ LΥk χ(S(θ)), then ∥φ∥ℶ ≤ LΥk ∥φ∥ℶ, implies that ∥φ∥ℶ = 0, thus φ(θ) = χ(S(θ)) = 0. Consequently S(θ) is relatively compact in H. Therefore, S is relatively compact in ℶℑ. Applying now Theorem 2.7, we conclude that Ξ̂ has at least one fixed point w∗. Then ϑ∗ = w∗ + x is a fixed point of the operator Ξ, which is a mild solution of problem (1.1). □ 4. Approximate controllability In this section we investigate the approximate controllability for System (1.1). First we provide a definition of the approximation controllability idea. Let ϑ(T, ζ0, ℘, u) be the state value of (1.1) at terminal time T corresponding to ζ0 ∈ H, ℘ ∈ B. To define the notion of approximate controllability we introduce the set R(T, ζ0, ℘) = {ϑ(T, ζ0, ℘, u), u(·) ∈ L2(∇ : Y)}, which is called the reachable set of system (1.1) at terminal time T . Its closure in H is denoted by R(T, ζ0, ℘). Definition 4.1. System (1.1) is said to be approximately controllable on the interval ∇ = [0, T ] if R(T, ζ0, ℘) is dense in H, i.e. R(T, ζ0, ℘) = H. To study the approximate controllability of system (1.1) we introduce the following operators: Γθk+1 εk = ∫ θk+1 εk Q(θk+1, ε)PP∗Q∗(θk+1 − ε)dε, R(λ,Γθk+1 εk ) = (λI + Γθk+1 εk )−1, EJDE-2025/54 SECOND-ORDER IMPULSIVE INTEGRO-DIFFERENTIAL EQUATIONS 11 where ε0 = 0, θk+1 = T k ∈ Nm 0 ; P∗ and Q∗ denote the adjoint of the operators P and Q respectively. It is straightforward to see that the operator Γ θk+1 εk is a linear bounded operator. So we assume that for all k ∈ Nm 0 , the operator R(λ,Γ θk+1 εk ) satisfies (A10) λR(λ,Γ θk+1 εk ) → 0 as λ→ 0+ in the strong operator topology. From [14], hypothesis (A10) is equivalent to the fact that the linear control system corresponding to system (1.1) is approximately controllable on [0, T ]. More precisely, we have the following theorem. Theorem 4.2. The following statements are equivalent: (i) The linear control system corresponding to system (1.1) is approximately controllable on [0, T ]. (ii) If P∗Q∗(θ)κ = 0 for all θ ∈ [0, T ], then κ = 0. (iii) The condition (C0) holds. The proof of this theorem is similar to that of [3, Theorem 2] and [14, Theorem 4.4.17], so we omit it here. We are now in a position to prove the approximate controllability of system (1.1). For any given ηθk+1 ∈ H and λ ∈ (0, 1], we take the control function uλ(θ) as follows: uλ(θ) = P∗Q∗(θk+1, ε)R(λ,Γ θk+1 εk )∆(ηθk+1 , θ); k ∈ Nm 0 . Where ∆(ηθk+1 , θ) = ηθk+1 −∆k(θ)− ∫ θ εk Q(θ − ε)K(ε, ϑℑ(ε,ϑε), (Ψϑ)(ε))dε, and ∆k(θ) = { −∂Q(θ,ε) ∂ε ∣∣ ε=0 ℘(0) +Q(θ, 0)ζ0, if k = 0, −∂Q(θ,ε) ∂ε ∣∣ ε=εk Υk(εk, ϑ(θ − k )) +Q(θ, εk)Θk(εk, ϑ(θ − k )), if k ∈ Nm 1 . . Theorem 4.3. Assume (A4)–(A10) hold, the function f is uniformly bounded, and the resolvent operator {Q(θ, ε)}θ≥ε is compact. Then, (1.1) is approximately controllable on [0, T ]. Proof. According to Theorem 3.3, we know that system (1.1) has at least one mild solution ξλ ∈ ℶℑ. Then we obtain ξλ(θk+1) = ∆k(θk+1) + ∫ θk+1 εk Q(θk+1, ε) ( K(ε, ϑλℑ(ε,ϑλ ε ) , (Ψϑλ)(ε)) + Pu(ε) ) dε = ∆k(θk+1) + ∫ θk+1 εk Q(θk+1, ε) ( K(ε, ϑλℑ(ε,ϑλ ε ) , (Ψϑλ)(ε)) ) dε + ∫ θk+1 εk Q(θk+1, ε)P(P∗Q∗(θk+1, ε)R(λ,Γ θk+1 εk )∆(ηθk+1 , θk+1))dε = ηθk+1 + (Γθk+1 εk R(λ,Γθk+1 εk )− I)∆(ηθk+1 , θk+1) = ηθk+1 + λR(λ,Γθk+1 εk )∆(ηθk+1 , θk+1). So ∥ξλ(θk+1)− ηθk+1∥ ≤ ∥R(λ,Γθk+1 εk )[ηθk+1 −∆k(θk+1]∥ + ∥∥R(λ,Γθk+1 εk ) [ ∫ θk+1 εk Q(θk+1, v)K(vλ, ϑℑ(v,ϑλ v ) , (Ψϑλ)(v))dv ]∥∥. We infer from the uniform boundedness of K(·, ·, ·) that there exists MK > 0, such that∫ T 0 ∥K(ε, ϑλℑ(ε,ϑλ ε ) , (Ψϑλ)(ε))∥2dε ≤ T (MK) 2. Therefore, the sequence {K(ε, ϑλℑ(ε,ϑλ ε ) , (Ψϑλ)(ε))}λ is bounded in L2(∇,H), then there exists subsequence still denoted by {K(ε, ϑλℑ(ε,ϑλ ε ) , (Ψϑλ)(ε))}λ that weakly converge to the limit K̃(ε) in 12 A. BENSALEM, A. SALIM, M. BENCHOHRA, G. N’GUÉRÉKATA EJDE-2025/54 L2(∇,H). The compactness of (Q(θ, ε))θ≥ε implies that ∥ ∫ T 0 Q(θ, ε) ( K(ε, ϑλℑ(ε,ϑλ ε ) , (Ψϑλ)(ε))− K̃(ε) ) dε∥ 0−−−→ λ→0 . Then we obtain ∥ξλ(θk+1)− ηθk+1∥ ≤ ∥R(λ,Γθk+1 εk )[ηθk+1 −∆k(θk+1)−Θ(θk+1, ϑθk+1 )]∥ + ∥∥R(λ,Γθk+1 εk ) [ ∫ θk+1 εk Q(θk+1, v) ( K(ε, ϑλℑ(ε,ϑλ ε ) , (Ψϑλ)(ε))− K̃(ε) ) dv ]∥∥ + ∥∥R(λ,Γθk+1 εk ) [ ∫ θk+1 εk Q(θk+1, v)K̃(ε)dε ]∥∥ −−−→ λ→0 0. Thus, ξλ(θk+1) → ηθk+1 holds. Therefore, we obtain the approximate controllability of system (1.1), and the proof is complete. □ Remark 4.4. We can eliminate the uniform boundedness condition on K. From the growth condition on K and the continuity conditions on ψ1 K and ψ2 K, we can deduce that K is uniformly bounded on each bounded subset of the space ℶ. This is sufficient to construct a sequence that converges weakly in L2(∇,H). 5. Existence of optimal controls In this section, we prove the existence of optimal state-control pairs of the Bolza problem corresponding to system (1.1). From now, we suppose that Y is a separable reflexive Banach space from which the controls u take its values. The multifunction ω : ∇ ⇒ 2Y has closed, convex and bounded values. ω(·) is graph measurable and ω(·) ⊂ B where B is a bounded set of U , the admissible control set Yad = {u ∈ L2(∇, B) : u(θ) ∈ ω(θ) a. e.}. Clearly, Yad ̸= ∅ (see [27]) and Yad ⊆ L2(∇, B) is bounded, closed and convex. Consider the following optimal controls Bolza problem (BP): Find an optimal pair (ϑ0, u0) ∈ X ×Yad, such that I(ϑ0, u0) ≤ I(ϑu, u), for all (ϑu, u) ∈ X ×Yad, (5.1) where the cost functional is I(u) = ∫ T 0 J̃ (ε, ϑuε (ε), ϑ u(ε), u(ε))dε+ Γ(ϑu(T )), where ϑu is the mild solution of system (1.1) corresponding to the control u ∈ Yad, and P ∈ L∞(∇, L(Y,H)). To establish the existence of optimal controls, we impose the following additional assumptions: (A11) (i1) The functional J̃ : ∇×G×H×Y → R ∪ {∞} is Borel measurable. (i2) J̃ (θ, ·, ·, ·) is sequentially lower semicontinuous on G×H×Y for almost all θ ∈ ∇. (i3) J̃ (θ,κ, ϑ, ·) is convex on Y for each κ ∈ G, ϑ ∈ H and almost all θ ∈ ∇. (i4) There exist constants r0 > 0, r1 ≥ 0, r2 > 0, and ג ∈ L1(∇,R), such that J̃ (θ,κ, ϑ, u) ≥ (θ)ג + r0∥κ∥G + r1∥ϑ∥+ r2∥u∥2U . (i5) The function Γ : H → R is continuous and non-negative. Now, we provide the following result on existence of optimal controls for problem (5.1). Theorem 5.1. Assume (A11) and the conditions of Theorem 4.3 hold. Then problem (5.1) admits at least one optimal pair on X ×Yad. EJDE-2025/54 SECOND-ORDER IMPULSIVE INTEGRO-DIFFERENTIAL EQUATIONS 13 Proof. If inf{I(u) : u ∈ Yad} = +∞, then there is nothing to verify. Now assume that inf{I(u) : u ∈ Yad} = ȷ < +∞. Using (A11), we obtain I(u) ≥ ∫ T 0 (ε)ג) + r0∥ϑuε∥+ r1∥ϑu(ε)∥)dε+ r2 ∫ T 0 ∥u(ε)∥2Y dε+ Γ(ϑu(T )) ≥ −ς > −∞, where ς > 0 is constant. Hence, ȷ ≥ −ς > −∞. By the definition of infimum there exists a minimizing sequence of feasible pairs (ϑp, up)p∈N ⊂ Aad such that I(up) → ȷ as p→ +∞, where Aad = {(ϑ, u) : ϑ is a mild solution of system (1.1) corresponding to the control u ∈ Yad}. Since (up)p∈N ⊆ Yad, {up}p∈N is bounded in L2(∇,Y) there exists a subsequence which is still represented by {up}, and u0 ∈ L2(∇,Y) such that up → u0 in L2(∇,Y). Since Yad is closed and convex, by Mazur’s Lemma, we obtain u0 ∈ Yad. Let ϑp denote the corresponding sequence of solutions of system (1.1) with respect to up and satisfying the integral equation ϑp(θ) =  −∂Q(θ,ε) ∂ε ∣∣ ε=0 ℘(0) +Q(θ, 0)ζ0 + ∫ θ 0 Q(θ, ε) ( K(ε, ϑpℑ(ε,ϑp ε) , (Ψϑp)(ε)) + Pup(ε) ) dε, if θ ∈ I0, −∂Q(θ,ε) ∂ε ∣∣ ε=εk Υk ( εk, ϑ p(θ−k ) ) +Q(θ, εk)Θk(εk, ϑ p(θ−k )) + ∫ θ εk Q(θ, ε)(K(ε, ϑpℑ(ε,ϑp ε) , (Ψϑp)(ε)) + Pup(ε))dε, if θ ∈ Ik, k ∈ Nm 1 , Υk(θ, ϑ p(θ−k )), if θ ∈ Jk, k ∈ Nm 1 , ℘(θ); if θ ∈ R−. Let Kp(θ) ≡ K(θ, ϑpℑ(θ,ϑp θ) , (Ψϑp)(θ)). Then by (A4), we deduce that Kp is a bounded continuous operator from ∇ to H. Hence, Kp(·) ∈ L2(∇,H). Furthermore, {Kp(·)} is bounded in L2(∇,H), and there exists a sub-sequence, relabeled as {Kp(·)}, and K̂(·) ∈ L2(∇,H) such that Kp(·) w→ K̂(·) in L2(∇,H). By Lemma 2.2, we have N1Kp(·) ε→ N1K̂(·) in X . Now, we consider the controlled system ϑ′′(θ) = Z(θ)ϑ(θ) + ∫ θ 0 Υ(θ, ε)ϑ(ε)dε+ K̂(θ) + Pu0(θ), if θ ∈ Ik, k ∈ Nm 0 , ϑ(θ) = Υk(θ, ϑ(θ − k )), if θ ∈ Jk, k ∈ Nm 1 , ϑ′(θ) = Θk(θ, ϑ(θ − k )), if θ ∈ Jk, k ∈ Nm 1 , ϑ′(0) = ζ0 ∈ H, ϑ(θ) = ℘(θ), if θ ∈ R−. (5.2) By Theorem 3.3, it is clear that system (1.1) has a mild solution ϑ̂(θ) =  −∂Q(θ,ε) ∂ε ∣∣ ε=0 ℘(0) +Q(θ, 0)ζ0 + ∫ θ 0 Q(θ, ε) ( K̂(ε) + Pu0(ε) ) dε, if θ ∈ I0, −∂Q(θ,ε) ∂ε ∣∣ ε=εk Υk(εk, ϑ̂(θ − k )) +Q(θ, εk)Θk(εk, ϑ̂(θ − k )) + ∫ θ εk Q(θ, ε) ( K̂(ε) + Pu0(ε) ) dε, if θ ∈ Ik, k ∈ Nm 1 , Υk(θ, ϑ̂(θ − k )), if θ ∈ Jk, k ∈ Nm 1 , 4pt]℘(θ), if θ ∈ R−. For each θ ∈ I0, ϑ p, ϑ̂ ∈ X , we have ∥ϑp(θ)− ϑ̂(θ)∥ ≤ ∫ θ 0 ∥Q(θ, ε)(Kp(ε)− K̂(ε))∥dε+MQT 1− 1 q (∫ θ 0 ∥(Pup(ε)− Pu0(ε))∥qdε )1/q . 14 A. BENSALEM, A. SALIM, M. BENCHOHRA, G. N’GUÉRÉKATA EJDE-2025/54 If θ ∈ Ik, we obtain ∥ϑp(θ)− ϑ̂(θ)∥ ≤ M̃Q∥Υk(εk, ϑ p(θ−k ))−Υk(εk, ϑ̂(θ − k ))∥+MQ∥Θk(εk, ϑ p(θ−k ))−Θk(εk, ϑ̂(θ − k ))∥ + ∫ θ εk ∥Q(θ, ε)Kp(ε)−Q(θ, ε)K̂(ε)∥dε +MQT 1− 1 q (∫ θ εk ∥(Pup(ε)− Pu0(ε))∥qdε )1/q . Now for θ ∈ Jk, we have ∥ϑp(θ)− ϑ̂(θ)∥ ≤ L∗ Υk ∥ϑp(θ)− ϑ̂(θ)∥. We keep in mind that g, K, Υk, and Θk are continuous, and L∗ Υk ∈ (0, 1). By strongly continuity of P, we have ∥Pup − Pu0∥Lq → 0, as p→ +∞. Thus ∥ϑp(θ)− ϑ̂(θ)∥ → 0, as p→ +∞. Furthermore, using (A4) and (A5), we obtain Kp(·) ε→ K̂(·, ϑ̂ℑ(·,ϑ̂·) , (Ψϑ̂)(·)), in X as p→ ∞. Hence, K̂(θ) = K̂(θ, ϑ̂ℑ(θ,ϑ̂θ) , (Ψϑ̂)(θ)). Thus, ϑ̂ can be given by ϑ̂(θ) =  −∂Q(θ,ε) ∂ε ∣∣ ε=0 ℘(0) +Q(θ, 0)ζ0 + ∫ θ 0 Q(θ, ε) ( K̂(ε) + Pu0(ε) ) dε, if θ ∈ I0, −∂Q(θ,ε) ∂ε ∣∣ ε=εk Υk ( εk, ϑ̂(θ − k ) ) +Q(θ, εk)Θk(εk, ϑ̂(θ − k ) ) + ∫ θ εk Q(θ, ε)Big(K̂(ε) + Pu0(ε) ) dε, if θ ∈ Ik, k ∈ Nm 1 , Υk(θ, ϑ̂(θ − k )), if θ ∈ Jk, k ∈ Nm 1 , ℘(θ), if θ ∈ R−, which is just a mild solution of system (1.1) corresponding to u0. Since X ↪→ L1(∇,H), using (A11) and Balder’s Theorem, we conclude that (ϑθ × ϑ, u) → ∫ T 0 J̃ (ε, ϑε(ε), ϑ(ε), u(ε))dε, is sequentially lower semicontinuous in the weak topology of L2(∇,H) ⊂ L1(∇,H), therefore, I is weakly lower semicontinuous on L1(∇,H). Thus ȷ = lim p→∞ ∫ T 0 J̃ (ε, ϑpε(ε), ϑ p(ε), u(ε))dε+ Γ(ϑp(T )) ≥ ∫ T 0 J̃ ( ε, ϑ̂ε, ϑ̂(ε), u 0(ε) ) dθ + Γ(ϑ̂(T )) = I(u0) ≥ ȷ. Which implies that I attains its minimum at (ϑ̂, u0) ∈ X ×Yad. □ EJDE-2025/54 SECOND-ORDER IMPULSIVE INTEGRO-DIFFERENTIAL EQUATIONS 15 6. An example In this section, we give an example to illustrate the above theoretical result. Consider the partial integro-differential system ∂2ζ(θ, x) ∂2θ = ∂2ζ(θ, x) ∂2x + ∫ θ 0 Γ(θ − ε) ∂2ζ(ε, x) ∂2x dε + ∫ −θ −∞ e−8ν∥ζ(θ + σ(θ, ζ(θ + ν, x)), x)∥2 155ϵ1 √ π((θ + ν)2 + 2θ + 1) dν + ∫ π 0 cosh(θ) sin(π + e−θ2 )(1 + ∥ζ(ε)∥2) 460ϵ2 √ π(1 + 2θ2 + ε2)e11θ dε+ σ̃(θ)ζ(θ, x) + U(θ, x), if θ ∈ I1 ∪ I2 ∪ I3, x ∈ (0, π), ζ(θ, x) = 1 63 cos( √ πθ)ζ(θ−, x), if θ ∈ J1 ∪ J2, x ∈ (0, π), ∂ζ(θ, x) ∂θ = 1 77 sin( √ πθ)ζ(θ−, x), if θ ∈ J1 ∪ J2, x ∈ (0, π), ζ(θ, 0) = ζ(θ, 1) = 0, for θ ∈ I, ∂ζ(θ, x) ∂θ ∣∣ θ=0 = ζ1(x), ζ(θ, x) = ℘(θ, x), if θ ∈ R− x ∈ (0, π), (6.1) where I = [0, π], k1 = 1 16 , k2 = 1 9 , k3 = 1 8 , k4 = 1 4 , I1 = (0, k1], I2 = (k2, k3], I3 = (k4, π], J1 = (k1, k2], J2 = (k3, k4], σ : ∇×R → R, U : [0, π]× [0, π] → H, and ϵ1, ϵ2 are positive constants. We consider the cost function I(u) = ∫ π 0 ∫ π 0 ( 1 π ∫ 0 −∞ ∥ζ(θ + ν)∥22dν + |ζ(θ, ε)|2 + |u(θ, ε)|2 ) dεdθ + ∫ π 0 |ζ(π, ε)|2dε. and the Hilbert space H = Y := L2(0, π) = { u : (0, π) → R : ∫ π 0 |u(x)|2dx <∞ } , with scalar product ⟨u, v⟩ = ∫ π 0 u(x)v(x)dx, and norm ∥u∥2 = (∫ π 0 |u(x)|2dx )1/2 . We define the control set Yad = {u ∈ L2([0, π]) : ∥u(·, θ)∥2 ≤ ϖ(θ) a.e.}, where ϖ ∈ L2(∇,R+). On the other hand, let the phase space G be BUC(R−,H), the space of bounded uniformly continuous functions endowed with the norm ∥ψ∥G = sup −∞<ν≤0 ∥ψ(ν)∥L2 , ψ ∈ G. It is well known that G satisfies the assumptions (A1) and (A2) withK = 1 and β2(θ) = β3(θ) = 1, (see [26]). We define the operator Ẑ induced on H as follows: Ẑκ = κ′′, and D(Z) = {κ ∈ H2(0, π) : κ(0) = κ(π) = 0}, Then Ẑ is the infinitesimal generator of a cosine function of operators (C0(θ))θ∈R on H associated with sine function (S0(θ))θ∈R. Additionally, Ẑ has discrete spectrum which consists of eigenvalues −n2 for n ∈ N, with corresponding eigenvectors wn(x) = 1√ 2π einx. The set {wn : n ∈ N} is an orthonormal basis of H. Applying this idea, we can write Ẑκ = ∞∑ n=1 −n2⟨κ, wn⟩wn, κ ∈ D(Z) The cosine family associated with Ẑ is given by (C0(θ))θ∈R is given by C0(θ)κ = ∞∑ n=1 cos(nθ)⟨κ, wn⟩wn, θ ∈ R, 16 A. BENSALEM, A. SALIM, M. BENCHOHRA, G. N’GUÉRÉKATA EJDE-2025/54 and the sine function is given by S0(θ)κ = ∞∑ n=1 sin(nθ) n ⟨κ, wn⟩wn, θ ∈ R. It is immediate from these representations that ∥C0(θ)∥ ≤ 1 and that S0(θ) is compact for all θ ∈ R. We define Z(θ)κ = Ẑκ + σ̃(θ)κ on D(Z). Clearly, Z(θ) is a closed linear operator. Therefore, Z(θ) generates (S(θ, ε))(θ,ε)∈∆ such that S(θ, ε) is compact and self-adjoint for all (θ, ε) ∈ ∆ = {(θ, ε) : 0 ≤ ε ≤ θ ≤ 1}, (see [23]). We define the operators Λ(θ, ε) : D(Z) ⊂ H 7→ H as follows: Λ(θ, ε)κ = Γ(θ − ε)Ẑκ, for 0 ≤ ε ≤ θ ≤ 1, κ ∈ D(Z). Then assumption (A7) holds under more suitable conditions on the operator Γ. Moreover, it is evident that conditions (1)–(3) of Υ are satisfied, indicating the existence of a resolvent operator that is compact. More details can be found in [23, 35]. Now let P : Y → H be defined by Pu(θ)(x) = U(θ, x), x ∈ [0, π], u ∈ Y, where U : [0, π]×[0, π] → H is linear continuous and for ℘ ∈ BUC(R−,H), we put ℑ(θ, ℘)(ζ) = σ(θ, ζ(θ + ν, x)), such that (A9) holds, and let θ → ℘θ be continuous on R(ℑ−). We put ζ(θ)(x) = ζ(θ, x) and define K(θ, ϑ1, ϑ2)(x) = ∫ −θ −∞ e−8ν∥ϑ1(θ + σ(θ, ϑ1(θ + ν, x)), x)∥2 155ϵ1 √ π((θ + ν)2 + 2θ + 1) dν + cosh(θ)ϑ2(θ)(x) 4ϵ2e11θ , ϑ2(θ)(x) = Ψ(ϑ1)(x) = ∫ π 0 sin(π + e−θ2 )(1 + ∥ϑ1(ε)∥2) 115 √ π(1 + 2θ2 + ε2) dε, Υk(θ, ϑ(θ − k ))(x) = 1 63 cos( √ πθ)ϑ(θ−, x), Θk(θ, ϑ(θ − k ))(x) = 1 77 sin( √ πθ)ϑ(θ−, x). These definitions allow us to depict the system (6.1) in the abstract form (1.1). Now, for θ ∈ [0, π], we have ∥K(θ, γ1(θ), γ2(θ))∥2 ≤ 1− e−16π 310ϵ1(θ + 1)2 (1 2 ∥γ1∥G4 ) + 1 4ϵ2 cosh(θ)e−11θ(∥γ2(θ)∥2). So, ψi+1 K (θ) = θ 2−i ; i = 0, 1 are continuous nondecreasing functions, and we have ξ1 = 1− e−16π 310ϵ1 , ξ2 = cosh(π) 4ϵ2 . And for any bounded set Π ⊂ H, and Πθ ∈ G, we obtain χ(K(θ,Πθ,Ψ(Π(θ)))) ≤ ξ1 sup ν∈(−∞,0] χ(Π(ν + θ)) + ξ2χ(Π(θ)). Now, about g, Υk, and Θk, we obtain ∥g(θ, ε, γ1)− g(θ, ε, γ2)∥2 ≤ 1 115 ∥γ1 − γ2∥2, ∥Υk(θ, γ1(θ − k ))−Υk(θ, γ2(θ − k ))∥2 ≤ 1 63 ∥γ1 − γ2∥2, ∥Θk(θ, γ1(θ − k ))−Θk(θ, γ2(θ − k ))∥2 ≤ 1 77 ∥γ1 − γ2∥2. Furthermore, we have M̃QLΥk +MQLΘk ≤ 0, 0289. And for ϵ1 > 1 + ∥γ1∥G4 , ϵ2 > 1 + ∥γ2∥2, for all γ1 ∈ G4, gamma2 ∈ H, we obtain ∥K(·, γ1(·), γ2(·))∥2 ≤ 1− e−16π 620 + cosh(π) 4ϵ2 . EJDE-2025/54 SECOND-ORDER IMPULSIVE INTEGRO-DIFFERENTIAL EQUATIONS 17 Thus under appropriate conditions on the operator U the corresponding linear system is approxi- mately controllable, then all the assumptions of Theorems 4.3 and 5.1 are fulfilled. 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Wang; Integral boundary value problems for nonlinear non-instataneous impulsive differential equations, J. Appl. Math. Comput., 55 (2017), 59-78. [40] K. Yosida; Functional Analysis, 6 Springer-Verlag, Berlin, 1980. Abdelhamid Bensalem Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbès, P.O. Box 89, Sidi Bel-Abbès 22000, Algeria Email address: bensalem.abdelhamid@yahoo.com Abdelkrim Salim Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbès, P.O. Box 89, Sidi Bel-Abbès 22000, Algeria. Faculty of Technology, Hassiba Benbouali University of Chlef, P.O. Box 151 Chlef 02000, Algeria Email address: salim.abdelkrim@yahoo.com Mouffak Benchohra Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbès, P.O. Box 89, Sidi Bel-Abbès 22000, Algeria Email address: benchohra@yahoo.com Gaston M. N’Guérékata NEERLab, Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21252, USA Email address: Gaston.NGuerekata@morgan.edu 1. Introduction 2. Preliminaries 3. Existence of mild solution 4. Approximate controllability 5. Existence of optimal controls 6. An example References