Second-Order Nonlinear Impulsive Integro-Differential Equations of Mixed Type and Optimal Controls in Fractional Power Spaces

Abstract

A class of second-order nonlinear impulsive integro-differential equations of mixed type whose principal part is given by time-varying generating operators in fractional power spaces is considered. We introduce the reasonable PC-𝛼-mild solution of second-order nonlinear impulsive integro-differential equations of mixed type and prove its existence. The existence of optimal controls for a Lagrange problem of systems governed by second-order nonlinear impulsive integro-equations of mixed type is also presented. An example is given for demonstration.

1. Introduction

Some interesting models of mathematical biology or population, mechanics of materials, nuclear physics, and so forth, can be written in terms of second-order nonlinear partial integro-differential equations. This is the case of the model proposed to describe viscoelastic problems with memory. The system is given by𝑥̈(𝑡)(𝑡)𝑥̇(𝑡)((𝑡),𝑥̇(𝑡),(𝑡),(𝑡(]𝑥(0)+𝐴=𝑓𝑡,𝑥𝐺𝑥)(𝐻𝑥)()),𝑡∈0,𝑇⧵Θ,=𝑥₀,Δ𝑥𝑡_𝑖=𝐽⁰_𝑖𝑥𝑡_𝑖,𝑥̇𝑡_𝑖,𝑡_𝑖𝑥̇(0)∈Θ,=𝑥₁,Δ𝑥̇𝑡_𝑖=𝐽¹_𝑖𝑥𝑡_𝑖,𝑥̇𝑡_𝑖,𝑡_𝑖∈Θ, (1.1) where {𝐴(𝑡)∣0≤𝑡≤𝑇} generates an evolution system {𝑈(𝑡,𝑠)∣0≤𝑠≤𝑡≤𝑇} in the parabolic case in Banach spaces 𝑋 (see [1–3]). 𝐺, 𝐻 are nonlinear integral operators given by (𝐺𝑥)(𝑡)=^𝑡₀𝑘(𝑡,𝜏)𝑔(𝜏,𝑥(𝜏),𝑥̇(𝜏))𝑑𝜏,(𝐻𝑥)(𝑡)=^𝑇₀𝑚(𝑡,𝜏)ℎ(𝜏,𝑥(𝜏),𝑥̇(𝜏))𝑑𝜏. (1.2)

Θ={𝑡_𝑖∈(0,𝑇)∣0=𝑡₀<𝑡₁<⋯<𝑡_𝑛<𝑡_𝑛+1=𝑇}, 𝐽⁰_𝑖, 𝐽¹_𝑖(𝑖=1,2,…,𝑛) are nonlinear maps, and Δ𝑥(𝑡_𝑖)=𝑥(𝑡_𝑖+0)−𝑥(𝑡_𝑖), Δ𝑥̇(𝑡_𝑖)=𝑥̇(𝑡_𝑖+0)−𝑥̇(𝑡_𝑖). This represents the jump in the state 𝑥, 𝑥̇ at time 𝑡_𝑖, respectively, with 𝐽⁰_𝑖, 𝐽¹_𝑖 determining the size of the jump at time 𝑡_𝑖.

In fact, since the end of last century, impulsive evolution equations on infinite-dimensional spaces have been investigated by many authors including us. Particularly, Ahmed and we considered optimal control problems of systems governed by first-order impulsive evolution equations and first-order impulsive integro-differential equations [4–7]. Recently, we discussed the second-order impulsive evolution equations and the second-order impulsive integro-differential equations and their optimal controls in general Banach spaces [8–11]. In addition, to our knowledge, the second-order impulsive functional differential equations and the second-order impulsive integro-differential equations whose principal operator is bounded have been deeply studied by many authors [12–16]. However, the second-order impulsive integro-differential evolutions equations of mixed type whose principle operator is unbounded in infinite dimensional fractional power spaces and corresponding optimal control problems have not been extensively considered in the literature.

Reducing the second-order evolution equations to the first-order evolution equations, we introduce a family of unbounded linear matrix operators 𝔄(𝑡), 0≤𝑡≤𝑇 and prove that {𝔄(𝑡)∣0≤𝑡≤𝑇} generates an evolution system {𝑈(𝑡,𝑠)∣0≤𝑠≤𝑡≤𝑇} which can be represented by 𝑈(𝑡,𝑠), 0≤𝑠≤𝑡≤𝑇. Based on the evolution system {𝑈(⋅,⋅)}, we introduce a reasonable PC-α-mild solution of (1.1). Using the interpolation space technique, we can overcome the difficulty brought by fractional power spaces. Next, by a virtue of the generalized Gronwall lemma with singularity, impulse, and integrals of mixed type given by us, one can overcome the difficulty brought by operator 𝐻 to get a priori estimate of PC-α-mild solution. By compactness condition of space PC¹([0,𝑇],𝑋_𝛼) and Leray-Schauder fixed point theory, we can obtain the existence of PC-α-mild solution for (1.1). Particularly, introducing new norm, we use the contraction mapping principle to give the uniqueness of α-mild solution for the second-order nonlinear integro-differential equation (1.1) without impulses (see Remark 3.4). A Lagrange problem of a system governed by (1.1) whose cost functional includes both 𝑥 and 𝑥̇ is investigated. By the structure of {𝑈(⋅,⋅)} and compactness of {𝑈(⋅,⋅)}, the existence of optimal controls is verified.

The rest of the paper is organized as follows. In Section 2, we give some associated notations and important lemmas. In Section 3, the existence of PC-α-mild solution for (1.1) is presented. In Section 4, we consider a Lagrange problem of system governed by (1.1) and prove the existence of optimal controls. At last, an example demonstrates the applicability of our results.

2. Preliminaries

Let 𝑋, 𝑌 denote a pair of Banach spaces. If 𝑋 is continuously embedded in 𝑌, we write 𝑋↪𝑌; if 𝑋 is compactly embedded in 𝑌, we write 𝑋↪↪𝑌. Set Δ={(𝑡,𝑠)∣0≤𝑠≤𝑡≤𝑇}, £(𝑋) is the class of (not necessarily bounded) linear operators in 𝑋. £_𝑏(𝑋) stands for the family of bounded linear operators in 𝑋. For 𝐴∈£(𝑋), let 𝜌(𝐴) denote the resolvent set and 𝑅(𝜆,𝐴) the resolvent corresponding to 𝜆∈𝜌(𝐴).

Assumption A. (P₁) Let {𝐴(𝑡)∣𝑡∈[0,𝑇]} be a family of closed linear operators in 𝑋, the domain 𝐷(𝐴(𝑡))=𝐷 of 𝐴(𝑡), and 𝑡∈[0,𝑇] dense in 𝑋 and independent of 𝑡.
(P₂) For 𝑡∈[0,𝑇], the resolvent 𝑅(𝜆,𝐴(𝑡)) of 𝐴(𝑡) exists for all 𝜆 with Re𝜆≤0 and there exists a constant 𝑀>0 such that ‖𝑅(𝜆,𝐴(𝑡))‖_£(𝑋)≤𝑀|𝜆|[]+1,for𝑡∈0,𝑇. (2.1)
(P₃) There exist constants 𝐿>0 and 0<𝛼≤1 such that (𝐴(𝑡)−𝐴(𝑠))𝐴⁻¹(𝜏)_£(𝑋)≤𝐿|𝑡−𝑠|^𝛼for𝑠,𝑡,𝜏∈[0,𝑇]. (2.2)
Let 𝑋₁={𝐷,‖⋅‖₁} where ‖𝑥‖₁=‖𝐴𝑥‖. 𝑋₁ is a Banach space and 𝑋₁↪𝑋. More generally, in a usual way we introduce the fractional power operator 𝐴^𝛼(𝑡) (𝛼∈(0,1)), which satisfies 𝐷(𝐴)=𝐷(𝐴(𝑡)) and 𝐷(𝐴^𝛼)=𝐷(𝐴^𝛼(𝑡)) for 𝑡∈[0,𝑇]. Let ‖𝑥‖_𝛼=‖𝐴^𝛼𝑥‖ for 𝑥∈𝐷(𝐴^𝛼(𝑡)) and denote the Banach space {𝐷(𝐴^𝛼),‖⋅‖_𝛼} as 𝑋_𝛼. Then it is clear that 𝑋_𝛽↪𝑋_𝛼 for 0≤𝛼≤𝛽≤1. Define PC([0,𝑇],𝑋_𝛼)≡{𝑥∶[0,𝑇]→𝑋_𝛼∣𝑥 as continuous at 𝑡∈[0,𝑇]⧵Θ; 𝑥 is continuous from left and has right-hand limits at 𝑡∈Θ}, PC¹([0,𝑇],𝑋_𝛼)≡{𝑥∈PC([0,𝑇],𝑋_𝛼)∣𝑥̇∈PC([0,𝑇],𝑋)}. It can be seen that PC([0,𝑇],𝑋_𝛼) and PC¹([0,𝑇],𝑋_𝛼) are Banach spaces, respectively, with the norms ‖𝑥‖_PC=maxsup_{[]𝑡∈0,𝑇}‖𝑥(𝑡+0)‖_𝛼,sup_{[]𝑡∈0,𝑇}‖𝑥(𝑡−0)‖_𝛼,‖𝑥‖_PC¹=‖𝑥‖_PC+‖𝑥̇‖_PC. (2.3)
For the initial value problem 𝑥̇(𝑡)(𝑡)𝑥(𝑡)(],𝑥(0)+𝐴=0,𝑡∈0,𝑇=𝑥₀, (2.4) it is well known that (2.4) has a unique classical solution 𝑥. Moreover, 𝑥∈𝐶¹([0,𝑇],𝑋) provided 𝑥₀∈𝐷(𝐴). Further, there exists a unique evolution operator 𝑈(𝑡,𝑠)∈𝐿_𝑏(𝑋), (𝑡,𝑠)∈Δ, such that every solution of (2.4) can be represented in the form 𝑥(𝑡)=𝑈(𝑡,0)𝑥₀. (2.5)
Consider the following second-order initial value problem: 𝑥̈(𝑡)(𝑠)𝑥̇(𝑡)(],𝑥(0)+𝐴=0,𝑡∈0,𝑇=𝑥₀,𝑥̇(0)=𝑥₁. (2.6) Set 𝑣(𝑡)=_̇^𝑥(𝑡)_𝑥(𝑡) and 𝑣₀=^𝑥0_𝑥1; (2.6) can be rewritten as 𝑣̇(𝑡)(𝑡)𝑣(𝑡)(],𝑣(0)+𝔄=0,𝑡∈0,𝑇=𝑣₀, (2.7) where 𝔄(𝑡)=^0−𝐼_0𝐴(𝑡). By [9, Theorem  2.A], we have the following theorem.

Theorem 2.1. Under the Assumption A, (2.7) has a unique evolution system {𝑈(𝑡,𝑠)∣(𝑡,𝑠)∈Δ} given by 𝑈()=𝐼𝑡,𝑠^𝑡𝑠𝑈()()𝜏,𝑠𝑑𝜏0𝑈𝑡,𝑠. (2.8)
In order to derive a priori estimates on the PC-α-mild solution of integro-differential equation of mixed type, we need the following generalized Gronwall inequality with impulses.

Theorem 2.2. Let 𝑥∈PC([0,𝑇],𝑋_𝛼) and satisfy the following inequality: ‖𝑥(𝑡)‖_𝛼≤𝑎+𝑏^𝑡₀|𝑡−𝜏|^−𝛾‖𝑥(𝜏)‖_𝛼𝑑𝜏+𝑐^𝑡₀|𝑡−𝜏|^−𝛾𝑥_𝜏𝐵𝑑𝜏+𝛿^𝑇₀‖𝑥(𝜏)‖^𝜆_𝛼𝑑𝜏+_{0<𝑡𝑘<𝑡}𝜃_𝑘𝑥𝑡_𝑘𝛼 (2.9)
for all 𝑡∈[0,𝑇], where 1>𝜆, 𝛾>0, and 𝑎, 𝑏, 𝑐, 𝛿, 𝜃_𝑘≥0 are constants, and ‖𝑥_𝑡‖_𝐵=sup_{0≤𝑠≤𝑡}‖𝑥(𝑠)‖_𝛼. Then there exists constant 𝑀>0 such that ‖𝑥(𝑡)‖_𝛼≤𝑀,∀𝑡∈[0,𝑇]. (2.10)

Proof. By the inequality (2.9) and [7, Lemma  2.1], there exist 𝑎₀>0, 𝑏₀>0 such that 𝑥_𝑡𝐵≤𝑎₀𝑎+𝛿^𝑇₀‖𝑥(𝜏)‖^𝜆_𝛼𝑑𝜏+𝑏₀^𝑡₀𝑥_𝜏𝐵𝑑𝜏+_{0<𝑡𝑘<𝑡}𝑎₀𝜃_𝑘𝑥𝑡_𝑘𝐵. (2.11) By [17, Lemma  1.7.1], we have 𝑥_𝑡𝐵≤𝑎₀𝑎+𝛿^𝑇₀‖𝑥(𝜏)‖^𝜆_𝛼𝑑𝜏_{0<𝑡𝑘<𝑡}1+𝑎₀𝜃_𝑘𝑒^𝑏0𝑡≤𝑎₀𝑎+𝛿^𝑇₀‖𝑥(𝜏)‖^𝜆_𝛼𝑑𝜏_{0<𝑡𝑘<𝑇}1+𝑎₀𝜃_𝑘𝑒^𝑏0𝑇. (2.12) Using the argument method [10, Lemma  3.1], there exists constant 𝑀>0 such that ‖𝑥(𝑡)‖_𝛼≤𝑀,∀𝑡∈[0,𝑇]. (2.13) This completes the proof.

Next, we extend the Ascoli-Arzela Theorem from 𝐶([0,𝑇],𝑋) to PC¹([0,𝑇],𝑋_𝛼).

Theorem 2.3. Suppose 𝒲⊂PC¹([0,𝑇],𝑋_𝛼) is a subset. If the following conditions are satisfied:(1)𝒲 is uniformly bounded subset of PC¹([0,𝑇],𝑋_𝛼),(2)𝒲 is equicontinuous in (𝑡_𝑖,𝑡_𝑖+1), 𝑖=0,1,2,…,𝑛, where 𝑡₀=0, 𝑡_𝑛+1=𝑇,(3)its 𝑡-sections 𝒲(𝑡)≡{𝑥(𝑡)∣𝑥∈𝒲,𝑡∈[0,𝑇]⧵Θ},  𝒲̇(𝑡)≡{𝑥̇(𝑡)∣𝑥∈𝒲,𝑡∈[0,𝑇]⧵Θ}, 𝒲(𝑡_𝑖+0)≡{𝑥(𝑡_𝑖+0)∣𝑥∈𝑊}, 𝒲(𝑡_𝑖−0)≡{𝑥(𝑡_𝑖−0)∣𝑥∈𝒲}, 𝒲̇(𝑡_𝑖+0)≡{𝑥̇(𝑡_𝑖+0)∣𝑥∈𝑊}, and 𝒲̇(𝑡_𝑖−0)≡{𝑥̇(𝑡_𝑖−0)∣𝑥∈𝒲} are relatively compact subsets of 𝑋_𝛼,
then 𝒲 is a relatively compact subset of PC¹([0,𝑇],𝑋_𝛼).

Proof. Let 𝑊=𝑥(⋅)⊂PC¹([0,𝑇],𝑋_𝛼) satisfy assumptions of Theorem 2.3 and {𝑥_𝑚(⋅)} any sequence of 𝑊. Define 𝑊¹=𝑊_[0,𝑡1−]={𝑥¹∈𝐶¹([0,𝑡₁],𝑋)|𝑥¹(𝑡)=𝑥(𝑡)for𝑡∈[0,𝑡₁),𝑥¹(𝑡₁)=𝑥(𝑡₁−0),𝑥∈𝑊}. By Ascoli-Arzela Theorem, 𝑊¹ is a relatively compact subset of 𝐶¹([0,𝑡₁],𝑋_𝛼). Then, there exists a subsequence of {𝑥_𝑚}, labeled {𝑥¹_𝑚}⊂𝑊, and 𝑥¹∈𝐶¹([0,𝑡₁],𝑋_𝛼), such that 𝑥¹_𝑚⟶𝑥¹in𝐶¹0,𝑡₁,𝑋as𝑚⟶+∞. (2.14) Define 𝑊²=𝑊_{[𝑡1+,𝑡2−]}={𝑥²∈𝐶¹([𝑡₁,𝑡₂],𝑋)∣𝑥²(𝑡)=𝑥(𝑡)for𝑡∈(𝑡₁,𝑡₂) and 𝑥²(𝑡₁)=𝑥(𝑡₁+0),𝑥²(𝑡₂)=𝑥(𝑡₂−0),𝑥∈𝑊}. It is not difficult to see that, due to Ascoli-Arzela Theorem again, 𝑊² is a relatively compact subset of 𝐶¹([𝑡₁,𝑡₂],𝑋_𝛼). There exists a subsequence of {𝑥¹_𝑚}, labeled {𝑥²_𝑚}⊂𝑊, and 𝑥²∈𝐶¹([𝑡₁,𝑡₂],𝑋_𝛼) such that 𝑥²_𝑚⟶𝑥²in𝐶¹𝑡₁,𝑡₂,𝑋as𝑚⟶+∞. (2.15) In general, define 𝑊^𝑖=𝑊_{[𝑡𝑖+,𝑡𝑖+1−]}={𝑥^𝑖∈𝐶¹([𝑡_𝑖,𝑡_𝑖+1],𝑋)∣𝑥²(𝑡)=𝑥(𝑡)for𝑡∈(𝑡_𝑖,𝑡_𝑖+1) and 𝑥^𝑖(𝑡_𝑖)=𝑥(𝑡_𝑖+0),𝑥^𝑖(𝑡_𝑖+1)=𝑥(𝑡_𝑖+1−0),𝑥∈𝑊}(𝑖=2,…,𝑛+1). Similarly, 𝑊ⁱ is a relatively compact subset of 𝐶¹([𝑡_𝑖,𝑡_𝑖+1],𝑋_𝛼). There exists a subsequence of {𝑥_𝑚^𝑖−1}, labeled {𝑥^𝑖_𝑚}⊂𝑊, and 𝑥^𝑖∈𝐶¹([𝑡_𝑖,𝑡_𝑖+1],𝑋) such that 𝑥^𝑖_𝑚⟶𝑥^𝑖in𝐶¹𝑡_𝑖,𝑡_𝑖+1,𝑋as𝑚⟶+∞. (2.16) Setting 𝑥(𝑡)=𝑥¹(𝑡),𝑡∈0,𝑡₁,𝑥²(𝑡),𝑡∈𝑡₁,𝑡₂,⋮𝑥^𝑛+1(𝑡),𝑡∈𝑡_𝑛,𝑇, (2.17) then 𝑥_𝑚^𝑛+1⟶𝑥inPC¹[0,𝑇],𝑋_𝛼as𝑚⟶+∞. (2.18) Thus, the set 𝑊 is a relatively compact set.

3. Solution of Second-Order Nonlinear Integro-Differential Equations of Mixed Type

We introduce reasonable mild solution for (1.1) and show the existence of PC-α-mild solution.

Definition 3.1. A function 𝑥∈PC¹([0,𝑇],𝑋_𝛼) is said to be a PC-α-mild solution of (1.1) if 𝑥 satisfies the following integral equation: 𝑥(𝑡)=𝑥₀+^𝑡₀𝑈(𝜏,0)𝑥₁𝑑𝜏+^𝑡₀^𝑡_𝜏𝑈(𝑠,𝜏)𝑓(𝜏,𝑥(𝜏),𝑥̇(𝜏),(𝐺𝑥)(𝜏),(𝐻𝑥)(𝜏))𝑑𝑠𝑑𝜏+_{0<𝑡𝑖<𝑡}𝐽⁰_𝑖𝑥𝑡_𝑖,𝑥̇𝑡_𝑖+_{0<𝑡𝑖<𝑡}^𝑡_𝑡𝑖𝑈𝑠,𝑡_𝑖𝐽¹_𝑖𝑥𝑡_𝑖,𝑥̇𝑡_𝑖𝑑𝑠. (3.1)
We introduce the following assumptions.

Assumption F. (1) The functions 𝑓∶[0,𝑇]×𝑋_𝛼×𝑋_𝛼×𝑋_𝛼×𝑋_𝛼→𝑋, 𝑔∶[0,𝑇]×𝑋_𝛼×𝑋_𝛼→𝑋_𝛼, and ℎ∶[0,𝑇]×𝑋_𝛼×𝑋_𝛼→𝑋_𝛼 are measurable in 𝑡∈[0,𝑇] and locally Lipschitz continuous, that is, for all 𝑥₁, 𝑥₂, 𝑦₁, 𝑦₂, 𝑧₁, 𝑧₂∈𝑋_𝛼, satisfying ‖𝑥₁‖_𝛼, ‖𝑥₂‖_𝛼, ‖𝑦₁‖_𝛼, ‖𝑦₂‖_𝛼, ‖𝑧₁‖_𝛼, ‖𝑧₂‖_𝛼, ‖𝑤₁‖_𝛼, ‖𝑤₂‖_𝛼≤𝜌, we have 𝑓𝑡,𝑥₁,𝑦₁,𝑧₁,𝑤₁−𝑓𝑡,𝑥₂,𝑦₂,𝑧₂,𝑤₂≤𝐿𝜌𝑥₁−𝑥_2𝛼+𝑦₁−𝑦_2𝛼+𝑧₁−𝑧_2𝛼+𝑤₁−𝑤_2𝛼,𝑔𝑡,𝑥₁,𝑦₁−𝑔𝑡,𝑥₂,𝑦_2𝛼≤𝐿𝜌𝑥₁−𝑥₂+𝑦₁−𝑦₂,ℎ𝑡,𝑥₁,𝑦₁−ℎ𝑡,𝑥₂,𝑦_2𝛼≤𝐿𝜌𝑥₁−𝑥₂+𝑦₁−𝑦₂. (3.2) (2) There exist a constant 0<𝜆<1 and a function 𝑞∈𝐿^𝑟([0,𝑇],[0,+∞))(𝑟>1) such that 𝑓𝑡,𝑥,𝑦,𝑧,𝑤≤𝑞(𝑡)1+‖𝑥‖_𝛼+𝑦_𝛼+‖𝑧‖_𝛼+‖𝑤‖_𝛼,∀𝑥,𝑦,𝑧,𝑤∈𝑋_𝛼,𝑔𝑡,𝑥,𝑦_𝛼≤𝑞(𝑡)1+‖𝑥‖_𝛼+𝑦_𝛼,∀𝑥,𝑦∈𝑋_𝛼,ℎ𝑡,𝑥,𝑦_𝛼≤𝑞(𝑡)1+‖𝑥‖^𝜆_𝛼+𝑦^𝜆_𝛼,∀𝑥,𝑦∈𝑋_𝛼. (3.3) (3) 𝑘, 𝑚∈𝐶([0,𝑇]²,𝑅).

Assumption J. There exists a constant 𝑏≥0 such that maps 𝐽⁰_𝑖∶𝑋_𝛼×𝑋_𝛼→𝑋_𝛼 and 𝐽¹_𝑖∶𝑋_𝛼×𝑋_𝛼→𝑋_𝛽𝑖  (0<𝛼<𝛽_𝑖<1) satisfy 𝐽⁰_𝑖𝑥₁,𝑦₁−𝐽⁰_𝑖𝑥₂,𝑦_2𝛼≤𝑏𝑥₁−𝑥_2𝛼+𝑦₁−𝑦_2𝛽𝑖(),𝐽𝑖=1,2,…,𝑛_1𝑖𝑥₁,𝑦₁−𝐽¹_𝑖𝑥₂,𝑦_2𝛽𝑖≤𝑏𝑥₁−𝑥_2𝛼+𝑦₁−𝑦_2𝛽𝑖(𝑖=1,2,…,𝑛). (3.4)

Theorem 3.2. Suppose that 𝐴(0) has a compact resolvent, (𝑥₀,𝑥₁)∈𝑋_𝛼×𝑋_𝛽, 0<𝛼<𝛽≤1,and(𝛼+𝛽)/2<(𝑟−1)/𝑟. Under the Assumptions A, F and J, the second-order impulsive integro-differential equation (1.1) has a PC-α-mild solution 𝑥∈PC¹([0,𝑇],𝑋_𝛼).

Proof. Let (𝑥₀,𝑥₁)∈𝑋_𝛼×𝑋_𝛽 be fixed; define the operator 𝑃 on PC¹([0,𝑇],𝑋_𝛼) given by (𝑃𝑥)(𝑡)=𝑥₀+^𝑡₀𝑈(𝜏,0)𝑥₁𝑑𝜏+^𝑡₀^𝑡_𝜏𝑈(𝑠,𝜏)𝑓(𝜏,𝑥(𝜏),𝑥̇(𝜏),(𝐺𝑥)(𝜏),(𝐻𝑥)(𝜏))𝑑𝑠𝑑𝜏+_{0<𝑡𝑖<𝑡}𝐽⁰_𝑖𝑥𝑡_𝑖,𝑥̇𝑡_𝑖+_{0<𝑡𝑖<𝑡}^𝑡_𝑡𝑖𝑈𝑠,𝑡_𝑖𝐽¹_𝑖𝑥𝑡_𝑖,𝑥̇𝑡_𝑖𝑑𝑠. (3.5) By virtue of the properties of evolution system 𝑈(⋅,⋅) and Assumption J, for 𝑡∈[0,𝑇]⧵Θ, we have (𝑃𝑥)(𝑡)∈𝑋_𝛼 and 𝑑(𝑡)()𝑥𝑑𝑡𝑃𝑥)(=𝑈𝑡,0₁+^𝑡₀𝑈()𝑓((𝜏),𝑥̇(𝜏),(𝜏),(𝜏+𝑡,𝜏𝜏,𝑥𝐺𝑥)(𝐻𝑥)())𝑑𝜏_{0<𝑡𝑖<𝑡}𝑈𝑡,𝑡_𝑖𝐽¹_𝑖𝑥𝑡_𝑖,𝑥̇𝑡_𝑖. (3.6) This means that 𝑃𝑥∈PC¹([0,𝑇],𝑋_𝛼) for 𝑥∈PC¹([0,𝑇],𝑋_𝛼).
For 𝑥,𝑦∈PC¹([0,𝑇],𝑋_𝛼) and ‖𝑥‖_PC¹, ‖𝑦‖_PC1≤𝜌 in which 𝜌 is a constant. Using Assumptions F(1) and J, we have (𝑃𝑥)(𝑡)−𝑃𝑦(𝑡)_𝛼+𝑑(𝑡)−𝑑𝑑𝑡𝑃𝑥)((𝑡𝑑𝑡𝑃𝑦)_𝛼≤^𝑡₀^𝑡_𝜏‖𝑈(𝑠,𝜏)‖_0,𝛼((𝜏),𝑥̇(𝜏),(𝜏),(𝜏(𝜏),𝑦̇(𝜏),(𝜏),(𝜏)+‖𝑓𝜏,𝑥𝐺𝑥)(𝐻𝑥)())−𝑓𝜏,𝑦𝐺𝑦𝐻𝑦‖𝑑𝑠𝑑𝜏_𝑡0‖𝑈(𝑡,𝜏)‖_0,𝛼((𝜏),𝑥̇(𝜏),(𝜏),(𝜏(𝜏),𝑦̇(𝜏),(𝜏),(𝜏)‖𝑓𝜏,𝑥𝐺𝑥)(𝐻𝑥)())−𝑓𝜏,𝑦𝐺𝑦𝐻𝑦‖𝑑𝜏+_{0<𝑡𝑖<𝑡}𝐽⁰_𝑖𝑥𝑡_𝑖,𝑥̇𝑡_𝑖−𝐽⁰_𝑖𝑦𝑡_𝑖,𝑦̇𝑡_𝑖𝛼+_{0<𝑡𝑖<𝑡}^𝑡_𝑡𝑖𝑈𝑠,𝑡_{𝑖𝛽𝑖,𝛼}𝐽¹_𝑖𝑥𝑡_𝑖,𝑥̇𝑡_𝑖−𝐽¹_𝑖𝑦𝑡_𝑖,𝑦̇𝑡_𝑖𝛽𝑖𝑑𝑠+_{0<𝑡𝑖<𝑡}𝑈𝑡,𝑡_{𝑖𝛽𝑖,𝛼}𝐽¹_𝑖𝑥𝑡_𝑖,𝑥̇𝑡_𝑖−𝐽¹_𝑖𝑦𝑡_𝑖,𝑦̇𝑡_𝑖𝛽𝑖≤𝐶𝛼,𝛾𝐿²𝜌‖𝑚‖𝑇𝑥−𝑦_PC1^𝑡₀^𝑡_𝜏|𝑠−𝜏|^−𝛾𝑑𝑠𝑑𝜏+𝐶𝛼,𝛾𝐿²𝜌‖𝑚‖𝑇𝑥−𝑦_PC1×^𝑡₀|𝑡−𝜏|^−𝛾𝑑𝜏𝑑𝑠+𝐶𝛼,𝛾1+𝐿𝜌²𝑘‖)×(‖𝑇+1_{𝑡0𝑡𝜏}|𝑠−𝜏|^−𝛾𝑥(𝜏)−𝑦(𝜏)_𝛼+𝑥̇(𝜏)−𝑦̇(𝜏)_𝛼𝑑𝑠𝑑𝜏+𝐶𝛼,𝛾(1+𝐿(𝜌))²(‖𝑚‖𝑇+1)^𝑡₀^𝑡_𝜏|𝑠−𝜏|^−𝛾𝑥−𝑦_𝜏𝐵+𝑥̇−𝑦̇_𝜏𝐵𝑑𝑠𝑑𝜏+𝐶𝛼,𝛾1+𝐿𝜌²(‖𝑘‖𝑇+1)^𝑡₀|𝑡−𝜏|^−𝛾𝑥(𝜏)−𝑦(𝜏)_𝛼+𝑥̇(𝜏)−𝑦̇(𝜏)_𝛼𝑑𝜏+𝐶𝛼,𝛾1+𝐿𝜌²(‖𝑚‖𝑇+1)^𝑡₀|𝑡−𝜏|^−𝛾𝑥−𝑦_𝜏𝐵+𝑥̇−𝑦̇_𝜏𝐵𝑑𝜏+𝑏_{0<𝑡𝑖<𝑡}𝑥𝑡_𝑖−𝑦𝑡_𝑖𝛼+𝑥̇𝑡_𝑖−𝑦̇𝑡_𝑖𝛼()×+𝑏𝑇+1_{0<𝑡𝑖<𝑡}𝐶𝛽_𝑖,𝛼𝑥𝑡_𝑖−𝑦𝑡_𝑖𝛼+𝑥̇𝑡_𝑖−𝑦(̇𝑡_𝑖)_𝛼≤𝑀𝑥−𝑦_PC1, (3.7) where 𝛼<𝛾<1, 𝜌=𝜌+2𝑞_𝐿1𝜌+𝜌^𝜆+1(‖𝑘‖+‖𝑚‖),𝑀=2𝐶𝛼,𝛾𝐿𝜌+1²𝑘‖‖𝑚‖)𝑇(‖𝑇+𝑇+1^1−𝛾+𝑇^2−𝛾()1−𝛾+𝑏𝑇+1^𝑛_𝑖=1𝐶𝛽_𝑖,𝛼+1. (3.8) Hence 𝑃∶PC¹([0,𝑇],𝑋_𝛼)→PC¹([0,𝑇],𝑋_𝛼) is a continuous operator.
Let 𝒳⊆PC¹([0,𝑇],𝑋_𝛼) be a bounded subset; there exists a constant 𝜇>0 such that ‖𝑥‖_PC¹≤𝜇 for all 𝑥∈𝒳. By Assumption F(2) and the properties of evolution operators, there exists constant 𝜔>0 such that 𝑓(𝑡,𝑥(𝑡),𝑥̇(𝑡),(𝐺𝑥)(𝑡),(𝐻𝑥)(𝑡))≤𝜔𝑞(𝑡),∀𝑥∈𝒳 (3.9) and 𝒦=𝑃𝒳 is a bounded subset of PC¹([0,𝑇],𝑋_𝛼). Define 𝒦(𝑡)=𝑡)},𝒦̇(𝑡)=𝑑{(𝑃𝑥)(∣𝑥∈𝒳(𝑡)[]𝑑𝑡𝑃𝑥)(∣𝑥∈𝒳for𝑡∈0,𝑇. (3.10) Clearly, 𝒦(0)={𝑥₀} and 𝒦̇(0)={𝑥₁} are compact, and hence, it is only necessary to consider 𝑡>0. Since 𝐴⁻¹(0) is a compact operator, 𝑋_𝛽↪↪𝑋_𝛼 for 0≤𝛼<𝛽≤1. For 0<𝜀<𝑡≤𝑇, define 𝒦_𝜀(𝑡)=𝑃_𝜀𝑥(𝑡)(),𝒦̇=𝑈𝑡,𝑡−𝜀)(𝑃𝑥)(𝑡−𝜀∣𝑥∈𝒳_𝜀(𝑡)=𝑑𝑃𝑑𝑡^𝜀𝑥𝑡)()𝑑(=𝑈𝑡,𝑡−𝜀().𝑑𝑡𝑃𝑥)(𝑡−𝜀∣𝑥∈𝒳 (3.11) By the properties of {𝑈(𝑡,𝑠)∣(𝑠,𝑡)∈Δ}, one can verify that (𝑃𝑥)(𝑡)−𝑃_𝜀𝑥(𝑡)_𝛼≤𝑥₀−𝑈(𝑡,𝑡−𝜀)𝑥_0𝛼+^𝑡_𝑡−𝜀‖𝑈(𝑠,0)‖_𝛽,𝛼𝑥_1𝛽+‖(𝑑𝑠𝐼−𝑈𝑡,𝑡−𝜀)‖_{(𝛼+𝛽)/2,𝛼0}^𝑡−𝜀‖𝑈(𝑠,0)‖_{𝛽,(𝛼+𝛽)/2}𝑥_1𝛽+𝑑𝑠_{𝑡𝑡−𝜀𝑡𝜏}‖𝑈(𝑠,𝜏)‖_0,𝛼𝑓((𝜏),𝑥̇(𝜏),(𝜏),(𝜏+𝜏,𝑥𝐺𝑥)(𝐻𝑥)())𝑑𝑠𝑑𝜏_{0𝑡−𝜀𝑡𝑡−𝜀}‖𝑈(𝑠,𝜏)‖_0,𝛼𝑓((𝜏),𝑥̇(𝜏),(𝜏),(𝜏+‖(𝜏,𝑥𝐺𝑥)(𝐻𝑥)())𝑑𝑠𝑑𝜏𝐼−𝑈𝑡,𝑡−𝜀)‖_{(𝛼+𝛽)/2,𝛼}×₀^𝑡−𝜀_𝜏^𝑡−𝜀‖𝑈(𝑠,𝜏)‖_{0,(𝛼+𝛽)/2}𝑓((𝜏),𝑥̇(𝜏),(𝜏),(𝜏≤𝑥𝜏,𝑥𝐺𝑥)(𝐻𝑥)())𝑑𝑠𝑑𝜏₀−𝑈(𝑡,𝑡−𝜀)𝑥_0𝛼+𝐶𝛽,𝛼𝑥_1𝛽𝜀+𝐶𝛼+𝛽2,𝛼,𝛾₁𝐶𝛽,𝛼+𝛽2𝑥_1𝛽𝜀^𝛾1+𝜔𝐶𝛼,𝛾𝑞1−𝛾^𝐿𝑟𝑟−1𝑟2−𝛾−1^{(𝑟−1)/𝑟}𝜀^{(𝑟(2−𝛾)−1)/𝑟}+𝜔𝑞_𝐿𝑟𝐶𝛼,𝛾𝑟−1𝑟−𝑟𝛾−1^{(𝑟−1)/𝑟}𝑇^{(𝑟−𝑟𝛾−1)/𝑟}𝜀+𝜔𝑞_𝐿𝑟𝐶𝛼+𝛽2,𝛼,𝛾₁𝐶𝛼+𝛽2,𝛾₂𝑟−1𝑟−1−𝑟𝛾₂^{(𝑟−1)/𝑟}𝑇^{(2𝑟−1−𝛾2)/𝑟}𝜀^𝛾1, (3.12) where 0<𝛾₁<(𝛽−𝛼)/2,  (𝛼+𝛽)/2<𝛾₂<1, and 𝑑(𝑡)−𝑑𝑑𝑡𝑃𝑥)(𝑃𝑑𝑡^𝜀𝑥(𝑡)_𝛼≤𝜔𝑞_𝐿𝑟𝐶𝛼,𝛾𝑟−1𝜀𝑟−1−𝑟𝛾^{(𝑟−1−𝑟𝛾)/(𝑟−1)},𝑥∈𝒳. (3.13) This means that the set 𝒦(𝑡)(𝒦̇(𝑡)) can be approximated to an arbitrary degree of accuracy by a relatively compact set for 𝑡∈(0,𝑡₁]. Hence 𝒦(𝑡)(𝒦̇(𝑡)) itself is relatively compact in 𝑋_𝛼 for 𝑡∈(0,𝑡₁]. Define 𝒦𝑡₁+0=(𝑃𝑥)(𝑡)|_𝑡=𝑡1+𝐽⁰₁(𝑃𝑥)𝑡₁,𝑑(𝑡𝑑𝑡𝑃𝑥)()|_𝑡=𝑡1,𝒦̇𝑡₁+0=𝑑(𝑡𝑑𝑡𝑃𝑥)()|_𝑡=𝑡1+𝐽¹₁(𝑃𝑥)𝑡₁,𝑑(𝑡𝑑𝑡𝑃𝑥)()|_𝑡=𝑡1. (3.14) Using relative compactness of 𝒦(𝑡) and 𝒦̇(𝑡) (𝑡∈[0,𝑡₁]) and Assumption J, one can show that 𝒦(𝑡₁+0) and 𝒦̇(𝑡₁+0) are relative compacts in 𝑋_𝛼.
By the same procedure, the compactness of 𝒦̇(𝑡) and 𝒦(𝑡) (𝑡∈[0,𝑇]) can also be proved.
Further, using representations of (𝑃𝑥)(𝑡) and (𝑑/𝑑𝑡)(𝑃𝑥)(𝑡), properties of {𝑈(𝑡,𝑠),0≤𝑠≤𝑡≤𝑇} (see Theorem 2.1), and those above, one can show that 𝑃𝑥 is piecewise equicontinuous. Summarily, 𝑃 is a compact operator in PC¹([0,𝑇],𝑋_𝛼).
By virtue of Gronwall inequality (see Theorem 2.2), one can verify that 𝒴={𝑥∈PC¹([0,𝑇],𝑋_𝛼)∣𝑥=𝛿𝑃𝑥,𝛿∈[0,1]} is a bounded subset of PC¹([0,𝑇],𝑋_𝛼). According to Leray-Schauder fixed point theorem, 𝑃 has a fixed point in PC¹([0,𝑇],𝑋_𝛼). It can be given by the representation (3.1) and 𝑥(𝑡)∈𝐷(𝐴) for 𝑡∈[0,𝑇].

Remark 3.3. Similarly, under suitable assumptions, the following second-order nonlinear impulsive integro-differential equation of mixed type 𝑥̈(𝑡)(𝑡)𝑥̇(𝑡)((𝑡),𝑥̇(𝑡),(𝑡),(𝑡(]⧵(),𝑥(0)+𝐴=𝑓𝑡,𝑥𝐺𝑥)(𝐻𝑥)()),𝑡∈0,𝑇Θ∪Λ=𝑥₀,Δ𝑥𝑡_𝑖=𝐽⁰_𝑖𝑥𝑡_𝑖,𝑥̇𝑡_𝑖,𝑡_𝑖𝑥̇(0)∈Θ,𝑖=1,2,…,𝑛,=𝑥₁,Δ𝑥̇𝑠_𝑗=𝐽¹_𝑗𝑥𝑠_𝑗,𝑥̇𝑠_𝑗,𝑠_𝑗∈Λ,𝑗=1,2,…,𝑚, (3.15) has a PC-α-mild solution 𝑥∈PC¹([0,𝑇],𝑋_𝛼) given by 𝑥(𝑡)=𝑥₀+^𝑡₀𝑈(𝜏,0)𝑥₁𝑑𝜏+^𝑡₀^𝑠_𝜏𝑈(𝑠,𝜏)𝑓(𝜏,𝑥(𝜏),𝑥̇(𝜏),(𝐺𝑥)(𝜏),(𝐻𝑥)(𝜏))𝑑𝑠𝑑𝜏+_{0<𝑡𝑖<𝑡}𝐽⁰_𝑖𝑥𝑡_𝑖,𝑥̇𝑡_𝑖+_{0<𝑠𝑗<𝑡}^𝑡_𝑠𝑗𝑖𝑈𝑠,𝑠_𝑗𝐽¹_𝑗𝑥𝑠_𝑗,𝑥̇𝑠_𝑗𝑑𝑠. (3.16)

Remark 3.4. Assume that 𝐴(0) has a compact resolvent, (𝑥₀,𝑥₁)∈𝑋_𝛼×𝑋_𝛽, 0<𝛼<𝛽≤1, and (𝛼+𝛽)/2<(𝑟−1)/𝑟.
Under the Assumptions A and F, the second-order nonlinear integro-differential equation of mixed type 𝑥̈(𝑡)(𝑡)𝑥̇(𝑡)((𝑡),𝑥̇(𝑡),(𝑡),(𝑡(],𝑥(0)+𝐴=𝑓𝑡,𝑥𝐺𝑥)(𝐻𝑥)()),𝑡∈0,𝑇=𝑥₀,𝑥̇(0)=𝑥₁ (3.17) has a unique α-mild solution 𝑥∈PC¹([0,𝑇],𝑋_𝛼) given by 𝑥(𝑡)=𝑥₀+^𝑡₀𝑈(𝜏,0)𝑥₁𝑑𝜏+^𝑡₀^𝑠_𝜏𝑈(𝑠,𝜏)𝑓(𝜏,𝑥(𝜏),𝑥̇(𝜏),(𝐺𝑥)(𝜏),(𝐻𝑥)(𝜏))𝑑𝑠𝑑𝜏. (3.18)

4. Existence of Optimal Controls

We suppose that 𝑌 is a reflexive Banach space from which the controls 𝑢 take the values. We denote a class of nonempty closed and convex subsets of 𝑌 by 𝑃_𝑓(𝑌). Assume that the multifunction 𝜔∶[0,𝑇]→𝑃_𝑓(𝑌) is measurable and 𝜔(⋅)⊂𝐸 where 𝐸 is a bounded set of 𝑌, the admissible control set 𝑈_ad={𝑢∈𝐿_𝑝(𝐸)∣𝑢(𝑡)∈𝜔(𝑡)𝑎.𝑒}≠∅ (see [18, Proposition  1.7, page  142, Lemma  3.2, page  175]).

Consider the following controlled system:𝑥̈(𝑡)(𝑡)𝑥̇(𝑡)((𝑡),𝑥̇(𝑡),(𝑡),(𝑡(𝑡)𝑢(𝑡)(]𝑥(0)+𝐴=𝑓𝑡,𝑥𝐺𝑥)(𝐻𝑥)())+𝐵,𝑡∈0,𝑇⧵Θ,=𝑥₀,Δ𝑥𝑡_𝑖=𝐽⁰_𝑖𝑥𝑡_𝑖,𝑥̇𝑡_𝑖,𝑡_𝑖𝑥̇(0)∈Θ,𝑖=1,2,…,𝑛,=𝑥₁,Δ𝑥̇𝑡_𝑖=𝐽¹_𝑖𝑥𝑡_𝑖,𝑥̇𝑡_𝑖,𝑡_𝑖∈Θ,𝑖=1,2,…,𝑛. (4.1)

Assumption B. 𝐵∈𝐿_∞([0,𝑇],£(𝑌,𝑋)).
It is easy to see that 𝐵𝑢∈𝐿_𝑝([0,𝑇],𝑋) for all 𝑢∈𝑈_ad. Define ℱ(𝑡,𝑥(𝑡),𝑥̇(𝑡),(𝐺𝑥)(𝑡),(𝐻𝑥)(𝑡))=𝑓(𝑡,𝑥(𝑡),𝑥̇(𝑡),(𝐺𝑥)(𝑡),(𝐻𝑥)(𝑡))+𝐵(𝑡)𝑢(𝑡). It is obvious that ℱ satisfies the Assumption F.

Theorem 4.1. Suppose 𝐴(0) has a compact resolvent. Under Assumptions A, B, F, and J, for every 𝑢∈𝑈_ad, the system (4.1) has a PC-α-mild solution corresponding to 𝑢 provided 0<(𝛼+𝛽)/2,𝛽<𝑚𝑖𝑛{(𝑝−1)/𝑝,(𝑟−1)/𝑟}.

Define 𝒯={(𝑥,𝑢)∈PC¹([0,𝑇],𝑋_𝛼)×𝑈_ad∣𝑥 as a PC-α-mild solution of system (4.1) corresponding to the control 𝑢∈𝑈_ad}. (𝑥,𝑢)∈𝒯 is called a feasible pair. We consider the Lagrange problem (P).

Find (𝑥⁰,𝑢⁰)∈𝒯 such that𝐽𝑥⁰,𝑢⁰≤𝐽(𝑥,𝑢),∀(𝑥,𝑢)∈𝒯, (4.2) where𝐽(𝑥,𝑢)=^𝑇₀𝑙(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))𝑑𝑡. (4.3)

We introduce some assumptions on 𝑙.

Assumption L. (1) The functional 𝑙∶[0,𝑇]×𝑋_𝛼×𝑋_𝛼×𝑌→𝑅∪∞ is Borel measurable.
(2) 𝑙(𝑡,⋅,⋅,⋅) is sequentially lower semicontinuous on 𝑋_𝛼×𝑋_𝛼× for almost all 𝑡∈[0,𝑇].
(3) 𝑙(𝑡,𝑥,𝑦,⋅) is convex on 𝑌 for each (𝑥,𝑦)∈𝑋_𝛼×𝑋_𝛼 and almost all 𝑡∈[0,𝑇].
(4) There exist constants 𝑏≥0, 𝑐≥0, 𝜛>0, and 𝜑∈𝐿₁([0,𝑇],𝑅) such that 𝑙𝑡,𝑥,𝑦,𝑢≥𝜑(𝑡)+𝑏‖𝑥‖_𝛼+𝑐𝑦_𝛼+𝜛‖𝑢‖^𝑝_𝑌,∀𝑥,𝑦∈𝑋_𝛼,𝑢∈𝑌. (4.4)

Now we can give the following result on the existence of optimal controls for problem (P).

Theorem 4.2. Under assumptions of Theorem 4.1 and assumption L, the problem (P) has a solution.

Proof. If inf{𝐽(𝑥,𝑢)∣𝑢∈𝑈_ad}=+∞, there is nothing to prove. Assume that inf{𝐽(𝑥,𝑢)∣𝑢∈𝑈_ad}=𝑚<+∞. By Assumption L, we have 𝑚>−∞. By definition of infimum, there exists a sequence {𝑢^𝑛}⊂𝑈_ad such that 𝐽(𝑥^𝑢,𝑢^𝑛)→𝑚. Since {𝑢^𝑛} is bounded in 𝐿_𝑝([0,𝑇],𝑌), there exists a subsequence, relabeled as {𝑢^𝑛}, and 𝑢⁰∈𝐿_𝑝([0,𝑇],𝑌) such that 𝑢⟶^𝑛𝑤𝑢⁰in𝐿_𝑝([0,𝑇],𝑌). (4.5) Since 𝑈_ad is closed and convex, from the Mazur Lemma, we have 𝑢⁰∈𝑈_ad.
Suppose 𝑥^𝑛 is the PC-α-mild solution of (4.1) corresponding to 𝑢^𝑛 (𝑛=0,1,2,…). Then 𝑥^𝑛 satisfies the following impulsive integral equation: 𝑥^𝑛(𝑡)=𝑥₀+^𝑡₀𝑈(𝜏,0)𝑥₁+𝑑𝜏_𝑡0𝑠0𝑈(𝑠,𝜏)𝑓𝜏,𝑥^𝑛(𝜏),𝑥̇^𝑛(𝜏),𝐺𝑥^𝑛(𝜏),𝐻𝑥^𝑛(𝜏)+𝐵(𝜏)𝑢^𝑛(𝜏)𝑑𝜏𝑑𝑠+_{0<𝑡𝑖<𝑡}𝐽⁰_𝑖𝑥^𝑛𝑡_𝑖,𝑥̇^𝑛𝑡_𝑖+_{0<𝑡𝑖<𝑡}^𝑡_𝑡𝑖𝑈𝑠,𝑡_𝑖𝐽¹_𝑖𝑥^𝑛𝑡_𝑖,𝑥̇^𝑛𝑡_𝑖𝑑𝑠. (4.6) Setting 𝐹_𝑛(𝑡)=𝑓(𝑡,𝑥^𝑛(𝑡),𝑥̇^𝑛(𝑡),(𝐺𝑥^𝑛)(𝑡),(𝐻𝑥^𝑛)(𝑡)), by Assumptions F(2) and J, and Lemma  3.1, we obtain that 𝐹_𝑛∈ℒ_𝑟([0,𝑇],𝑋) (𝑟>1). Furthermore, there exists a subsequence, relabeled as {𝐹_𝑛}, and 𝐹∈𝐿_𝑟([0,𝑇],𝑋) such that 𝐹−→_𝑛^𝑤𝐹 in 𝐿_𝑟([0,𝑇],𝑋) as 𝑛→∞.
Define operators 𝑄∶𝐿_𝑟([0,𝑇],𝑋)→𝐶([0,𝑇],𝑋), given by 𝑄𝑦(𝑡)=^𝑡₀𝑈(𝑡,𝑠)𝑦(𝑠)𝑑𝑠. (4.7) By [7, Lemma  4.1], we have 𝑄𝐹_𝑛⟶𝑄𝐹in𝐶([0,𝑇],𝑋)as𝑛⟶∞. (4.8)
Consider the following impulsive differential equation: 𝑥̈(𝑡)+𝐴(𝑡)𝑥̇(𝑡)=𝐹(𝑡)+𝐵(𝑡)𝑢⁰(𝑡)(]𝑥(0),𝑡∈0,𝑇⧵Θ,=𝑥₀,Δ𝑥𝑡_𝑖=𝐽⁰_𝑖𝑥𝑡_𝑖,𝑥̇𝑡_𝑖,𝑡_𝑖𝑥̇(0)∈Θ,𝑖=1,2,…,𝑛,=𝑥₁,Δ𝑥̇𝑡_𝑖=𝐽¹_𝑖𝑥𝑡_𝑖,𝑥̇𝑡_𝑖,𝑡_𝑖∈Θ,𝑖=1,2,…,𝑛,. (4.9) By Theorem 3.2, we know that (4.9) have a PC-α-mild solution 𝑥(𝑡)=𝑥₀+^𝑡₀𝑈(𝜏,0)𝑥₁𝑑𝜏+^𝑡₀^𝑠₀𝑈(𝑠,𝜏)𝐹(𝜏)+𝐵(𝜏)𝑢⁰(𝜏)𝑑𝜏𝑑𝑠+_{0<𝑡𝑖<𝑡}𝐽⁰_𝑖𝑥𝑡_𝑖,𝑥̇𝑡_𝑖+_{0<𝑡𝑖<𝑡}^𝑡_𝑡𝑖𝑈𝑠,𝑡_𝑖𝐽¹_𝑖𝑥𝑡_𝑖,𝑥̇𝑡_𝑖𝑑𝑠. (4.10) Define 𝜂_𝑛(𝑡)=^𝑡₀𝑈(𝑡,𝜏)𝐹_𝑛(𝜏)−𝐹(𝜏)+𝐵(𝜏)𝑢^𝑛(𝜏)−𝐵(𝜏)𝑢⁰(𝜏)𝑑𝜏, (4.11) then,  𝜂_𝑛→0 in 𝐶([0,𝑇],𝑋_𝛼) as 𝑛→∞. Using Assumption J, we obtain 𝑥^𝑛(𝑡)−𝑥(𝑡)_𝛼+𝑥̇^𝑛(𝑡)−𝑥̇(𝑡)_𝛼≤(𝑇+1)𝜂_{𝑛𝐶([0,𝑇],𝑋𝛼)}+𝑏(𝑇+1)_{0<𝑡𝑖<𝑡}𝐶𝛽_𝑖,𝛼+1𝑥^𝑛𝑡_𝑖−𝑥𝑡_𝑖𝛼+𝑥̇^𝑛𝑡_𝑖−𝑥̇𝑡_𝑖𝛼. (4.12) Using the Gronwall lemma with impulse (see [17, Lemma  1.7.1]), we have 𝑥^𝑛(𝑡)−𝑥(𝑡)_𝛼+𝑥̇^𝑛(𝑡)−𝑥̇(𝑡)_𝛼≤𝑏(𝑇+1)^2𝑛_𝑖=1𝐶𝛽_𝑖,𝛼+1𝜂_{𝑛𝐶([0,𝑇],𝑋)}, (4.13) that is, 𝑥^𝑛→𝑥 in PC¹([0,𝑇],𝑋_𝛼) as 𝑛→∞. Further, 𝐹_𝑛(⋅)⟶𝑓⋅,𝑥(⋅),𝑥̇(⋅),(𝐺𝑥)(⋅),(𝐻𝑥)(⋅)in𝐿_𝑟([0,𝑇],𝑋)as𝑛⟶∞. (4.14) By the uniqueness of limit, we have 𝐹(𝑡)=𝑓𝑡,𝑥(𝑡),𝑥̇(𝑡),(𝐺𝑥)(𝑡),(𝐻𝑥)(𝑡), (4.15) Furthermore, 𝑥(𝑡)=𝑥₀+^𝑡₀𝑈(𝜏,0)𝑥₁+𝑑𝜏_𝑡0𝑠0𝑈(𝑠,𝜏)𝑓𝜏,𝑥(𝜏),𝑥̇(𝜏),(𝐺𝑥)(𝜏),(𝐻𝑥)(𝜏)+𝐵(𝜏)𝑢⁰(𝜏)𝑑𝜏𝑑𝑠+_{0<𝑡𝑖<𝑡}𝐽⁰_𝑖𝑥𝑡_𝑖,𝑥̇𝑡_𝑖+_{0<𝑡𝑖<𝑡}^𝑡_𝑡𝑖𝑈𝑠,𝑡_𝑖𝐽¹_𝑖𝑥𝑡_𝑖,𝑥̇𝑡_𝑖𝑑𝑠. (4.16) Thus, 𝑥 is a PC-α-mild solution of (4.1) corresponding to 𝑢⁰.
Since PC¹([0,𝑇],𝑋_𝛼)↪𝐿₁([0,𝑇],𝑋_𝛼), using the Assumption L, we can obtain 𝑚=lim_𝑛→∞^𝑇₀𝑙𝑡,𝑥^𝑛(𝑡),𝑥̇^𝑛(𝑡),𝑢^𝑛(𝑡)𝑑𝑡≥^𝑇₀𝑙𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢⁰(𝑡)𝑑𝑡=𝐽𝑥,𝑢⁰≥𝑚. (4.17) This means that 𝐽 attains its minimum at 𝑢⁰∈𝑈_ad.

At the end of this section, an example is given to illustrate our theory. We consider the following problem:𝑥̈+()Δ𝑥̇=𝑥̇𝑡,𝑦𝑡+1𝑡,𝑦𝑡,𝑦+𝑥𝑡,𝑦+^𝑡₀(𝑡−𝜏)𝑥²𝜏,𝑦+𝑥̇²+𝜏,𝑦+1𝑑𝜏₁₀(𝑡+𝜏)²𝑥²𝜏,𝑦+𝑥̇²(]⧵1𝜏,𝑦+1𝑑𝜏+𝑢𝑡,𝑦,𝑦∈Ω,𝑡∈0,1,210910,…,,𝑥10𝑡,𝑦_{[0,1]×𝜕Ω}𝜕=0,𝑥0,𝑦=0,𝑥𝜕𝑡𝑡,𝑦_{[0,1]×𝜕Ω}𝜕=0,𝑥𝜕𝑡𝑡,𝑦_𝑡=0𝑥𝑖=0,𝑦∈Ω,𝑖10+0,𝑦−𝑥=110−0,𝑦2𝑥𝑖𝑥̇𝑖10−0,𝑦,𝑖=1,…,9,−𝑥̇𝑖10+0,𝑦=110−0,𝑦2𝑥̇𝑖10−0,𝑦,𝑖=1,2,…,9, (4.18) where 𝑥̈(𝑡,𝑦)=(𝜕²/𝜕𝑡²)𝑥(𝑡,𝑦), 𝑥̇(𝑡,𝑦)=(𝜕/𝜕𝑡)𝑥(𝑡,𝑦), Ω⊂𝑅³ is bounded domain, and 𝜕Ω∈𝐶³.

Define 𝑋=𝑌=𝐿²(Ω), 𝐷(𝐴(𝑡))=𝐻²(Ω)∩𝐻¹₀(Ω), and 𝐴(𝑡)𝑥=(𝑡+1)(𝜕²𝑥/𝜕𝑦²₁+𝜕²x/𝜕𝑦²₂+𝜕²𝑥/𝜕𝑦²₃) for 𝑥∈𝐷(𝐴(𝑡)); 𝑈_ad is a nonempty bounded closed subset of 𝑌. Let 𝐽(𝑢)=¹_0Ω𝑥(𝑡,𝜉)²𝑑𝜉𝑑𝑡+¹_0Ω𝜕𝑥(𝜕𝑡𝑡,𝜉)²𝑑𝜉𝑑𝑡+¹_0Ω𝑢(𝑡,𝜉)²𝑑𝜉𝑑𝑡. (4.19)

Define 𝑉(⋅)[𝑢(⋅)](𝑦)=𝑢(⋅,𝑦), [𝑥(⋅)](𝑦)=𝑥(⋅,𝑦),  [(𝐺𝑥)(⋅)](𝑦)=∫^⋅₀(⋅−𝜏)𝑥²(𝜏,𝑦)+𝑥̇²(𝜏,𝑦)+1𝑑𝜏, [(𝐻𝑥)(⋅)](𝑦)=∫¹₀(⋅+𝜏)²𝑥²(𝜏,𝑦)+𝑥̇²(𝜏,𝑦)+1𝑑𝜏, [𝐹(⋅,𝑥(⋅),𝑥̇(⋅),(𝐺𝑥)(⋅)),(𝐻𝑥)(⋅))](𝑦)=∫^⋅₀(⋅−𝜏)𝑥²(𝜏,𝑦)+𝑥̇²(𝜏,𝑦)+1𝑑𝜏+∫¹₀(⋅+𝜏)²𝑥²(𝜏,𝑦)+𝑥̇²(𝜏,𝑦)+1𝑑𝜏+𝑥(⋅,𝑦)+𝑥̇(⋅,𝑦), [𝑔⁰_𝑖(𝑥(𝑖/10))](𝑦)=1/2𝑥(𝑖/10,𝑦), [𝑔¹_𝑖(𝑥̇(𝑖/10))](𝑦)=(1/2)𝑥̇(𝑖/10,𝑦); then 𝐹 satisfies Assumption F. Thus the problem (4.18) can be rewritten as𝑥̈(𝑡)(𝑡)𝑥̇(𝑡)((𝑡),𝑥̇(𝑡),(𝑡),(𝑡(𝑡)𝑢(𝑡)(]⧵1+𝐴=𝐹𝑡,𝑥𝐺𝑥)(𝐻𝑥)())+𝑉,𝑡∈0,1910,…,,𝑥(0)𝑖10=0,𝑥+𝑖10−𝑥−10=𝑔⁰_𝑖𝑥𝑖𝑥̇(0)𝑥̇𝑖10,𝑖=1,2,…,9,=0,+−𝑥̇𝑖10−10=𝑔¹_𝑖𝑥̇𝑖10,𝑖=1,2,…,9, (4.20) with the cost function𝐽(𝑢)=¹₀‖𝑥(𝑡)‖²_𝑋+‖𝑥̇(𝑡)‖²_𝑋+‖𝑢(𝑡)‖²_𝑋𝑑𝑡. (4.21)Obviously, satisfying all the assumptions given in our former Theorem 4.2, our results can be used in (4.20).

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant no. 10961009, the Fok Ying Tung Education Foundation under Grant no. 121104 and Introducing Talents Foundation for the Doctor of Guizhou University.