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

Authors: Y. Peng
Journal: Abstract and Applied Analysis
 (2010)

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## 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.


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## Body

## 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 by(1.1)ẍ(t)+A(t)ẋ(t)=f(t,x(t),ẋ(t),(Gx)(t),(Hx)(t)),t∈(0,T]∖Θ,x(0)=x0,Δx(ti)=Ji0(x(ti),ẋ(ti)),ti∈Θ,ẋ(0)=x1,Δẋ(ti)=Ji1(x(ti),ẋ(ti)),ti∈Θ,
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 (1.2)(Gx)(t)=∫0tk(t,τ)g(τ,x(τ),ẋ(τ))dτ,(Hx)(t)=∫0Tm(t,τ)h(τ,x(τ),ẋ(τ))dτ.Θ={ti∈(0,T)∣0=t0<t1<⋯<tn<tn+1=T}, Ji0, Ji1(i=1,2,…,n) are nonlinear maps, and Δx(ti)=x(ti+0)-x(ti), Δẋ(ti)=ẋ(ti+0)-ẋ(ti). This represents the jump in the state x, ẋ at time ti, respectively, with Ji0, Ji1 determining the size of the jump at time ti.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 operators𝔄(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],Xα) 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 Section2, 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

LetX, 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
(2.1)‖R(λ,A(t))‖£(X)≤M|λ|+1,fort∈[0,T].
(P3) There exist constants L>0 and 0<α≤1 such that
(2.2)‖(A(t)-A(s))A-1(τ)‖£(X)≤L|t-s|αfors,t,τ∈[0,T].
LetX1={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 Aα(t) (α∈(0,1)), which satisfies D(A)=D(A(t)) and D(Aα)=D(Aα(t)) for t∈[0,T]. Let ∥x∥α=∥Aαx∥ for x∈D(Aα(t)) and denote the Banach space {D(Aα),∥·∥α} as Xα. Then it is clear that Xβ↪Xα for 0≤α≤β≤1. Define PC([0,T],Xα)≡{x:[0,T]→Xα∣x as continuous at t∈[0,T]∖Θ; x is continuous from left and has right-hand limits at t∈Θ}, PC1([0,T],Xα)≡{x∈PC([0,T],Xα)∣ẋ∈PC([0,T],X)}. It can be seen that PC([0,T],Xα) and PC1([0,T],Xα) are Banach spaces, respectively, with the norms
(2.3)‖x‖PC=max⁡{sup⁡t∈[0,T]‖x(t+0)‖α,sup⁡t∈[0,T]‖x(t-0)‖α},‖x‖PC1=‖x‖PC+‖ẋ‖PC.
For the initial value problem(2.4)ẋ(t)+A(t)x(t)=0,t∈(0,T],x(0)=x0,
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)∈Lb(X), (t,s)∈Δ¯, such that every solution of (2.4) can be represented in the form
(2.5)x(t)=U(t,0)x0.
Consider the following second-order initial value problem:(2.6)ẍ(t)+A(s)ẋ(t)=0,t∈(0,T],x(0)=x0,ẋ(0)=x1.
Set v(t)=(x(t)ẋ(t)) and v0=(x0x1); (2.6) can be rewritten as
(2.7)v̇(t)+A(t)v(t)=0,t∈(0,T],v(0)=v0,
where 𝔄(t)=(0-I0A(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
(2.8)U¯(t,s)=(I∫stU(τ,s)dτ0U(t,s)).
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.
Letx∈PC([0,T],Xα) and satisfy the following inequality:
(2.9)‖x(t)‖α≤a+b∫0t|t-τ|-γ‖x(τ)‖αdτ+c∫0t|t-τ|-γ‖xτ‖Bdτ+δ∫0T‖x(τ)‖αλdτ+∑0<tk<tθk‖x(tk)‖α
for allt∈[0,T], where 1>λ, γ>0, and a, b, c, δ, θk≥0 are constants, and ∥xt∥B=sup0≤s≤t∥x(s)∥α. Then there exists constant M>0 such that
(2.10)‖x(t)‖α≤M,∀t∈[0,T].Proof.
By the inequality (2.9) and [7, Lemma  2.1], there exist a0>0, b0>0 such that
(2.11)‖xt‖B≤a0(a+δ∫0T‖x(τ)‖αλdτ)+b0∫0t‖xτ‖Bdτ+∑0<tk<ta0θk‖x(tk)‖B.
By [17, Lemma  1.7.1], we have
(2.12)‖xt‖B≤a0(a+δ∫0T‖x(τ)‖αλdτ)∏0<tk<t(1+a0θk)eb0t≤a0(a+δ∫0T‖x(τ)‖αλdτ)∏0<tk<T(1+a0θk)eb0T.
Using the argument method [10, Lemma  3.1], there exists constant M>0 such that
(2.13)‖x(t)‖α≤M,∀t∈[0,T].
This completes the proof.Next, we extend the Ascoli-Arzela Theorem fromC([0,T],X) to PC1([0,T],Xα).Theorem 2.3.
Suppose𝒲⊂PC1([0,T],Xα) is a subset. If the following conditions are satisfied:(1)
𝒲 is uniformly bounded subset of PC1([0,T],Xα),(2)
𝒲 is equicontinuous in (ti,ti+1), i=0,1,2,…,n, where t0=0, tn+1=T,(3)
itst-sections 𝒲(t)≡{x(t)∣x∈𝒲,t∈[0,T]∖Θ},  𝒲̇(t)≡{ẋ(t)∣x∈𝒲,t∈[0,T]∖Θ}, 𝒲(ti+0)≡{x(ti+0)∣x∈W}, 𝒲(ti-0)≡{x(ti-0)∣x∈𝒲}, 𝒲̇(ti+0)≡{ẋ(ti+0)∣x∈W}, and 𝒲̇(ti-0)≡{ẋ(ti-0)∣x∈𝒲} are relatively compact subsets of Xα,
then𝒲 is a relatively compact subset of PC1([0,T],Xα).Proof.
LetW=x(·)⊂PC1([0,T],Xα) satisfy assumptions of Theorem 2.3 and {xm(·)} 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],Xα). Then, there exists a subsequence of {xm}, labeled {xm1}⊂W, and x1∈C1([0,t1],Xα), such that
(2.14)xm1⟶x1inC1([0,t1],X)asm⟶+∞.
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],Xα). There exists a subsequence of {xm1}, labeled {xm2}⊂W, and x2∈C1([t1,t2],Xα) such that
(2.15)xm2⟶x2inC1([t1,t2],X)asm⟶+∞.
In general, define Wi=W[ti+,ti+1-]={xi∈C1([ti,ti+1],X)∣x2(t)=x(t)fort∈(ti,ti+1) and xi(ti)=x(ti+0),xi(ti+1)=x(ti+1-0),x∈W}(i=2,…,n+1). Similarly, Wi is a relatively compact subset of C1([ti,ti+1],Xα). There exists a subsequence of {xmi-1}, labeled {xmi}⊂W, and xi∈C1([ti,ti+1],X) such that
(2.16)xmi⟶xiinC1([ti,ti+1],X)asm⟶+∞.
Setting
(2.17)x(t)={x1(t),t∈[0,t1],x2(t),t∈(t1,t2],⋮xn+1(t),t∈(tn,T],
then
(2.18)xmn+1⟶xinPC1([0,T],Xα)asm⟶+∞.
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 functionx∈PC1([0,T],Xα) is said to be a PC-α-mild solution of (1.1) if x satisfies the following integral equation:
(3.1)x(t)=x0+∫0tU(τ,0)x1dτ+∫0t∫τtU(s,τ)f(τ,x(τ),ẋ(τ),(Gx)(τ),(Hx)(τ))dsdτ+∑0<ti<tJi0(x(ti),ẋ(ti))+∑0<ti<t∫titU(s,ti)Ji1(x(ti),ẋ(ti))ds.
We introduce the following assumptions.Assumption F.
(1) The functionsf:[0,T]×Xα×Xα×Xα×Xα→X, g:[0,T]×Xα×Xα→Xα, and h:[0,T]×Xα×Xα→Xα are measurable in t∈[0,T] and locally Lipschitz continuous, that is, for all x1, x2, y1, y2, z1, z2∈Xα, satisfying ∥x1∥α, ∥x2∥α, ∥y1∥α, ∥y2∥α, ∥z1∥α, ∥z2∥α, ∥w1∥α, ∥w2∥α≤ρ, we have
(3.2)‖f(t,x1,y1,z1,w1)-f(t,x2,y2,z2,w2)‖≤L(ρ)(‖x1-x2‖α+‖y1-y2‖α+‖z1-z2‖α+‖w1-w2‖α),‖g(t,x1,y1)-g(t,x2,y2)‖α≤L(ρ)(‖x1-x2‖+‖y1-y2‖),‖h(t,x1,y1)-h(t,x2,y2)‖α≤L(ρ)(‖x1-x2‖+‖y1-y2‖).
(2) There exist a constant 0<λ<1 and a function q∈Lr([0,T],[0,+∞))(r>1) such that
(3.3)‖f(t,x,y,z,w)‖≤q(t)(1+‖x‖α+‖y‖α+‖z‖α+‖w‖α),∀x,y,z,w∈Xα,‖g(t,x,y)‖α≤q(t)(1+‖x‖α+‖y‖α),∀x,y∈Xα,‖h(t,x,y)‖α≤q(t)(1+‖x‖αλ+‖y‖αλ),∀x,y∈Xα.
(3) k, m∈C([0,T]2,R).Assumption J.
There exists a constantb≥0 such that maps Ji0:Xα×Xα→Xα and Ji1:Xα×Xα→Xβi  (0<α<βi<1) satisfy
(3.4)‖Ji0(x1,y1)-Ji0(x2,y2)‖α≤b(‖x1-x2‖α+‖y1-y2‖βi)(i=1,2,…,n),‖Ji1(x1,y1)-Ji1(x2,y2)‖βi≤b(‖x1-x2‖α+‖y1-y2‖βi)(i=1,2,…,n).Theorem 3.2.
Suppose thatA(0) has a compact resolvent, (x0,x1)∈Xα×Xβ, 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],Xα).Proof.
Let(x0,x1)∈Xα×Xβ be fixed; define the operator P on PC1([0,T],Xα) given by
(3.5)(Px)(t)=x0+∫0tU(τ,0)x1dτ+∫0t∫τtU(s,τ)f(τ,x(τ),ẋ(τ),(Gx)(τ),(Hx)(τ))dsdτ+∑0<ti<tJi0(x(ti),ẋ(ti))+∑0<ti<t∫titU(s,ti)Ji1(x(ti),ẋ(ti))ds.
By virtue of the properties of evolution system U(·,·) and Assumption J, for t∈[0,T]∖Θ, we have (Px)(t)∈Xα and
(3.6)ddt(Px)(t)=U(t,0)x1+∫0tU(t,τ)f(τ,x(τ),ẋ(τ),(Gx)(τ),(Hx)(τ))dτ+∑0<ti<tU(t,ti)Ji1(x(ti),ẋ(ti)).
This means that Px∈PC1([0,T],Xα) for x∈PC1([0,T],Xα).
Forx,y∈PC1([0,T],Xα) and ∥x∥PC1, ∥y∥PC1≤ρ in which ρ is a constant. Using Assumptions F(1) and J, we have
(3.7)‖(Px)(t)-(Py)(t)‖α+‖ddt(Px)(t)-ddt(Py)(t)‖α≤∫0t∫τt‖U(s,τ)‖0,α∥f(τ,x(τ),ẋ(τ),(Gx)(τ),(Hx)(τ))-f(τ,y(τ),ẏ(τ),(Gy)(τ),(Hy)(τ))∥dsdτ+∫0t‖U(t,τ)‖0,α∥f(τ,x(τ),ẋ(τ),(Gx)(τ),(Hx)(τ))-f(τ,y(τ),ẏ(τ),(Gy)(τ),(Hy)(τ))∥dτ+∑0<ti<t‖Ji0(x(ti),ẋ(ti))-Ji0(y(ti),ẏ(ti))‖α+∑0<ti<t∫tit‖U(s,ti)‖βi,α‖Ji1(x(ti),ẋ(ti))-Ji1(y(ti),ẏ(ti))‖βids+∑0<ti<t‖U(t,ti)‖βi,α‖Ji1(x(ti),ẋ(ti))-Ji1(y(ti),ẏ(ti))‖βi≤C(α,γ)L2(ρ¯)‖m‖T‖x-y‖PC1∫0t∫τt|s-τ|-γdsdτ+C(α,γ)L2(ρ¯)‖m‖T‖x-y‖PC1×∫0t|t-τ|-γdτds+C(α,γ)(1+L(ρ¯))2(‖k‖T+1)×∫0t∫τt|s-τ|-γ[‖x(τ)-y(τ)‖α+‖ẋ(τ)-ẏ(τ)‖α]dsdτ+C(α,γ)(1+L(ρ¯))2(‖m‖T+1)∫0t∫τt|s-τ|-γ[‖(x-y)τ‖B+‖(ẋ-ẏ)τ‖B]dsdτ+C(α,γ)(1+L(ρ¯))2(‖k‖T+1)∫0t|t-τ|-γ[‖x(τ)-y(τ)‖α+‖ẋ(τ)-ẏ(τ)‖α]dτ+C(α,γ)(1+L(ρ¯))2(‖m‖T+1)∫0t|t-τ|-γ[‖(x-y)τ‖B+‖(ẋ-ẏ)τ‖B]dτ+b∑0<ti<t[‖x(ti)-y(ti)‖α+‖ẋ(ti)-ẏ(ti)‖α]+b(T+1)×∑0<ti<tC(βi,α)[‖x(ti)-y(ti)‖α+‖ẋ(ti)-ẏ(ti)‖α]≤M¯‖x-y‖PC1,
where α<γ<1,
(3.8)ρ¯=ρ+2‖q‖L1(ρ+ρλ+1)(‖k‖+‖m‖),M¯=2C(α,γ)(L(ρ¯)+1)2(‖k‖T+‖m‖T+1)T1-γ+T2-γ1-γ+b(T+1)∑i=1n(C(βi,α)+1).
Hence P:PC1([0,T],Xα)→PC1([0,T],Xα) is a continuous operator.
Let𝒳⊆PC1([0,T],Xα) 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
(3.9)‖f(t,x(t),ẋ(t),(Gx)(t),(Hx)(t))‖≤ωq(t),∀x∈X
and 𝒦=P𝒳 is a bounded subset of PC1([0,T],Xα). Define
(3.10)K(t)={(Px)(t)∣x∈X},K̇(t)={ddt(Px)(t)∣x∈X}fort∈[0,T].
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, Xβ↪↪Xα for 0≤α<β≤1. For 0<ɛ<t≤T, define
(3.11)Kɛ(t)={(Pɛx)(t)=U(t,t-ɛ)(Px)(t-ɛ)∣x∈X},K̇ɛ(t)={ddt(Pɛx)(t)=U(t,t-ɛ)ddt(Px)(t-ɛ)∣x∈X}.
By the properties of {U(t,s)∣(s,t)∈Δ¯}, one can verify that
(3.12)‖(Px)(t)-(Pɛx)(t)‖α≤‖x0-U(t,t-ɛ)x0‖α+∫t-ɛt‖U(s,0)‖β,α‖x1‖βds+‖I-U(t,t-ɛ)‖(α+β)/2,α∫0t-ɛ‖U(s,0)‖β,(α+β)/2‖x1‖βds+∫t-ɛt∫τt‖U(s,τ)‖0,α‖f(τ,x(τ),ẋ(τ),(Gx)(τ),(Hx)(τ))‖dsdτ+∫0t-ɛ∫t-ɛt‖U(s,τ)‖0,α‖f(τ,x(τ),ẋ(τ),(Gx)(τ),(Hx)(τ))‖dsdτ+‖I-U(t,t-ɛ)‖(α+β)/2,α×∫0t-ɛ∫τt-ɛ‖U(s,τ)‖0,(α+β)/2‖f(τ,x(τ),ẋ(τ),(Gx)(τ),(Hx)(τ))‖dsdτ≤‖x0-U(t,t-ɛ)x0‖α+C(β,α)‖x1‖βɛ+C(α+β2,α,γ1)C(β,α+β2)‖x1‖βɛγ1+ωC(α,γ)1-γ‖q‖Lr(r-1r(2-γ)-1)(r-1)/rɛ(r(2-γ)-1)/r+ω‖q‖LrC(α,γ)(r-1r-rγ-1)(r-1)/rT(r-rγ-1)/rɛ+ω‖q‖LrC(α+β2,α,γ1)C(α+β2,γ2)(r-1r-1-rγ2)(r-1)/rT(2r-1-γ2)/rɛγ1,
where 0<γ1<(β-α)/2,  (α+β)/2<γ2<1, and
(3.13)‖ddt(Px)(t)-ddt(Pɛx)(t)‖α≤ω‖q‖LrC(α,γ)r-1r-1-rγɛ(r-1-rγ)/(r-1),x∈X.
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 Xα for t∈(0,t1]. Define
(3.14)K(t1+0)={(Px)(t)|t=t1+J10((Px)(t1),ddt(Px)(t)|t=t1)},K̇(t1+0)={ddt(Px)(t)|t=t1+J11((Px)(t1),ddt(Px)(t)|t=t1)}.
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 Xα.
By the same procedure, the compactness of𝒦̇(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],Xα).
By virtue of Gronwall inequality (see Theorem2.2), one can verify that 𝒴={x∈PC1([0,T],Xα)∣x=δPx,δ∈[0,1]} is a bounded subset of PC1([0,T],Xα). According to Leray-Schauder fixed point theorem, P has a fixed point in PC1([0,T],Xα). 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(3.15)ẍ(t)+A(t)ẋ(t)=f(t,x(t),ẋ(t),(Gx)(t),(Hx)(t)),t∈(0,T]∖(Θ∪Λ),x(0)=x0,Δx(ti)=Ji0(x(ti),ẋ(ti)),ti∈Θ,i=1,2,…,n,ẋ(0)=x1,Δẋ(sj)=Jj1(x(sj),ẋ(sj)),sj∈Λ,j=1,2,…,m,
has a PC-α-mild solution x∈PC1([0,T],Xα) given by
(3.16)x(t)=x0+∫0tU(τ,0)x1dτ+∫0t∫τsU(s,τ)f(τ,x(τ),ẋ(τ),(Gx)(τ),(Hx)(τ))dsdτ+∑0<ti<tJi0(x(ti),ẋ(ti))+∑0<sj<t∫sjitU(s,sj)Jj1(x(sj),ẋ(sj))ds.Remark 3.4.
Assume thatA(0) has a compact resolvent, (x0,x1)∈Xα×Xβ, 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(3.17)ẍ(t)+A(t)ẋ(t)=f(t,x(t),ẋ(t),(Gx)(t),(Hx)(t)),t∈(0,T],x(0)=x0,ẋ(0)=x1
has a unique α-mild solution x∈PC1([0,T],Xα) given by
(3.18)x(t)=x0+∫0tU(τ,0)x1dτ+∫0t∫τsU(s,τ)f(τ,x(τ),ẋ(τ),(Gx)(τ),(Hx)(τ))dsdτ.


## 4. Existence of Optimal Controls

We suppose thatY 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 Pf(Y). Assume that the multifunction ω:[0,T]→Pf(Y) is measurable and ω(·)⊂E where E is a bounded set of Y, the admissible control set Uad={u∈Lp(E)∣u(t)∈ω(t)a.e}≠∅ (see [18, Proposition  1.7, page  142, Lemma  3.2, page  175]).Consider the following controlled system:(4.1)ẍ(t)+A(t)ẋ(t)=f(t,x(t),ẋ(t),(Gx)(t),(Hx)(t))+B(t)u(t),t∈(0,T]∖Θ,x(0)=x0,Δx(ti)=Ji0(x(ti),ẋ(ti)),ti∈Θ,i=1,2,…,n,ẋ(0)=x1,Δẋ(ti)=Ji1(x(ti),ẋ(ti)),ti∈Θ,i=1,2,…,n.Assumption B.
B∈L∞([0,T],£(Y,X)).
It is easy to see thatBu∈Lp([0,T],X) for all u∈Uad. 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.
SupposeA(0) has a compact resolvent. Under Assumptions A, B, F, and J, for every u∈Uad, 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}.Define𝒯={(x,u)∈PC1([0,T],Xα)×Uad∣x as a PC-α-mild solution of system (4.1) corresponding to the control u∈Uad}. (x,u)∈𝒯 is called a feasible pair. We consider the Lagrange problem (P).Find(x0,u0)∈𝒯 such that(4.2)J(x0,u0)≤J(x,u),∀(x,u)∈T,
where(4.3)J(x,u)=∫0Tl(t,x(t),ẋ(t),u(t))dt.We introduce some assumptions onl.Assumption L.
(1) The functionall:[0,T]×Xα×Xα×Y→R∪∞ is Borel measurable.
(2)l(t,·,·,·) is sequentially lower semicontinuous on Xα×Xα× for almost all t∈[0,T].
(3)l(t,x,y,·) is convex on Y for each (x,y)∈Xα×Xα and almost all t∈[0,T].
(4) There exist constantsb≥0, c≥0, ϖ>0, and φ∈L1([0,T],R) such that
(4.4)l(t,x,y,u)≥φ(t)+b‖x‖α+c‖y‖α+ϖ‖u‖Yp,∀x,y∈Xα,u∈Y.Now we can give the following result on the existence of optimal controls for problem (P).Theorem 4.2.
Under assumptions of Theorem4.1 and assumption L, the problem (P) has a solution.Proof.
Ifinf{J(x,u)∣u∈Uad}=+∞, there is nothing to prove. Assume that inf{J(x,u)∣u∈Uad}=m<+∞. By Assumption L, we have m>-∞. By definition of infimum, there exists a sequence {un}⊂Uad such that J(xu,un)→m. Since {un} is bounded in Lp([0,T],Y), there exists a subsequence, relabeled as {un}, and u0∈Lp([0,T],Y) such that
(4.5)un⟶wu0inLp([0,T],Y).
Since Uad is closed and convex, from the Mazur Lemma, we have u0∈Uad.
Supposexn is the PC-α-mild solution of (4.1) corresponding to un (n=0,1,2,…). Then xn satisfies the following impulsive integral equation:
(4.6)xn(t)=x0+∫0tU(τ,0)x1dτ+∫0t∫0sU(s,τ)[f(τ,xn(τ),ẋn(τ),(Gxn)(τ),(Hxn)(τ))+B(τ)un(τ)]dτds+∑0<ti<tJi0(xn(ti),ẋn(ti))+∑0<ti<t∫titU(s,ti)Ji1(xn(ti),ẋn(ti))ds.
Setting Fn(t)=f(t,xn(t),ẋn(t),(Gxn)(t),(Hxn)(t)), by Assumptions F(2) and J, and Lemma  3.1, we obtain that Fn∈ℒr([0,T],X) (r>1). Furthermore, there exists a subsequence, relabeled as {Fn}, and F¯∈Lr([0,T],X) such that Fn→wF¯ in Lr([0,T],X) as n→∞.
Define operatorsQ:Lr([0,T],X)→C([0,T],X), given by
(4.7)(Qy)(t)=∫0tU(t,s)y(s)ds.
By [7, Lemma  4.1], we have
(4.8)QFn⟶QF¯inC([0,T],X)asn⟶∞.
Consider the following impulsive differential equation:(4.9)ẍ(t)+A(t)ẋ(t)=F¯(t)+B(t)u0(t),t∈(0,T]∖Θ,x(0)=x0,Δx(ti)=Ji0(x(ti),ẋ(ti)),ti∈Θ,i=1,2,…,n,ẋ(0)=x1,Δẋ(ti)=Ji1(x(ti),ẋ(ti)),ti∈Θ,i=1,2,…,n,.
By Theorem 3.2, we know that (4.9) have a PC-α-mild solution
(4.10)x¯(t)=x0+∫0tU(τ,0)x1dτ+∫0t∫0sU(s,τ)[F¯(τ)+B(τ)u0(τ)]dτds+∑0<ti<tJi0(x¯(ti),x¯̇(ti))+∑0<ti<t∫titU(s,ti)Ji1(x¯(ti),x¯̇(ti))ds.
Define
(4.11)ηn(t)=∫0tU(t,τ)[Fn(τ)-F¯(τ)+B(τ)un(τ)-B(τ)u0(τ)]dτ,
then,  ηn→0 in C([0,T],Xα) as n→∞. Using Assumption J, we obtain
(4.12)‖xn(t)-x¯(t)‖α+‖ẋn(t)-x¯̇(t)‖α≤(T+1)‖ηn‖C([0,T],Xα)+b(T+1)∑0<ti<t(C(βi,α)+1)(‖xn(ti)-x¯(ti)‖α+‖ẋn(ti)-x¯̇(ti)‖α).
Using the Gronwall lemma with impulse (see [17, Lemma  1.7.1]), we have
(4.13)‖xn(t)-x¯(t)‖α+‖ẋn(t)-x¯̇(t)‖α≤b(T+1)2∏i=1n(C(βi,α)+1)‖ηn‖C([0,T],X),
that is, xn→x¯ in PC1([0,T],Xα) as n→∞. Further,
(4.14)Fn(⋅)⟶f(⋅,x¯(⋅),x¯̇(⋅),(Gx¯)(⋅),(Hx¯)(⋅))inLr([0,T],X)asn⟶∞.
By the uniqueness of limit, we have
(4.15)F¯(t)=f(t,x¯(t),x¯̇(t),(Gx¯)(t),(Hx¯)(t)),
Furthermore,
(4.16)x¯(t)=x0+∫0tU(τ,0)x1dτ+∫0t∫0sU(s,τ)[f(τ,x¯(τ),x¯̇(τ),(Gx¯)(τ),(Hx¯)(τ))+B(τ)u0(τ)]dτds+∑0<ti<tJi0(x¯(ti),x¯̇(ti))+∑0<ti<t∫titU(s,ti)Ji1(x¯(ti),x¯̇(ti))ds.
Thus, x¯ is a PC-α-mild solution of (4.1) corresponding to u0.
SincePC1([0,T],Xα)↪L1([0,T],Xα), using the Assumption L, we can obtain
(4.17)m=lim⁡n→∞⁡∫0Tl(t,xn(t),ẋn(t),un(t))dt≥∫0Tl(t,x¯(t),x¯̇(t),u0(t))dt=J(x¯,u0)≥m.
This means that J attains its minimum at u0∈Uad.At the end of this section, an example is given to illustrate our theory. We consider the following problem:(4.18)ẍ(t,y)+(t+1)Δẋ(t,y)=ẋ(t,y)+x(t,y)+∫0t(t-τ)x2(τ,y)+ẋ2(τ,y)+1dτ+∫01(t+τ)2x2(τ,y)+ẋ2(τ,y)+1dτ+u(t,y),y∈Ω,t∈(0,1]∖{110,210,…,910},x(t,y)|[0,1]×∂Ω=0,x(0,y)=0,∂∂tx(t,y)|[0,1]×∂Ω=0,∂∂tx(t,y)|t=0=0,y∈Ω,x(i10+0,y)-x(i10-0,y)=12x(i10-0,y),i=1,…,9,ẋ(i10+0,y)-ẋ(i10-0,y)=12ẋ(i10-0,y),i=1,2,…,9,
where ẍ(t,y)=(∂2/∂t2)x(t,y), ẋ(t,y)=(∂/∂t)x(t,y), Ω⊂R3 is bounded domain, and ∂Ω∈C3.DefineX=Y=L2(Ω), D(A(t))=H2(Ω)∩H01(Ω), and A(t)x=(t+1)(∂2x/∂y12+∂2x/∂y22+∂2x/∂y32) for x∈D(A(t)); Uad is a nonempty bounded closed subset of Y. Let (4.19)J(u)=∫01∫Ω|x(t,ξ)|2dξdt+∫01∫Ω|∂∂tx(t,ξ)|2dξdt+∫01∫Ω|u(t,ξ)|2dξdt.DefineV(·)[u(·)](y)=u(·,y), [x(·)](y)=x(·,y),  [(Gx)(·)](y)=∫0·(·-τ)x2(τ,y)+ẋ2(τ,y)+1dτ, [(Hx)(·)](y)=∫01(·+τ)2x2(τ,y)+ẋ2(τ,y)+1dτ, [F(·,x(·),ẋ(·),(Gx)(·)),(Hx)(·))](y)=∫0·(·-τ)x2(τ,y)+ẋ2(τ,y)+1dτ+∫01(·+τ)2x2(τ,y)+ẋ2(τ,y)+1dτ+x(·,y)+ẋ(·,y), [gi0(x(i/10))](y)=1/2x(i/10,y), [gi1(ẋ(i/10))](y)=(1/2)ẋ(i/10,y); then F satisfies Assumption F. Thus the problem (4.18) can be rewritten as(4.20)ẍ(t)+A(t)ẋ(t)=F(t,x(t),ẋ(t),(Gx)(t),(Hx)(t))+V(t)u(t),t∈(0,1]∖{110,…,910},x(0)=0,x(i10+)-x(i10-)=gi0(x(i10)),i=1,2,…,9,ẋ(0)=0,ẋ(i10+)-ẋ(i10-)=gi1(ẋ(i10)),i=1,2,…,9,
with the cost function(4.21)J(u)=∫01(‖x(t)‖X2+‖ẋ(t)‖X2+‖u(t)‖X2)dt.Obviously, satisfying all the assumptions given in our former Theorem 4.2, our results can be used in (4.20).



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*Source: 213503-2011-01-23.xml*