# Existence of Solution and Approximate Controllability for Neutral Differential Equation with State Dependent Delay

Authors: Sanjukta Das; Dwijendra N. Pandey; N. Sukavanam
Journal: International Journal of Partial Differential Equations
 (2014)

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## Abstract

This paper is divided in two parts. In the first part we study a second order neutral partial differential equation with state dependent delay and noninstantaneous impulses. The conditions for existence and uniqueness of the mild solution are investigated via Hausdorff measure of noncompactness and Darbo Sadovskii fixed point theorem. Thus we remove the need to assume the compactness assumption on the associated family of operators. The conditions for approximate controllability are investigated for the neutral second order system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. A simple range condition is used to prove approximate controllability. Thereby, we remove the need to assume the invertibility of a controllability operator used by authors in (Balachandran and Park, 2003), which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by the authors in (Dauer and Mahmudov, 2002), which are practically difficult to verify and apply. Examples are provided to illustrate the presented theory.


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## Body

## 1. Introduction

Neutral differential equations appear as mathematical models in electrical networks involving lossless transmission, mechanics, electrical engineering, medicine, biology, ecology, and so forth. Neutral differential equations are functional differential equations in which the highest order derivative of the unknown function appears both with and without derivatives. Second order neutral differential equations model variational problems in calculus of variation and appear in the study of vibrating masses are attached to an electric bar.Impulsive differential equations are known for their utility in simulating processes and phenomena subject to short term perturbations during their evolution. Discrete perturbations are negligible to the total duration of the process which have been studied in [1–6].However noninstantaneous impulses are recently studied by Ahmad [7]. Stimulated by their numerous applications in mechanics, electrical engineering, medicine, ecology, and so forth, noninstantaneous impulsive differential equations are recently investigated.Recently, much attention is paid to partial functional differential equation with state dependent delay. For details see [7–12]. As a matter of fact, in these papers their authors assume severe conditions on the operator family generated by A, which imply that the underlying space X has finite dimension. Thus the equations treated in these works are really ordinary and not partial equations. The literature related to state dependent delay mostly deals with functional differential equations in which the state belongs to a finite dimensional space. As a consequence, the study of partial functional differential equations with state dependent delay is neglected. This is one of the motivations of our paper.The papers [13, 14] study existence of differential equation via measure of noncompactness. Measure of noncompactness significantly removes the need to assume Lipschitz continuity of nonlinear functions and operators.In recent years, controllability of infinite dimensional systems has been extensively studied for various applications. In the papers [15, 16] the authors discuss the exact controllability results by assuming that the semigroup associated with the linear part is compact. However, if the operator B is compact or C0-semigroup T(t) is compact then the controllability operator is also compact. Hence the inverse of it does not exist if the state space X is infinite dimensional [17].Another available method in the literature involves the invertibility of operator(αI+Γ0T), where Γ0T is the controllability Gramian and a limit condition which is difficult to check and apply in practical real world problems. See for details [18]. Also it is practically difficult to verify their condition directly. This is one of the motivations of our paper.However our work is a continuation of coauthor Sukavanam's novel approach in article [19]. We extend our work [20–22] in this paper.Controllability results are available in overwhelming majority for abstract differential delay systems (see [1, 3–6, 9–12, 14–17, 19–34]), rather than for neutral differential with state dependent delay.The organization of the paper is as follows. In Section3 we study the existence and uniqueness of mild solution of the second order equation modelled in the form
(1)ddt(x′(t)+g(t,xt))=Ax(t)+f(t,xρ(t,xt)),t∈(si,ti+1],i=0,…,n,x0=ϕ∈B,x′(0)=z∈X,x(t)=Ji1(t,xt),t∈(ti,si],i=1,2,…,n,x′(t)=Ji2(t,xt),t∈(ti,si],i=1,2,…,n,
where A is the infinitesimal generator of a strongly continuous cosine family  {C(t):t∈R} of bounded linear operators on a Banach space X. The history valued function xt:(-∞,0]→X,xt(θ)=x(t+θ) belongs to some abstract phase space B defined axiomatically; g,f,Ji1,Ji2,i=1,…,n are appropriate functions. 0=t0=s0<t1≤s1≤t2,<⋯,<tn≤sn≤tn+1=a are prefixed numbers. In Section 5 we study the approximate controllability of
(2)ddt(x′(t)+g(t,xt))=Ax(t)+f(t,xρ(t,xt))+Bu(t),t∈J=[0,a],x0=ϕ∈B,x′(0)=w∈X,
where A is the infinitesimal generator of a strongly continuous cosine family {C(t):t∈R} of bounded linear operators on a Hilbert space X. The history valued function xt:(-∞,0]→X,xt(θ)=x(t+θ) belongs to some abstract phase space B defined axiomatically; g, f are appropriate functions. B is a bounded linear operator on a Hilbert space U.


## 2. Preliminaries

In this section some definitions, notations, and lemmas that are used throughout this paper are stated. The family{C(t):t∈R} of operators in B(X) is a strongly continuous cosine family if the following are satisfied:(a)
C(0)=I (I is the identity operator in X);(b)
C(t+s)+C(t-s)=2C(t)C(s) for all t,s∈R;(c)
the mapt→C(t)x is strongly continuous for each x∈X.{S(t):t∈R} is the strongly continuous sine family associated to the strongly continuous cosine family {C(t):t∈R}. It is defined as S(t)x=∫0t‍C(s)xds, x∈X, t∈R.The operatorA is the infinitesimal generator of a strongly continuous cosine function of bounded linear operators (C(t))t∈R and S(t) is the associated sine function. Let N, N~ be certain constants such that ∥C(t)∥≤N and ∥S(t)∥≤N~ for every t∈J=[0,a]. For more details see book by Fattorini [28] and articles [35–37]. In this work we use the axiomatic definition of phase space B, introduced by Hale and Kato [30].Definition 1 (see [30]).
LetB be a linear space of functions mapping (-∞,0] into X endowed with seminorm ∥·∥B and satisfy the following conditions:(A)
Ifx:(-∞,σ+b]→X,b>0, such that xθ∈B and x|[σ,σ+b]∈C([σ,σ+b]:X), then for every t∈[σ,σ+b) the following conditions:
(i)
xt is in B,(ii)
∥x(t)∥≤H∥xt∥B,(iii)
∥xt∥B≤K(t-σ)sup⁡{∥x(s)∥:σ≤s≤t}+M(t+σ)∥xσ∥B,whereH>0 is a constant K,M:[0,∞)→[1,∞),  K is continuous, M is locally bounded and H, K, M are independent of x(·).(B)
The spaceB is complete.Definition 2 (see [31]).
Hausdorff’s measure of noncompactnessχY for a bounded set B in any Banach space Y is defined by χY(B)=inf⁡{r>0,B can be covered by finite number of balls with radii r.}Lemma 3 (see [31]).
LetY be a Banach space and B,C⊂Y be bounded, then the following properties hold: (1)
B is precompact if and only if χY(B)=0;(2)
χY(B)=χY(B¯)=χY(convB), where B¯ and convB are closure and convex hull of B, respectively;(3)
χY(B)≤χY(C) when B⊂C;(4)
χY(B+C)≤χY(B)+χY(C) where B+C={x+y;x∈B,y∈C};(5)
χY(B∪C)=max⁡{χY(B),χY(C)};(6)
χY(λB)=∥λ∥χY(B) for any λ∈R;(7)
if the mapQ:D(Q)⊂Y→Z is Lipschitz continuous with constant k then χZ(QB)≤kχY(B) for any bounded subset B⊂D(Q), where Z is a Banach space;(8)
if{Wn}n=1+∞ is a decreasing sequence of bounded closed nonempty subset of Y and lim⁡n→∞χY(Wn)=0, then ⋂n=1+∞Wn is nonempty and compact in Y.Definition 4 (see [31]).
The mapQ:W⊂Y→Y is said to be a χ-contraction if there exists a positive constant k<1 such that χYQ(C)≤kχY(C) for any bounded close subset C⊂W where Y is a Banach space.Lemma 5 (Darbo-Sadovskii [31]).
IfW⊂Y is closed and convex and 0∈W, the continuous map Q:W→W is χ-contraction, then the map Q has at least one fixed point.PC
(
[
0
,
a
]
,
X
) is the space formed by normalized piecewise continuous function from [0,b] into X. In particular it is the space PC formed by all functions u:[0,b]→X such that u is continuous at t≠ti, u(ti-)=u(ti) and u(ti+) exists for all i=1,2,…,n. It is clear that PC endowed with the norm ∥x∥PC=supt∈J∥x(t)∥ is a Banach space. For any x∈PC(3)x~i(t)={x(t),t∈(ti,ti+1];x(ti+),t=ti,i=1,2,…,n.
So, x~∈C([ti,ti+1],X).Lemma 6 (see [31]).
(1) IfW⊂PC([a,b];X) is bounded, then χ(W(t))≤χPC(W) for any t∈[a,b] where W(t)={u(t):u∈W}⊂X.
(2) IfW is piecewise equicontinuous on [a,b], then χ(W(t)) is piecewise continuous for t∈[a,b], and
(4)χPC(W)=sup⁡{χ(W(t)),t∈[a,b]}.
(3) IfW⊂PC([a,b];X) is bounded and piecewise equicontinuous, then χ(W(t)) is piecewise continuous for t∈[a,b] and
(5)χ(∫at‍W(s)ds)≤∫at‍χ(W(s))dst∈[a,b].Lemma 7 (see [35]).
If the semigroupS(t) is equicontinuous and η∈L([0,a];R+), then the set {∫0t‍S(t-s)u(s)ds:∥u(s)∥≤η(s)fora.es∈[0,a]} is equicontinuous for t∈[0,b].


## 3. Existence and Uniqueness of Mild Solution

We define mild solution of problem (1) as follows.Definition 8.
A functionx:(-∞,a]→X is a mild solution of the problem (1) if x0=ϕ;x(·)|[0,a]∈PC(X), x(t)=Ji1(t,xt)∀t∈(ti,si]i=1,…,n, and
(6)x(t)=C(t)ϕ(0)+S(t)[z+g(0,ϕ)]-∫0t‍C(t-s)g(s,xs)ds+∫0t‍S(t-s)f(s,xρ(s,xs))ds,t∈[0,t1],x(t)=C(t-si)Ji1(si,xsi)+S(t-si)(Ji2(si,xsi)+g(si,xsi))-∫sit‍C(t-s)g(s,xs)ds+∫sit‍S(t-s)f(s,xρ(s,xs))dsfort∈[si,ti+1]i=1,…,n.To prove our result we always assumeρ:J×B→(-∞,a] is a continuous function. The following hypotheses are used.(Hϕ)
The functiont→ϕt is continuous from R(ρ-)={ρ(s,ψ):ρ(s,ψ)≤0} into B and there exists a continuous bounded function Jϕ:R(ρ-)→(0,∞) such that ∥ϕt∥B≤Jϕ(t)∥ϕ∥B for every t∈R(ρ-).(Hf)
f:J×B→X satisfies the following.
(1)
For everyx:(-∞,a]→X,x0∈B and x|J∈PC, the function f(·,ψ):J→X is strongly measurable for every ψ∈B and f(·,t) is continuous for a.e. t∈J.(2)
There exists an integrable functionα:J→[0,+∞) and a monotone continuous nondecreasing function Ω:[0,+∞)→(0,+∞) such that ∥f(t,v)∥≤α(t)Ω(∥v∥B)∀t∈J and v∈B.(3)
There exists an integrable functionη:J→[0,∞) such that χ(S(s)f(t,D))≤η(t)sup⁡-∞<θ<0χ(D(θ)) for a.e. s,t,∈J, where D(θ)={v(θ):v∈D}.(Hg)
The functiong(·) is continuous ∀t,v∈J×B and g(t,·) is Lipschitz continuous such that there exists positive constant Lg such that
(7)∥g(t,v1)-g(t,v2)∥≤Lg∥v1-v2∥B,(t,vi)∈J×B,i=1,2.(HJ)(1) There exist positive constantsci1,ci2,di1,di2 such that ∥Ji1(t,v)∥≤ci1∥v∥B+ci2 and ∥Ji2(t,v)∥≤di1∥v∥+di2.(2)∥Jij(t,u)-Jij(t,v)∥≤LJij∥u-v∥B for all u,v∈Bi=1,…,n,j=1,2.(1)Ka(NaLg+N~∫0a‍α(s)dslim⁡τ→∞sup⁡⁡(Ω(τ)/τ)+∑i=1n‍(Nci1+N~(di1+Lg)))+∑i=1n‍ci1Ka<1;(2)(KaNLga+∑i=1n‍{NLJi1+N~(LJi2+Lg)}Ka+∑i=0n‍∫sit‍η(s)ds)+∑i=1n‍{LJi1}Ka<1.(H1)(1)Ka(NaLg+N~∫0a‍α(s)dslim⁡τ→∞sup⁡⁡(Ω(τ)/τ)+∑i=1n‍(Nci1+N~(di1+Lg)))+∑i=1n‍ci1Ka<1;(2)(KaNLga+∑i=1n‍{NLJi1+N~(LJi2+Lg)}Ka+∑i=0n‍∫sit‍η(s)ds)+∑i=1n‍{LJi1}Ka<1.Lemma 9 (see [9]).
Ify:(-∞,a]→X is a function such that y0=ϕ and y|J∈PC(X) then
(8)∥yρ(s,ys)∥B≤(Ma+Jϕ~)∥ϕ∥B+Kasup⁡{∥y(θ)∥;θ∈[0,max⁡{0,s}]},s∈R(ρ-)∪[0,a],
where Jϕ~=sup⁡t∈R(ρ-)Jϕ(t),Ma=sup⁡t∈JM(t)andKa=max⁡t∈JK(t).In this sectiony:(-∞,a]→X is the function defined by y0=ϕ and y(t)=C(t)ϕ(0)+S(t)(z+g(0,ϕ)) on J1=[0,t1]. Clearly ∥yt∥B≤Ka∥y∥a+Ma∥ϕ∥B where ∥y∥b=sup0≤t≤b∥y(t)∥.Theorem 10.
If the hypotheses (Hf), (Hg), (HI), (H1) are satisfied, then the initial value problem (1) has at least one mild solution.Proof.
LetS(a) be the space S(a)={x:(-∞,a]→X∣x0=0,x|J∈PC} endowed with supremum norm ∥·∥a.
LetΓ:S(a)→S(a) be the map defined by (Γx)0=0 and Γ=∑i=1n‍Γi1+∑i=1n‍Γi2:
(9)(Γi1x)(t)={Ji1(t,x¯t),t∈(ti,si];i=1,…,nC(t-si)Ji1(si,x¯si)+S(t-si)(Ji2(si,x¯si)+g(si,xsi)),t∈(si,ti+1];i=1,…,n;(10)(Γi2x)(t)={∫sit‍C(t-s)g(s,xs¯)ds+∫sit‍S(t-s)×f(s,x¯ρ(s,xs¯))ds,t∈(si,ti+1];i=0,…,n0,t∉(si,ti+1],i=0,…,n,
where x¯0=ϕ and x¯=x+y on J. It is easy to see that
(11)∥x¯t∥B≤Ka∥y∥a+Ma∥ϕ∥B+Ka∥x∥t,
where ∥x∥t=sup0≤s≤t∥x(s)∥(12)∥x¯ρ(s,x¯s)∥B≤k*∶=(Ma+Jϕ~)∥ϕ∥B+Ka∥y∥a+Ka∥x∥a.
Thus Γ is well defined and has values in S(a). Also by axioms of phase space, the Lebesgue dominated convergence theorem, and the conditions (Hf), (Hg) it can be shown that Γ is continuous.
Step
1. There exists k>0 such that Γ(Bk)⊂Bk, where Bk={x∈S(a):∥x∥a≤k}. In fact, if we assume that the assertion is false, then for k>0 there exist xk∈Bk and tk∈(si,ti+1] such that k<∥Γxk(tk)∥:
(13)k≤∑i=0n‍∥Γi2xk(tk)∥+∑i=1n‍∥Γi1xk(tk)∥≤∑i=0n‍N∫sitk‍Lg(∥xk¯s∥B+∥g(s,0)∥)ds+∑i=1n‍N~∫sitk‍α(s)Ω(∥xk¯ρ(s,xk¯s)∥B)+∑i=1n‍N(ci1∥xk¯s∥+ci2)+∑i=1n‍N~(∥xk¯si-0∥di1∥xk¯s∥+di2+Lg∥xk¯si-0∥+∥g(s,0)∥)≤∑i=0n‍N∫sitk‍Lg(Ka∥y∥a+Ma∥ϕ∥B+Kak+∥g(s,0)∥)ds+∑i=0n‍N~∫sitk‍α(s)ds×Ω(Ka∥y∥a+(Ma+Jϕ~)∥ϕ∥B+Kak)+∑i=1n‍N(ci1(Ka∥y∥a+Ma∥ϕ∥B+Kak)+ci2)+N~(di1(Ka∥y∥a+Ma∥ϕ∥B+Kak)+di2+Lg(Ka∥y∥a+Ma∥ϕ∥B+Kak)+∥g(s,0)∥).
Hence
(14)1<(∫0a‍α(s)dslim⁡k→∞sup⁡Ω(Ka∥y∥a+(Ma+Jϕ~)∥ϕ∥B+Kak)kN~∫0a‍α(s)ds×lim⁡k→∞sup⁡Ω(Ka∥y∥a+(Ma+Jϕ~)∥ϕ∥B+Kak)k+NaKaLg∫0a‍α(s)dslim⁡k→∞sup⁡Ω(Ka∥y∥a+(Ma+Jϕ~)∥ϕ∥B+Kak)k)+Ka∑i=1n‍(Nci1+N~(di1+Lg))≤Ka(Ω(τ)τ+∑i=1n‍(Nci1+N~(di1+Lg))NaLg+N~∫0a‍α(s)ds×lim⁡τ→∞sup⁡Ω(τ)τ+∑i=1n‍(Nci1+N~(di1+Lg)))
which is a contradiction to the hypothesis (H1). Similarly (Γx)(t)<k, for tk∈(ti,si]∀i=1,2,…,n. Suppose on the contrary,
(15)k<∑i=1n‍(Γi1xk)(tk)=∑i=1n‍∥Ji1(tk,xk¯tk)∥≤∑i=1n‍{ci1∥xk¯tk∥B+ci2}≤∑i=1n‍{ci1(Ka∥y∥a+Ma∥ϕ∥B+Kak)+ci2}.
Hence,
(16)1<∑i=1n‍ci1Ka,
which is a contradiction.
Step
2. To prove that Γ is a χ-contraction. Let Γ=∑i=1n‍Γi1+∑i=0n‍Γi2 be split into Γ=∑i=1n‍Γi1+∑i=0n‍{Γi12+Γi22} for t>0(17)Γi12x(t)=∫sit‍C(t-s)g(s,xs¯)ds,Γi22x(t)=∫sit‍S(t-s)f(s,x¯ρ(s,x¯s))ds.
For arbitrary x1,x2∈Bk, and t∈(si,ti+1](18)∑i=0n‍∥Γi12x1(t)-∑i=0n‍Γi21x2(t)∥≤∑i=0n‍∥∫sit‍C(t-s)(g(s,x1s+ys)-g(s,x2s+ys))ds∥≤∑i=0n‍NLga∥x1t-x2t∥B≤KaNLga∥x1-x2∥a.
So,Γi12∀i=0,…,n is Lipschitz continuous with Lipschitz constant NLgaKa.
For anyW⊂Γi12(Bk), W is piecewise equicontinuous since S(t) is equicontinuous. Hence from the fact that ρ(s,xs¯)≤s, s∈[0,a] and Lemma 6 and χPC(W)=sup⁡{χ(W(t)),t∈J} we have
(19)χ(∑i=0n‍Γi12W(t))=∑i=0n‍χ(∫sit‍S(t-s)f(s,Wρ(s,x¯s)+ys)ds)≤∑i=0n‍∫sit‍η(s)sup⁡-∞<θ≤0⁡χ(W(ρ(s,x¯s)+θ)+y(s+θ))ds≤∑i=0n‍∫sit‍η(s)sup⁡-∞<θ≤0⁡χ(W(s+θ)+y(s+θ))ds≤∑i=0n‍∫sit‍η(s)sup⁡-∞<τ≤0⁡χW(τ)ds≤χPC(W)∑i=0n‍∫sit‍η(s)ds.
For arbitraryx1,x2∈Bk and t∈(si,ti+1](20)∑i=1n‍∥(Γi1x1)(t)-∑i=1n‍(Γi1x2)(t)∥≤∑i=1n‍{NLJi1∥x1¯si-x2¯si∥+N~(LJi2∥x1¯si-x2¯si∥+Lg∥x2si-x1si∥)}≤∑‍{NLJi1+N~(LJi2+Lg)}∥x1si+ys-x2si-ys∥≤∑‍{NLJi1+N~(LJi2+Lg)}∥x1si-x2si∥B≤∑‍{NLJi1+N~(LJi2+Lg)}Ka∥x1-x2∥a.
So, Γi1∀i=1,…,n is Lipschitz continuous with Lipschitz constant (NLJi1+N~LJi2)Ka.
For arbitraryx1,x2∈Bk and t∈(ti,si],
(21)∑i=1n‍∥(Γi1x1)(t)-∑i=1n‍(Γi1x2)(t)∥≤∑i=1n‍LJi1∥x1t-x2t∥B≤∑i=1n‍KaLJi1∥x1-x2∥a.
For each bounded set W∈PC(J;X) and t∈(si,ti+1],   ∀i=0,…,n we have
(22)χPC(ΓW)≤∑i=1n‍χPC(Γi1W)+∑i=0n‍χPC(Γi12W+Γi22W)≤(∑i=0n‍∫sit‍η(s)dsKaNLga+∑‍{NLJi1+N~(LJi2+Lg)}Ka+∑i=0n‍∫sit‍η(s)ds)χPC(W).
For each bounded set W∈PC(J;X) and t∈(ti,si]∀i=1,2,…,n we have
(23)χPC(ΓW)≤∑i=1n‍χPC(Γi1W)+∑i=0n‍χPC(Γi12W+Γi22W)≤(∑i=1n‍{LJi1}Ka+0+0)χPC(W).
Therefore, Γ is a χ-contraction. So, by Darbo-Sadovskii fixed point theorem we conclude that Γ has a fixed point in S(a). hence, z=x+y is a mild solution of (1).


## 4. Approximate Controllability

In this section the approximate controllability of the control system (1) without the impulsive conditions is studied. We consider
(24)ddt(x′(t)+g(t,xt))=Ax(t)+f(t,xρ(t,xt))+Bu(t),t∈J=[0,a],x0=ϕ∈B,x′(0)=w∈X,
where A is the infinitesimal generator of a strongly continuous cosine family {C(t):t∈R} of bounded linear operators on a Hilbert space X. The history valued function xt:(-∞,0]→X,xt(θ)=x(t+θ) belongs to some abstract phase space B defined axiomatically; g, f are appropriate functions. B is a bounded linear operator on a Hilbert space U. We define mild solution of problem (24) as follows.Definition 11.
A functionx:(-∞,a]→X is a mild solution of the problem (24) if x0=ϕ;x(·)|[0,a]∈C(J,X), the functions f(s,xρ(s,xs)) and g(s,xs) are integrable and the integral equation is satisfied:
(25)x(t)=C(t)ϕ(0)+S(t)[w+g(0,ϕ)]-∫0t‍C(t-s)g(s,xs)ds+∫0t‍S(t-s)[f(s,xρ(s,xs))+Bu(s)]ds,t∈[0,a].Lemma 12 (see [11]).
Under the assumption thath:[0,a]→X is an integrable function, such that
(26)x′′(t)=Ax(t)+h(t),t∈J,x(0)=x0,x′(0)=x1
and h is a function continuously differentiable, then
(27)∫0t‍C(t-s)h(s)ds=S(t)h(0)+∫0t‍S(t-s)h′(s)ds.Seta∶=T.Definition 13.
The set given byRT(f)={x(T)∈X:x is the mild solution of (24)} is called reachable set of the system (24). RT(0) is the reachable set of the corresponding linear control system (31).Definition 14.
The system (24) is said to be approximately controllable on [0,T] if RT(f) is dense in X. The corresponding linear system is approximately controllable if R(0) is dense in X.Lemma 15.
LetX be Hilbert space and X1,   X2 closed subspaces such that X=X1+X2. Then there exists a bounded linear operator P:X→X2 such that for each x∈X,x=x-Px∈X1 and ∥x1∥=min⁡{∥y∥:y∈X1,(1-Q)(y)=(1-Q)(x)} where Q denotes the orthogonal projection on X2.Let us define a continuous linear operatorL:L2([0,T];X)→C([0,T];X) as
(28)Lp=∫0T‍S(T-s)p(s)ds,p∈L2([0,T];X).
Let us denote the kernel of the operator L by N which is a closed subspace of L2([0,T];X). Let N0⊥ denote the corresponding orthogonal subspace of L2([0,T];X). Let P be a projection on L2([0,T];X) with range N0⊥. Let R(B)¯ denote the closure of the range of operator B. The following hypothesis is required to prove the approximate controllability (HR)
∀ϵ>0 and p(·)∈L2([0,T];X),   ∃u(·)∈U such that ∥Lp-LBu∥X<ϵ.It is easily seen that hypothesis (HR) is equivalent to the L2([0,T];X)=R(B)¯+N0 or PR(B)¯=N0⊥. Theorem 16 shows that (HR) implies approximate controllability of the system (29). It is also known that approximate controllability of (31) implies L2([0,T];X)=R(B)+N0¯. Hence the closeness of the product space implies that (HR) is equivalent to approximate controllability of (29).Theorem 16.
If the assumptions (Hg) and (HR) hold then the corresponding neutral system(29)d(x′(t)+g(t,xt))dt=Ax(t)+Bu(t),t∈J,x(0)=ϕ(0),x′(0)=w
with f≡0 is approximately controllable.Proof.
It is sufficient to prove thatD(A)⊂RT(0)¯ since D(A) is dense in X. Let h(T,ϕ)=C(t)ϕ(0)+S(t)[w+g(0,ϕ(0))]-∫0T‍C(t-s)g(s,xs)ds for any chosen ξ∈D(A), then ξ-h(T,ϕ)∈D(A). It can be easily seen from Lemma 12 and [28] that there exists some p∈C1([0,T];X) such that
(30)η=ξ-h(T,ϕ)=∫0T‍S(T-s)p(s)ds.
By hypothesis (HR) there exists a control function u(·)∈L2([0,T];U) such that ∥η-LBu∥<ϵ. As ϵ is arbitrary it implies that KT(0)⊂D(A). Since the D(A) is dense in X,KT(0) is dense in X. Hence the neutral system with f≡0 is approximately controllable.We state the corresponding linear control system(31)x′′(t)=Ax(t)+Bu(t),t∈J,x(0)=x0,x′(0)=x1.
Both exact and approximate controllability of the above system are studied extensively in [33, 38] and so forth.Assume thatf,  g satisfy the following conditions with μf,μg,νf,νg∈L2(J). For a fixed ϕ∈B and x∈C(J,X) such that x(0)=ϕ(0), we define maps F,G:C0(J,X)→L2(J,X) by F(z)(t)=f(t,zt+xt) and G(z)(t)=g(t,zt+xt). Here xt(θ)=x(t+θ), for t+θ≥0 and xt(θ)=ϕ(t+θ) for t+θ≤0 and zt(θ)=z(t+θ) for t+θ≥0 and zt(θ)=0 for t+θ≤0. Clearly, F,  G are continuous maps.(C1)
The functionF(t,·):B→X is continuous for almost all t∈I and F(·,z):J→X is strongly measurable, ∀z∈B.(C2)
There exists integrable functionsμF,νF:I→[0,∞) and a continuous nondecreasing function WF:[0,∞)→(0,∞) such that ∥F(t,z)∥2≤μF(t)WF(∥z∥1)+νF(t),   (t,z)∈J×B.(C3)
The functionf(·) is continuous ∀t,v∈J×B and f(t,·) is Lipschitz continuous such that there exists positive constant Lf such that
(32)∥f(t,v1)-f(t,v2)∥≤Lf∥v1-v2∥B,(t,vi)∈J×B,i=1,2.The above same conditions also hold for G.Also,y:(-∞,a]→X is the function defined by y0=ϕ and y(t)=C(t)ϕ(0)+S(t)(z+g(0,ϕ)) on J. Clearly ∥yt∥B≤Ka∥y∥a+Ma∥ϕ∥B where ∥y∥b=sup0≤t≤b∥y(t)∥.The operatorsΛi:L2(J,X)→Xi=1,2 are defined as
(33)Λ1x(t)=∫0a‍S(t-s)x(s)ds,Λ2x(t)=∫0a‍C(t-s)x(s)ds.
Clearly Λi are bounded linear operators. We set Ni=ker⁡(Λi),Λ=(Λ1,Λ2) and N=ker⁡(Λ). Let C0(J,X) denote the space consisting of continuous functions x:J→X such that x(0)=0, endowed with the norm of uniform convergence. Let Ji:L2(J,X)→C0(J,X),i=1,2 be maps defined as follows:
(34)J1x(t)=∫0t‍S(t-s)x(s)ds,J2x(t)=∫0t‍C(t-s)x(s)ds.
So, Jix(a)=Λi(x),i=1,2.As a continuation of coauthor Sukavanam’s work [19] and from hypothesis (B1) in [39] we assume that L2(J,X)=Ni+R(B)¯,i=1,2.By using Lemma15 we denote Pi the map associated to this decomposition and construct X2=Ni and X1=R(B)¯. Also set ci=∥Pi∥.We introduce the space(35)Z={z∈C0(J,X):z=J1(n1)+J2(n2),ni∈Ni,i=1,2}
and we define the map Γ:Z¯→C0(J,X) by
(36)Γ=J1∘P1∘F-J2∘P2∘G.Lemma 17.
If the hypothesis (Hϕ)–(Hg) and conditions (C1)-(C2) hold for f,g and aKa(c1N~Lf+c2NLg)<2 then Γ has a fixed point.Proof.
Forz1,z2∈Z¯ let Δf(s)=f(s,zρ(s,z2(s))2+xρ(s,x(s)))-f(s,zρ(s,z1(s))1+xρ(s,x(s))) and Δg(s)=g(s,zs2+xs)-f(s,zs1+xs).  ∀0≤t≤a(37)∥(Γz2-Γz2)(t)∥≤∥∫0t‍S(t-s)[P1(Δf)](s)ds∥+∥∫0t‍C(t-s)[P2(Δg)](s)ds∥≤N~∫0t‍∥[P1(Δf)](s)∥ds+N∫0t‍∥[P2(Δg)](s)∥ds≤N~t1/2c1∥Δf∥2+Nt1/2c2∥Δg∥2.
Now
(38)∥Δf∥22=∫0a‍∥f(s,zρ(s,z2(s))2+xρ(s,x(s)))-f(s,zρ(s,z1(s))1+xρ(s,x(s)))∥2ds≤Lf2∫0a‍∥zρ(s,z2(s))2-zρ(s,z1(s))1∥B2ds≤Lf2∫0a‍∥zs2-zs1∥B2ds≤aLf2Ka2∥z2-z1∥∞2ds.
Similarly we find for g. So,
(39)∥(Γz2-Γz1)(t)∥≤bt1/2∥z2-z1∥∞,
where b=a1/2Ka(c1N~Lf+c2NLg). Repeating this we get
(40)∥(Γnz2-Γnz1)(t)∥∞≤(bt1/2)n2(n-1)/(2n)∥z2-z1∥∞.
As b=aKa(c1N~Lf+c2NLg)<2 and 2(n-1)/2n→2 as n→∞, the map Γn is a contraction for n sufficiently large and therefore Γ has a fixed point.Theorem 18.
If the associated linear control system (31) is approximately controllable on J, the space L2([0,a],X)=Ni+R(B)¯,i=1,2 and condition of the preceding Lemma 17 hold then the semilinear control system (24) with state dependent delay is approximately controllable on J.Proof.
Assumex(·) to be the mild solution and u(·) to be an admissible control function of system (31) with initial conditions x(0)=ϕ(0) and x′(0)=w+g(0,ϕ). Let z be the fixed point of Γ. So, z(0)=0 and z(a)=Λ1(P1(F(z)))-Λ2(P2(G(z)))=0. By Lemma 12 we can split the functions F(z),G(z) with respect to the decomposition L2(J,X)=Ni+R(B)¯i=1,2, respectively, by setting q1=F(z)-P1(F(z)) and q2=G(z)-P2(G(z)). We define the function y(t)=z(t)+x(t) for t∈J and y0=ϕ. So, x(a)=y(a). Thus by the properties of x and z(41)y(t)=∫0t‍S(t-s)(f(s,yρ(s,y(s)))-q1(s)+Bu(s))ds-∫0t‍C(t-s)(g(s,ys)-q2(s))ds+C(t)x(0)+S(t)x′(0).
As C01(J,U) is dense in L2(J,U) we can choose a sequence vn1∈L2(J,U) and a sequence vn2∈L2(J,X) such that Bvn1→q1 and Bvn2→q2 as n→∞. By Lemma 15 we get
(42)yn(t)=∫0t‍S(t-s)(f(s,yρ(s,y(s))n)-Bvn1(s)+Bu(s))ds-∫0t‍C(t-s)(g(s,ysn)-Bvn2(s))ds+C(t)ϕ(0)+S(t)(w-g(0,ϕ))=∫0t‍S(t-s)(ddsvn2(s)f(s,yρ(s,y(s))n)-Bvn1(s)+Bddsvn2(s)+Bu(s))ds-∫0t‍C(t-s)g(s,ysn)ds+C(t)ϕ(0)+S(t)(w+g(0,ϕ)).
Hence by Definition 11 and the last expression we conclude that yn is the mild solution of the following equation:
(43)ddt(y′(t)+g(t,xt))=Ay(t)+f(t,yρ(t,y(t)))+B(-vn1(t)+ddtvn2(t)+u(t))x(0)=ϕ∈Bx′(0)=w.
Hence yn(a)∈RT(a,f,g,ϕ,w). Since the solution map is generally continuous, yn→y as n→∞. Thus y(a)∈RT(a,f,g,ϕ,w). Therefore RT(0)(a,ϕ(0),w+g(0,ϕ))⊂RT(a,f,g,ϕ,w)¯, which means RT(a,f,g,ϕ,w) is dense in X. Thus the system (1) is controllable.


## 5. Examples

Example 1.
In this section we discuss a partial differential equation applying the abstract results of this paper. In this application,B is the phase space C0×L2(h,X)  (see [10]).Consider the second order neutral differential equation:(44)∂∂t(∂u(t,ξ)∂t+∫-∞t‍∫0π‍b(t-s,η,ξ)u(s,η)dηds)=∂2u(t,ξ)∂ξ2+∫-∞t‍a(t-s)u(s-ρ1(t)ρ2(∥u(t)∥),ξ)ds,t∈(si,ti+1],i=0,…,n,ξ∈[0,π],u(t,0)=u(t,π)=0,t∈[0,a],u(τ,ξ)=ϕ(τ,ξ)τ≤0,0≤ξ≤π,u′(τ,ξ)=ω(t,ξ)τ≤0,0≤ξ≤π,u(t)(ξ)=∫∞ti‍ai1(ti-s)u(s,ξ)dst∈(ti,si],i=1,2,…,n,u′(t)(ξ)=∫∞ti‍ai2(ti-s)u(s,ξ)dst∈(ti,si],i=1,2,…,n,
where ϕ∈C0×L2(h,X),0<t1<,…,tn<a. For y∈D(A), y=∑n=1∞‍<y,ϕn>ϕn and Ay=-∑n=1∞‍n2<y,n>ϕn, where ϕn(x)=2/πsinnx,0≤x≤π,n=1,2,3,… is the eigenfunction corresponding to the eigenvalue λn=-n2 of the operator A. ϕn is an orthonormal base. A will generate the operators S(t),C(t) such that S(t)y=∑n=1∞‍((sin⁡(nt))/n)<y,ϕn>ϕn,n=1,2,…∀y∈X, and the operator C(t)y=∑n=1∞‍cos⁡(nt)<y,ϕn>ϕn,n=1,2,…∀y∈X. To find a solution to this problem we will assume that h(·) satisfies the conditions (g-5)–(g-7) in [34]. From Theorems 1.37 and 7.1.1 in [34] we conclude that Ca((-∞,0],X) is continuously included in B. Let us suppose that the functions ρi:R→[0,∞),a:R→R are piecewise continuous. By defining maps ρ,G,F:[0,a]×B→X by
(45)ρ(t,ψ)∶=ρ1(t)ρ2(∥ψ(0)∥),g(t,ψ)(ξ)∶=∫-∞0‍∫0π‍b(s,υ,ξ)ψ(s,υ)dυds,f(t,ψ)(ξ)∶=∫-∞0‍a(s)ψ(s,ξ)ds,Jij(ψ)(ξ)∶=∫-∞0‍aij(s)ψ(s,ξ)dsi=1,…,nj=1,2
the system (51) can be transformed into system (1). Assume that the following conditions hold: (a)
the functionsb(s,η,ξ),∂b(s,η,ξ)/∂ξ are measurable, b(s,η,π)=b(s,η,0)=0 and
(46)Lg∶=max⁡{(∫0π‍∫-∞0‍∫0π‍1h(s)(∂ib(s,η,ξ)∂ξi)2dηdsdξ)1/2:i=0,1(∫0π‍∫-∞0‍∫0π‍1h(s)(∂ib(s,η,ξ)∂ξi)2dηdsdξ)1/2}<∞
such that ∥g∥L(X)≤Lg.(b)
The functionF:R×R→R is continuous and there is continuous function Lf=∫-∞0‍(a(s)2/h(s))ds<∞ and ∥F∥L(X)≤Lf.(c)
The functionsaij∈C([0,∞);R) and Lij:=(∫-∞0‍((aij(s))2/h(s))ds)1/2<∞ for all i=1,2,…,nj=1,2. Moreover g(t,·),Jij,i=1,…,n,j=1,2 are bounded linear operators.Hence by assumptions (a)–(c) and Theorem10 it is ensured that mild solution to the problem (51) exists.Now let us consider a particular example from the point of view of an application:(47)∂∂t(∂u(t,ξ)∂t+∫-∞t‍∫0π‍b(t-s,η,ξ)u(s,η)dηds)=∂2u(t,ξ)∂ξ2+a(t)b(u(t-μ(u(t,0)),ξ)),t∈(si,ti+1],i=0,…,n,ξ∈[0,π],u(t,0)=u(t,π)=0,t∈[0,a],u(τ,ξ)=ϕ(τ,ξ)τ≤0,0≤ξ≤π,u′(τ,ξ)=ω(τ,ξ)τ≤0,0≤ξ≤π,u(t)(ξ)=di1sin|u(t,ξ)|,t∈(ti,si],i=1,2,…,n,u′(t)(ξ)=di2cos⁡|u(t,ξ)|,t∈(ti,si],i=1,2,…,n,
where ϕ∈B=Ch0(X). The functions a:J→R,b:R×J→R,μ:R→R+ are piecewise continuous. We assume the existence of positive constants b1,b2 such that
(48)|b(t)|≤b1|t|+b2,∀t∈R.
If we define maps
(49)f(t,ψ)(ξ)=a(t)b(ψ(0,ξ)),ρ(t,ψ)=t-μ(ψ(0,0)),
and g(t,ψ)(ξ) as in the problem (51) we can transform (47) into (1). Also a simple estimate shows that ∥f(t,ψ)∥≤a(t)[b1∥ψ∥B+b2π1/2]∀(t,ψ)∈J×B.Also if we defineJi1(t,u(t))=di1sin|u(t)| and Ji2=di2cos⁡|u(t)| for all i=1,…,n then the hypotheses (HJ) can be easily proved. For instance,
(50)∥Ji1(t,u(t))∥=∥di1sin|u(t)|∥≤di1∥u(t)∥,∥Ji1(t,u1(t))-Ji1(t,u2(t))∥=∥di1sin|u1(t)|-di1sin|u2(t)|∥≤∥di1[|u1(t)|-|u2(t)|]∥.
Similarly it is easily seen for Ji2. Now, if ϕ satisfies the hypothesis (Hϕ) then ∃ a mild solution of (47).Example 2.
Consider the second order neutral differential equation:(51)∂∂t(∂u(t,ξ)∂t+∫-∞t‍∫0π‍b(t-s,η,ξ)u(s,η)dηds)=∂2u(t,ξ)∂ξ2+∫-∞t‍a(t-s)u(s-ρ1(t)ρ2(∥u(t)∥),ξ)ds+Bv(t)t∈[0,a],ξ∈[0,π],u(t,0)=u(t,π)=0,t∈[0,a],u(t,ξ)=ϕ(t,ξ)τ≤0,0≤ξ≤π,
where ϕ∈C0×L2(h,X),0<t1<,…,tn<a. For y∈D(A), y=∑n=1∞‍<y,ϕn>ϕn, and Ay=-∑n=1∞‍n2<y,n>ϕn, where ϕn(x)=2/πsinnx,0≤x≤π,n=1,2,3,… is the eigenfunction corresponding to the eigenvalue λn=-n2 of the operator A. ϕn is an orthonormal base. A will generate the operators S(t),C(t) such that S(t)y=∑n=1∞‍((sin⁡(nt))/n)<y,ϕn>ϕn,n=1,2,…∀y∈X, and the operator C(t)y=∑n=1∞‍cos⁡(nt)<y,ϕn>ϕn,n=1,2,…∀y∈X. Let the infinite dimensional control space be defined as U={u:u=∑n=2∞‍unϕn,∑n=2∞‍un2<∞} with norm ∥u∥U=(∑n=2∞‍un2)1/2. Thus U is a Hilbert space. By defining maps ρ,G,F:[0,a]×B→X by
(52)ρ(t,ψ)∶=ρ1(t)ρ2(∥ψ(0)∥),G(ψ)(ξ)∶=∫-∞0‍∫0π‍b(s,υ,ξ)ψ(s,υ)dυds,F(ψ)(ξ)∶=∫-∞0‍a(s)ψ(s,ξ)ds,
the system (51) can be transformed into system (1). Assume that the functions ρi:R→[0,∞),a:R→R are continuous and satisfy the following conditions. (a)
The functionsb(s,η,ξ),∂b(s,η,ξ)/∂ξ are measurable, b(s,η,π)=b(s,η,0)=0 and
(53)Lg∶=max⁡{(∫0π‍∫-∞0‍∫0π‍1h(s)(∂ib(s,η,ξ)∂ξi)2dηdsdξ)1/2:i=0,1(∫0π‍∫-∞0‍∫0π‍1h(s)(∂ib(s,η,ξ)∂ξi)2dηdsdξ)1/2}<∞
such that ∥g∥L(X)≤Lg.(b)
The functionF:R×R→R is continuous and there is continuous function Lf=∫-∞0‍(a(s)2/h(s))ds<∞ and ∥F∥L(X)≤Lf.(c)
The functionsaij∈C([0,∞);R) and Lij:=(∫-∞0‍((aij(s))2/h(s))ds)1/2<∞ for all i=1,2,…,nj=1,2.Moreover g(t,·) is a bounded linear operator.Here we examine the conditions (HR) for this control system. Then by using Theorem18 we show its approximate controllability. Let B~:U→X:B~u=2u2ϕ1+∑n=2∞‍unϕn for u=∑n=2∞‍unϕn∈U. The bounded linear operator B:L2([0,T];U)→L2([0,T];X) is defined by (Bu)(t)=B~u(t).Letα∈N⊂L2(0,T:X),N is the null space of Γ. Also α=∑1∞‍αn(s)ϕn. Therefore
(54)∫0T‍S(T-s)α(s)ds=0.
This implies that
(55)∫0T‍sinn(T-s)nαn(s)ds=0,n∈N.
The Hilbert space L2(0,T) can be written as
(56)L2(0,T)=Sp{sins}⊥+Sp{sin4s}⊥.
Thus for h1,h2∈L2(0,T) there exists α1∈{sins}⊥,α2∈{sin4s}⊥ such that h1-2h2=α1-2α2. So let u2=h2-α2. Then h1=α1+2u2,h2=α2+u2 also let un=hn,n=3,4,… and αn=0,n=3,4,….. Thus we see that hypothesis (HR) is satisfied as U={u:u=∑n=2∞‍unϕn,∑n=2∞‍un2<∞} and B~:U→X:B~u=2u2ϕ1+∑n=2∞‍unϕn.Hence by assumptions (a)–(c) and Theorem18 it is ensured that the problem (51) is approximately controllable.



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