International Journal of Partial Differential Equations
Volume 2014 (2014), Article ID 787092, 12 pages
http://dx.doi.org/10.1155/2014/787092

Research Article

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

Sanjukta Das, Dwijendra N. Pandey, and N. Sukavanam

Department of Mathematics, IIT Roorkee, Roorkee, Uttarakhand 247667, India

Received 12 June 2014; Revised 10 September 2014; Accepted 15 September 2014; Published 2 October 2014

Academic Editor: Athanasios N. Yannacopoulos

Copyright © 2014 Sanjukta Das et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 , which imply that the underlying space 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 is compact or -semigroup is compact then the controllability operator is also compact. Hence the inverse of it does not exist if the state space is infinite dimensional [17].

Another available method in the literature involves the invertibility of operator , where 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 Section 3 we study the existence and uniqueness of mild solution of the second order equation modelled in the form

where is the infinitesimal generator of a strongly continuous cosine family   of bounded linear operators on a Banach space . The history valued function belongs to some abstract phase space defined axiomatically; are appropriate functions. are prefixed numbers. In Section 5 we study the approximate controllability of

where is the infinitesimal generator of a strongly continuous cosine family of bounded linear operators on a Hilbert space . The history valued function belongs to some abstract phase space defined axiomatically; ,  are appropriate functions. is a bounded linear operator on a Hilbert space .

2. Preliminaries

In this section some definitions, notations, and lemmas that are used throughout this paper are stated. The family of operators in is a strongly continuous cosine family if the following are satisfied:

(a) ( is the identity operator in );(b) for all ;(c)the map is strongly continuous for each .

is the strongly continuous sine family associated to the strongly continuous cosine family . It is defined as , , .

The operator is the infinitesimal generator of a strongly continuous cosine function of bounded linear operators and is the associated sine function. Let ,  be certain constants such that and for every . For more details see book by Fattorini [28] and articles [35–37]. In this work we use the axiomatic definition of phase space , introduced by Hale and Kato [30].

Definition 1 (see [30]). Let be a linear space of functions mapping into endowed with seminorm and satisfy the following conditions:

(A)If , such that and , then for every the following conditions:

(i) is in ,(ii),(iii),

 where is a constant ,   is continuous, is locally bounded and , ,  are independent of .(B)The space is complete.

Definition 2 (see [31]). Hausdorff’s measure of noncompactness for a bounded set in any Banach space is defined by can be covered by finite number of balls with radii .}

Lemma 3 (see [31]). Let be a Banach space and be bounded, then the following properties hold:

(1) is precompact if and only if ;(2), where and are closure and convex hull of , respectively;(3) when ;(4) where ;(5);(6) for any ;(7)if the map is Lipschitz continuous with constant then for any bounded subset , where is a Banach space;(8)if is a decreasing sequence of bounded closed nonempty subset of and , then is nonempty and compact in .

Definition 4 (see [31]). The map is said to be a -contraction if there exists a positive constant such that for any bounded close subset where is a Banach space.

Lemma 5 (Darbo-Sadovskii [31]). If is closed and convex and , the continuous map is -contraction, then the map has at least one fixed point.

is the space formed by normalized piecewise continuous function from into . In particular it is the space formed by all functions such that is continuous at , and exists for all . It is clear that endowed with the norm is a Banach space. For any

So, .

Lemma 6 (see [31]). (1) If is bounded, then for any where .
(2) If is piecewise equicontinuous on , then is piecewise continuous for , and

(3) If is bounded and piecewise equicontinuous, then is piecewise continuous for and

Lemma 7 (see [35]). If the semigroup is equicontinuous and , then the set is equicontinuous for .

3. Existence and Uniqueness of Mild Solution

We define mild solution of problem (1) as follows.

Definition 8. A function is a mild solution of the problem (1) if , , and

To prove our result we always assume is a continuous function. The following hypotheses are used.

()The function is continuous from into and there exists a continuous bounded function such that for every .(Hf) satisfies the following.

(1)For every and , the function is strongly measurable for every and is continuous for a.e. .(2)There exists an integrable function and a monotone continuous nondecreasing function such that and .(3)There exists an integrable function such that for a.e. , where .

(Hg)The function is continuous and is Lipschitz continuous such that there exists positive constant such that

 (HJ) 

 (1) There exist positive constants such that and . (2) for all . (1) ; (2) .

 (H1) 

 (1) ; (2) .

Lemma 9 (see [9]). If is a function such that and then

where .

In this section is the function defined by and on . Clearly where .

Theorem 10. If the hypotheses (Hf), (Hg), (HI), (H1) are satisfied, then the initial value problem (1) has at least one mild solution.

Proof. Let be the space endowed with supremum norm .
Let be the map defined by and :

where and on . It is easy to see that

where

Thus is well defined and has values in . Also by axioms of phase space, the Lebesgue dominated convergence theorem, and the conditions (Hf), (Hg) it can be shown that is continuous.
Step 1. There exists such that , where . In fact, if we assume that the assertion is false, then for there exist and such that :

Hence

which is a contradiction to the hypothesis (H1). Similarly , for . Suppose on the contrary,

Hence,

which is a contradiction.
Step 2. To prove that is a -contraction. Let be split into for

For arbitrary , and

So, is Lipschitz continuous with Lipschitz constant .
For any , is piecewise equicontinuous since is equicontinuous. Hence from the fact that ,  and Lemma 6 and we have

For arbitrary and

So, is Lipschitz continuous with Lipschitz constant .
For arbitrary and ,

For each bounded set and ,    we have

For each bounded set and we have

Therefore, is a -contraction. So, by Darbo-Sadovskii fixed point theorem we conclude that has a fixed point in . hence, 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

where is the infinitesimal generator of a strongly continuous cosine family of bounded linear operators on a Hilbert space . The history valued function belongs to some abstract phase space defined axiomatically; ,  are appropriate functions. is a bounded linear operator on a Hilbert space . We define mild solution of problem (24) as follows.

Definition 11. A function is a mild solution of the problem (24) if , the functions and are integrable and the integral equation is satisfied:

Lemma 12 (see [11]). Under the assumption that is an integrable function, such that

and is a function continuously differentiable, then

Set .

Definition 13. The set given by is the mild solution of (24)} is called reachable set of the system (24). is the reachable set of the corresponding linear control system (31).

Definition 14. The system (24) is said to be approximately controllable on if is dense in . The corresponding linear system is approximately controllable if is dense in .

Lemma 15. Let be Hilbert space and ,    closed subspaces such that . Then there exists a bounded linear operator such that for each and where denotes the orthogonal projection on .

Let us define a continuous linear operator as

Let us denote the kernel of the operator by which is a closed subspace of . Let denote the corresponding orthogonal subspace of . Let be a projection on with range . Let denote the closure of the range of operator . The following hypothesis is required to prove the approximate controllability

(HR) and ,    such that .

It is easily seen that hypothesis (HR) is equivalent to the or . Theorem 16 shows that (HR) implies approximate controllability of the system (29). It is also known that approximate controllability of (31) implies . 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

with is approximately controllable.

Proof. It is sufficient to prove that since is dense in . Let for any chosen , then . It can be easily seen from Lemma 12 and [28] that there exists some such that

By hypothesis (HR) there exists a control function such that . As is arbitrary it implies that . Since the is dense in is dense in . Hence the neutral system with is approximately controllable.

We state the corresponding linear control system

Both exact and approximate controllability of the above system are studied extensively in [33, 38] and so forth.

Assume that ,   satisfy the following conditions with . For a fixed and such that , we define maps by and . Here , for and for and for and for . Clearly, ,   are continuous maps.

(C1)The function is continuous for almost all and is strongly measurable, .(C2) There exists integrable functions and a continuous nondecreasing function such that ,   .(C3) The function is continuous and is Lipschitz continuous such that there exists positive constant such that

The above same conditions also hold for .

Also, is the function defined by and on . Clearly where .

The operators are defined as

Clearly are bounded linear operators. We set and . Let denote the space consisting of continuous functions such that , endowed with the norm of uniform convergence. Let be maps defined as follows:

So, .

As a continuation of coauthor Sukavanam’s work [19] and from hypothesis in [39] we assume that .

By using Lemma 15 we denote the map associated to this decomposition and construct and . Also set .

We introduce the space

and we define the map by

Lemma 17. If the hypothesis ()–(Hg) and conditions (C1)-(C2) hold for and then has a fixed point.

Proof. For let and .  

Now

Similarly we find for . So,

where . Repeating this we get

As and as , the map is a contraction for sufficiently large and therefore has a fixed point.

Theorem 18. If the associated linear control system (31) is approximately controllable on , the space and condition of the preceding Lemma 17 hold then the semilinear control system (24) with state dependent delay is approximately controllable on .

Proof. Assume to be the mild solution and to be an admissible control function of system (31) with initial conditions and . Let be the fixed point of . So, and . By Lemma 12 we can split the functions with respect to the decomposition , respectively, by setting and . We define the function for and . So, . Thus by the properties of and

As is dense in we can choose a sequence and a sequence such that and as . By Lemma 15 we get

Hence by Definition 11 and the last expression we conclude that is the mild solution of the following equation:

Hence . Since the solution map is generally continuous, as . Thus . Therefore , which means is dense in . 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, is the phase space   (see [10]).

Consider the second order neutral differential equation:

where . For , and , where is the eigenfunction corresponding to the eigenvalue of the operator . is an orthonormal base. will generate the operators such that , and the operator . To find a solution to this problem we will assume that satisfies the conditions (g-5)–(g-7) in [34]. From Theorems and in [34] we conclude that is continuously included in . Let us suppose that the functions are piecewise continuous. By defining maps by

the system (51) can be transformed into system (1). Assume that the following conditions hold:

(a)the functions are measurable, and

such that .(b)The function is continuous and there is continuous function and .(c)The functions and for all .

Moreover are bounded linear operators.

Hence by assumptions (a)–(c) and Theorem 10 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:

where . The functions are piecewise continuous. We assume the existence of positive constants such that

If we define maps

and as in the problem (51) we can transform (47) into (1). Also a simple estimate shows that .

Also if we define and for all then the hypotheses (HJ) can be easily proved. For instance,

Similarly it is easily seen for . Now, if satisfies the hypothesis () then a mild solution of (47).

Example 2. Consider the second order neutral differential equation:

where . For , , and , where is the eigenfunction corresponding to the eigenvalue of the operator . is an orthonormal base. will generate the operators such that , and the operator . Let the infinite dimensional control space be defined as with norm . Thus is a Hilbert space. By defining maps by

the system (51) can be transformed into system (1). Assume that the functions are continuous and satisfy the following conditions.

(a)The functions are measurable, and

such that .(b)The function is continuous and there is continuous function and .(c)The functions and for all .

Moreover is a bounded linear operator.

Here we examine the conditions (HR) for this control system. Then by using Theorem 18 we show its approximate controllability. Let for . The bounded linear operator is defined by .

Let is the null space of . Also . Therefore

This implies that

The Hilbert space can be written as

Thus for there exists such that . So let . Then also let and .. Thus we see that hypothesis (HR) is satisfied as and .

Hence by assumptions (a)–(c) and Theorem 18 it is ensured that the problem (51) is approximately controllable.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express sincere gratitude to the reviewer for his valuable suggestions. The first author would like to thank Ministry of Human Resource and Development with Grant no. MHR-02-23-200-429/304 for their funding.

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