Controllability of Neutral Impulsive Differential Inclusions with Non-local Conditions

In this short article, we have studied the controllability result for neutral impulsive differential inclusions with nonlocal conditions by using the fixed point theorem for condensing multi-valued map due to Martelli [1]. The system considered here follows the P.D.E involving spatial partial derivatives with α-norms.


Introduction
In this paper we have discussed the controllability of nonlocal Cauchy problem for neutral impulsive differential inclusions of the form where the linear operator ( ) A  generates an analytic semigroup ; is a multi-valued map and xt h represent the right and left limits of is a multi-valued map [ is the family of all subsets of X] and , a Banach space of admissible control functions with U as a Banach space.B is a bounded linear operator from to X and X is a separable Banach space with norm  .: ( ) As a model we consider the following system of heat equations; Since F and involve spatial partial derivative, the results obtained by other authors cannot be applied to our system even if .This is the main motivation of this paper.
Here authors have proved exact controllability by using fixed point theorem for condensing multi-valued maps due to Martelli.In this paper, we have discussed controllability results with α-norms as in [3] with de-) viating arguments in terms involving spatial partial derivatives.
As indicated in [4], and reference therein, the nonlocal Cauchy problem 0 (0) ( ) = x g x x  can be applied in different fields with better effect than the classical initial condition 0 (0) x x  .For example in [5], the author described the diffusion phenomenon of a small amount of gas in a transparent tube by using the formula =0 ( ) = ( ),  where are given constants and 0 1 In this case the above equation allows the additional measurement at i , .In the past several years theorems about controllability of differential, integro-differential, fractional differential systems and inclusions with nonlocal conditions have been studied by , Benchohra and Ntouyas [10,11], and Hernandez, Rabello and Henriquez [12] and the references therein.In [13], Chalishajar discussed exact controllability of third order nonlinear integro-differential dispersion system without compactness of semigroup.
Xianlong Fu and Yueju Cao [14], has discussed the existence of mild solution for neutral partial differential inclusions involving spatial partial derivative with - norms in Banach space.However in their work authors impose some severe assumptions on the operator family generator by ( ) A  , i.e. ( ): ( ) is an infinitesimal generator of a compact analytic semigroup of a uniformly bounded linear operator 0 t  , which imply that underlying space X has finite dimension and so the example considered in [14], and subsequently in Section 4 is ordinary differential equation but not partial differential equation which shows lack of existence (exact controllability) in abstract (control) system (refer [15]).This fact and several other applications of neutral equation (inclusions) are the main motivation of this paper.
In Section 3 (followed by Preliminaries) of present paper we discuss the controllability of neutral impulsive differential inclusion with nonlocal condition with deviating arguments with α-norm, which is the generalization of [14], in a finite dimensional space.The example is given in Section 4 to support the theory.In Section 5 we study exact controllability of same system in infinite dimension space by dropping the compactness assumption of semigroup   0 ( ) t T t  e generalized the result proved in Section 3.
. Here w

Preliminaries
In this section, we shall introduce some basic definitions, notations and lemmas which are used throughout this paper.
Let   , X  be a Banach space.
is the Banach space of continuous functions from ( , ) C J X J into X with the norm defined by   := sup ( ) : .

J x x t t J 
Let be the Banach space of bounded linear operators from is Bochner integrable if and only if x is Lebesgue integrable.(For properties of the Bochner integral see [16]).Let denotes the Banach space of Bochner integrable functions We use the notations G is called upper semi-continuous (u.s.c.) on X if for each 0 x X  , the set where is measurable.An upper semi-continuous map is said to be condensing, if for any bounded subset , with We remark that a completely continuous multi-valued map is the easiest example of a condensing map.For more details on multivalued maps see the books of Deimling [17].
Throughout this paper, : ( )  and the imbedding is compact whenever the resolvent operator of A is compact.
Semigroup satisfies the following properties: there exists a positive constant For more details about the above preliminaries, we refer to ( [18,19]).
In order to define the solution of the system (1) we shall consider the space   0 = : [0, ] ; ( , ); = 0,1, , and there exist ( ) and ( ); = 0,1, , with ( ) = ( ), (0) ( ) = , which is a Banach space with the norm where k x is the restriction of x to For the system (1) we assume that the following hypotheses are satisfied for some (0,1) : be the linear operator defined by induces a bounded invertible operator and there exists positive con- positive number , there exists a po ii) for each l N  on l sitive function ( )  w l dependent such that Now we define the mild solu system (1).
lo tion for the DEFINITION 2.1 The system (1) is said to be non cally controllable on the interval J if for every The following lem ma [2 et be a bounded and convex be an upper semi-continuous and conde ed map.If for every x   , ( ) nsing multi-valu F x is closed and convex set in  , then F has a fixed point in  .

Controlla
) define the con Using the above control, define a multi-valued map : 2 where denotes t is dependent on l .However on e other hand we have, , where

t x g x A A F x h A A F t x h t A T t s A F s x h s s
T t s v s s T t s Bu s s Dividing on both sides by and taking the lower limit as we get This is a contradiction with Formula (2).Hence for some positive integer Step 2: is c Indeed if ) then ther ( ) v S  x such that for ever

t t k F t x h t AT t s F s x h s s T t s v s s T t s Bu s s
Usin has compac , we m y pass bsequence if necessa t that v n conver-ges to 1 ( , that th is u.s. and condensing.
For this purpose, we decompose N as N = N 1 + N 2 , where the operators N 1 , N 2 are defined on H l respectively by . ( We will verify that N 1 is a contraction while 2 is a completely continuous operator.

A A F t x h t A F t x h t T t A A
N is a contraction.Next we show that .Therefore by assump s.c. and condensing.
The right hand side tends to zero as 0   , compactness in the uniform opeuous on .l since is strongly continuous and the of implies the continuity rator .Thus is equi-contin

t k y t T t x g x T t s v s s T t s Bu s s T t t I x t
( 1) ( 1)d ( ) and g Clearly, since are continuous we have that

t k y t T t x g x T t s Bu s s T t t I x t y x T t x g x T t s Bu s s T t t I x t n
From Lemma (H3) it follows that is a closed graph operator.
Moreover, we obtain that it follows from (H3) that has a i u -val p value, c.On the other hand is a contraction.nce is u.s.c. and ng By L a fixed point 0 by using Sadovskii's fixed-point theorem for condensing map, we can analogously study the controllability of the system (4).
(H3) ' The function satisfies the following conditions: i) for each Proof The mild solution of the system (4) is given by where p is a positive integer, and is defi Then A generates a strongly continuous semigroup which is compact, analytic and self-adjoint.a') Also A has a discrete spectrum re resentation A is given by he desire of is similar to Step 4 of Theorem 3.1.

Example
As an application of Theorem 3.2, we study the following impulsive partial function differential system with nonlocal condition d pro Copyright © 2011 SciRes.AM iii) For the function R the following three conditions are satisfied: 1) For each is continuous.2) For each 3) There is a positive number 1 such that c , and there exist cons- Here we choose 1 = = .2

 
According to paper [21], we know that, if , G t z x h t z x z x  and 1 ( ( )) = ( ), It is easy to see that for each This inequality alongwith condition (ii) says that (H2) is satisfied.Also G satisfies and g satisfies (H4).

,
controllability of the following system is studied by Benchohra and Ntouyas [2 le to state and prove our main contro-If the hypotheses (H1)-(H5) are satisfied, then the system (1) is controllable provided terms of properties (a') and (b'), and therefor function g.Since, for any 1 It has been observed that the example in ([2-11,22]) overed as special case of the abstract result.up is compact then the assumption (H1) in Vol. 15, 2002, pp.45-52.doi:10.1155/S1048953302000059Second Or linear Neutral Impulsive Diff Inclusions on Unbounded Domain with Infinite Delay in Banach Spaces," Bulletin of Korean Mathematical Society, Vol.48, No. 4, 2011, pp.813-838.