Boundary controllability of impulsive nonlinear fractional delay integro-differential system

Abstract By using the strongly continuous semigroup theory and the Banach contraction principle, we study the boundary controllability of time varying delay impulsive nonlinear fractional integrodifferential system in Banach spaces. An example is provided to illustrate the theory.


Introduction
Recently, there has been increasing interest in studying the problem of controllability of nonlinear systems. Zhang (2000) obtained results on local exact controllability of semilinear integrodifferential systems in abstract spaces by means of Banach fixed point theorem. Balasubramanian, Dauer, and Loganathan (2002) considered a class of semilinear functional integrodifferential equations in Banach space setting and provided sufficient conditions for the controllability. Balachandran, Balasubramaniam, and Dauer (1996) established sufficient conditions for the local null controllability of nonlinear functional differential systems. Dauer and Balasubramaniam (1997) established sufficient conditions for the null controllability of semilinear integrodifferential systems in Banach space. Kwun, Park, and Ryu (1991) discussed the approximate controllability for delay Volterra systems while Balachandran and Sakthivel (1998) established a set of sufficient conditions for the controllability of delay integrodifferential systems in Banach spaces. Several abstract settings have

PUBLIC INTEREST STATEMENT
The notion of controllability is of great importance in mathematical control theory. Many fundamental problems of control theory such as pole-assignment, stabilizability and optimal control may be solved under the assumption that the system is controllable. The problem of controllability is to show the existence of control function, which steers the solution of the system from its initial state to final state. Also, impulsive differential equations, i.e. differential equations involving impulse effect, appear as a natural description of observed evolution phenomena of several real world problems. Therefore, in this paper, we study the boundary controllability of time varying delay impulsive nonlinear fractional integrodifferential system in Banach spaces using the strongly continuous semigroup theory and the Banach contraction principle.
been developed to describe the distributed control system on a domain in which the control is enacted through the boundary. Fattorini (1968) developed a semigroup approach for boundary control systems. Balakrishnan (1976) showed that the solution of a parabolic boundary control equation with L 2 controls can be expressed as a mild solution to an operator equation using semigroup theory. Barbu (1980) discussed the general theory of boundary control systems and the existence of solutions for boundary control systems governed by parabolic equations with nonlinear boundary conditions. Balachandran and Anandhi (2000, 2001a, 2001b discussed the boundary controllability of semilinear systems and delay integrodifferential systems in Banach spaces. Hamdy (see Ahmed, 2010Ahmed, , 2012 discussed the boundary controllability of nonlinear fractional integrodifferential systems. In this paper we study the boundary controllability of delay nonlinear fractional integrodifferential system. Let E and U be a two real Banach spaces with ‖ ⋅ ‖ and | ⋅ |, respectively. Let be a closed, linear and densely defined operator in E. In addition, let be a linear operator (the boundary operator) with domain in E and range in some Banach space X. We consider the following boundary control delay nonlinear fractional integrodifferential system of the form where c D is the Caputo fractional derivative of order 0 < < 1, the delay. i (t):J → J, i = 1, 2, are continuous functions, the state x(⋅) takes values in the Banach space E, B 1 :U → X is a linear continuous operator, the control function u ∈ L 2 (J, U), a Banach space of admissible control functions, h:J × J → R is a continuous function, Δx| t=t k = I k (x(t − k )), where x(t + k ) and x(t − k ) represent the right and left limits of x(t) at t = t k , respectively and the nonlinear operators The operator A is the infinitesimal generator of an analytic semigroup T(t) on E and there exists a constant M > 0 such that ‖T(t)‖ ≤ M. We assume without loss of generality that 0 ∈ (A). This allows us to define the fractional power (−A) , for 0 < ≤ 1, as a closed linear operator on its domain D((−A) ) with inverse (−A) − .
(4) If 0 < < ≤ 1, then E ↪ E and the embedding is compact whenever the resolvent operator of A is compact.

Preliminaries
Let us recall the following known definitions.
Definition 2.1 (see Podlubny, & EI-Sayed, 1996;Podlubny, 1999;Miller & Ross, 1993;Samko, Kilbas, & Marichev, 1993). The fractional integral of order with the lower limit zero for a function f can be defined as Definition 2.2 (see Miller & Ross, 1993;Podlubny & EI-Sayed, 1996;Podlubny, 1999;Samko et al., 1993). The Caputo derivative of order with the lower limit zero for a function f can be written as If f is an abstract function with values in X, then the integrals appearing in the above definitions are taken in Bochner's sense.
Let Y = C(J, B r ) and B r = {y ∈ Y:‖y‖ ≤ r} for some r > 0.
We assume the following hypotheses to prove the controllability of the system (1.1): (H1) D( ) ⊂ D( ) and the restriction of to D( ) is continuous relative to graph norm of D( ). (H5) There exists a constant q such that for all x 1 ,

(H2) There exists a linear continuous operator
(H6) The linear operator W from L 2 (J, U) into E is defined by has an induced inverse operator W −1 which takes values in L 2 (J, U)∕kerW and there exists a positive constant K, K 1 > 0 and K 2 > 0 such that ‖(−A) − ‖ ≤ K, 0 < ≤ 1, ‖B 1 ‖ ≤ K 1 and ‖W −1 ‖ ≤ K 2 (see Quinn & Carmichael 1984, 1988. H7) There exists a constant r > 0 such that Let x(t) be the solution of the system (1.1). Then we can define a function z(t) = x(t) − Bu(t) and from our assumption we see that z(t) ∈ D(A). Hence (1.1) can be written in terms of A and B as From (2.1) and fractional calculus, the integral form of the system (1.1) can be written in the form and hence, the mild solution of the system (1.1) is given by (see El-Borai, 2002, 2006Zhou, Jiao, & Li, 2010) where is a probability density function defined on (0, ∞) and Zhou et al. (2010)).

Definition 2.3 The system (1.1) is said to be controllable on the interval J if for every
x 0 , x 1 ∈ E, there exists a control u ∈ L 2 (J, U) such that the solution x(⋅) of the system (1.1) satisfies x(b) = x 1 .

Theorem 3.1 If the hypotheses (H1)-(H7) are satisfied, then the problem (1.1) is controllable on J provided that
Proof Using the hypothesis (H6), for an arbitrary function x(⋅) define the control We shall show that the operator Φ defined by has a fixed point. This fixed point is then a solution of (1.1). Clearly Φx(b) = x 1 , which means that the control u steers the impulsive fractional delay integrodifferential system (1.1) from the initial state x 0 to final state x 1 in time b provided we can obtain a fixed point of the nonlinear operator Φ. .
Thus Φ maps Y into itself.
Next for x 1 , x 2 ∈ Y we obtain Since 0 ≤ Λ < 1 then, Φ is a contraction mapping and hence there exists unique fixed point x ∈ Y such that Φx(t) = x(t). Any fixed point of Φ is a mild solution of (1.1) on J which satisfies x(b) = x 1 .
Thus the system (1.1) is controllable on J. ✷

Application
Let Ω be a bounded, open subset of R n , and let Γ be a sufficiently smooth boundary of Ω.
The operator is the trace operator such that z = z| Γ is well defined and belongs to H −1∕2 (Γ) for each z ∈ D( ).