Controllability of fractional integrodifferential systems in Banach spaces

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Abstract

In this paper we study the controllability of fractional integrodifferential systems in Banach spaces. The results are obtained by using fractional calculus, semigroup theory and the fixed point theorem.

Introduction

Recently fractional differential equations emerged as a new branch of applied mathematics which have been used for many mathematical models in science and engineering. In fact fractional differential equations are considered as an alternative model to nonlinear differential equations [1]. The theory of fractional differential equations has been extensively studied by Delbosco and Rodino [2] and Lakshmikantham et al. [3], [4], [5], [6]. Stability problems for fractional differential systems are discussed in [7], [8]. Apart from stability another most important qualitative behaviour of a dynamical system is controllability. It means that it is possible to steer any initial state of the system to any final state in some finite time using an admissible control. The concept of controllability plays a major role in both finite and infinite dimensional spaces, that is, systems represented by ordinary differential equations and partial differential equations respectively. So it is natural to extend this concept to dynamical systems represented by fractional differential equations. Many partial fractional differential equations and integrodifferential equations can be expressed as fractional differential and integrodifferential equations in some Banach spaces [9].

There exist many papers on finite dimensional controllability of linear systems [10] and infinite dimensional systems in abstract spaces [11]. The problem of controllability of nonlinear systems and integrodifferential systems including delay systems has been studied by many researchers in both finite and infinite dimensional spaces [12], [13]. Controllability of fractional differential systems in finite dimensional space has been discussed in [14], [15], [16]. In this paper we study the controllability problem for nonlinear fractional integrodifferential systems by using the semigroup theory and a fixed point theorem.

Section snippets

Preliminaries

We need some basic definitions and properties of fractional calculus which are used in this paper.

Definition 2.1

A real function f(t) is said to be in the space Cα,α∈R if there exists a real number p>α, such that f(t)=tpg(t), where g∈C[0,∞) and it is said to be in the space Cαm iff f(m)∈Cα,m∈N.

Definition 2.2

The Riemann–Liouville fractional integral operator of order β>0 of function f∈Cα,α≥−1 is defined as Iβf(t)=1Γ(β)∫0t(t−s)β−1f(s)ds.

Definition 2.3

If the function f∈C−1m and m is a positive integer then we can define the fractional

Controllability result

Theorem 3.1

If the hypotheses (H1)–(H6) are satisfied, then the fractional integrodifferential system(1)is controllable on J .

Proof

Let Z=C(J,Br). Using the hypothesis (H2), for an arbitrary function x(⋅) define the control u(t)=W̃−1[x1−T(b)x0−1Γ(q)∫0b(b−s)q−1T(b−s)f(s,x(s),Hx(s))ds](t). We shall show that when using the above control the operator Φ:Z→Z defined by Φx(t)=T(t)x0+1Γ(q)∫0t(t−s)q−1T(t−s)BW̃−1[x1−T(b)x0−1Γ(q)∫0b(b−θ)q−1T(b−θ)f(θ,x(θ),Hx(θ))dθ](s)ds+1Γ(q)∫0t(t−s)q−1T(t−s)f(s,x(s),Hx(s))ds has a fixed

Nonlocal problem

Now we discuss the controllability of the fractional integrodifferential system (1) with a nonlocal condition of the form x(0)+g(x)=x0 where g:Z→X is a given function which satisfies the following condition.

  • (H6)

    g:Z→X is continuous and there exists a constant G>0 such that ‖g(x)−g(y)‖≤G‖x−y‖,for x,y∈C(J,Br).

  • (H4)′

    M[‖x0‖+Gr+‖g(0)‖]+γMK[‖x1‖+M‖x0‖+Nγ]+Nγ≤r.

  • (H5)′

    Let p′=MG+γ[MKN1γ+N1][M+MH1b] be such that 0≤p′<1.

Definition 4.1

The fractional system (1), (7) is said to be controllable on the interval J if for every x0,x1∈X,

Example

Consider the following nonlinear partial integrodifferential equation of the form ∂α∂tαz(t,y)=∂2∂y2z(t,y)+μ(t,y)+k0(y)sinz(t,y)+k1∫0te−z(s,y)ds,z(0,y)+∑i=1mciϕ(ti,y)=z0(y),ϕ∈Z,0≤y≤π,z(t,0)=z(t,π)=0,t∈J=[0,b], where 0<α<1, k0(y) is continuous on [0,π] and ci>0,k1>0.

Let us take X=U=L2[0,π],Z=C([0,b],Br),Br={y∈L2[0,π]:‖y‖≤r}. Put x(t)=z(t,⋅) and u(t)=μ(t,⋅) where μ:J×[0,π]→[0,π] is continuous, g(ϕ(t,⋅))=∑i=1mciϕ(ti,⋅)andf(t,x,Hx)=k0(⋅)sinz(t,⋅)+Hx,h(t,s,x)=k1e−z(s,⋅). Let A:D(A)⊂X→X be the

Acknowledgement

The work of the first author is supported by Korea Brain Pool Program of 2008.

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