Approximate Controllability of Fractional Neutral Evolution Equations in Banach Spaces

N I Mahmudov

doi:10.1155/2013/531894

Abstract

We discuss the approximate controllability of semilinear fractional neutral differential systems with infinite delay under the assumptions that the corresponding linear system is approximately controllable. Using Krasnoselkii's fixed-point theorem, fractional calculus, and methods of controllability theory, a new set of sufficient conditions for approximate controllability of fractional neutral differential equations with infinite delay are formulated and proved. The results of the paper are generalization and continuation of the recent results on this issue.

Highlights

Introduction and PreliminariesMany social, physical, biological and engineering problems can be described by fractional partial differential equations
Fractional differential equations are considered as an alternative model to nonlinear differential equations
The overwhelming majority of the approximate controllability results have only been available for semilinear evolution differential systems in Hilbert spaces, with the exception of the case of [11]

Summary

Introduction and Preliminaries

Physical, biological and engineering problems can be described by fractional partial differential equations. Note that our results are new even for the approximate controllability of fractional neutral differential equations with infinite delay in Hilbert spaces. It should be mentioned that (approximate) controllability results for first- and second-order partial neutral functional differential equations with infinite delay were considered by Sakthivel et al [18], Chalishajar [8], and Chalishajar and Acharya [9]. From Lemma 1(iv), since A−β is a bounded linear operator for 0 ≤ β ≤ 1, there exists a constant Cβ such that ‖A−β‖ ≤ Cβ for 0 ≤ β ≤ 1. (H1) A is the infinitesimal generator of an analytic semigroup of bounded linear operators S(t) in X, 0 ∈ ρ(A), S(t) is compact for t > 0, and there exists a positive constant M such that ‖S(t)‖ ≤ M.

Existence Theorem

Main Results

Applications

Conclusion