Approximate controllability of fractional integro-differential equations involving nonlocal initial conditions

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Approximate controllability of fractional integro-differential equations involving nonlocal initial conditions NI Mahmudov* and S Zorlu *

Correspondence: [email protected] Eastern Mediterranean University, via Mersin 10, Gazimagusa, T.R. North Cyprus, Turkey

Abstract We discuss the approximate controllability of nonlinear fractional integro-differential system under the assumptions that the corresponding linear system is approximately controllable. Using the fixed-point technique, fractional calculus and methods of controllability theory, a new set of sufficient conditions for approximate controllability of fractional integro-differential equations are formulated and proved. The results in this paper are generalization and continuation of the recent results on this issue. An example is provided to show the application of our result.

1 Introduction Controllability is one of the fundamental concepts in mathematical control theory, which plays an important role in control systems. The controllability of nonlinear systems represented by evolution equations or inclusions in abstract spaces and qualitative theory of fractional differential equations has been extensively studied by several authors. An extensive list of these publications can be found in [–] and the references therein. Recently, the approximate controllability for various kinds of (fractional) differential equations has generated considerable interest. A pioneering work on the approximate controllability of deterministic and stochastic systems has been reported by Bashirov and Mahmudov [], Dauer and Mahmudov [] and Mahmudov []. Sakthivel et al. [] studied the approximate controllability of nonlinear deterministic and stochastic evolution systems with unbounded delay in abstract spaces. On the other hand, the fractional differential equation has gained more attention due to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. Yan [] derived a set of sufficient conditions for the controllability of fractional-order partial neutral functional integrodifferential inclusions with infinite delay in Banach spaces. Debbouche and Baleanu [] established the controllability result for a class of fractional evolution nonlocal impulsive quasi-linear delay integro-differential systems in a Banach space using the theory of fractional calculus and fixed point technique. However, there exists only a limited number of papers on the approximate controllability of the fractional nonlinear evolution systems. Sakthivel et al. [] studied the approximate controllability of deterministic semilinear fractional differential equations in Hilbert spaces. Wang [] investigated the nonlocal controllability of fractional evolution systems. Surendra Kumar and Sukavanam [] obtained a new set of sufficient conditions for the approximate controllability of a class of © 2013 Mahmudov and Zorlu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attributi