Approximate controllability and optimal controls of fractional dynamical systems of order (Formula presented.) in Banach spaces

Abstract

This paper investigates the approximate controllability and optimal controls of fractional dynamical systems of order (Formula presented.) in Banach spaces. We research a class of fractional dynamical systems governed by fractional integrodifferential equations with nonlocal initial conditions. Using the Krasnosel’skii fixed point theorem and the Schauder fixed point theorem, the approximate controllability results are obtained under two cases of the nonlinear term. We also present the existence results of optimal pairs of the corresponding fractional control systems with a Bolza cost function. Finally, an application is given to illustrate the effectiveness of our main results.