Abstract
Abstract In this paper, under the assumption that the corresponding linear system is approximately controllable, we obtain the approximate controllability of semilinear fractional evolution systems in Hilbert spaces. The approximate controllability results are proved by means of the Hölder inequality, the Banach contraction mapping principle, and the Schauder fixed point theorem. We also discuss the existence of optimal controls for semilinear fractional controlled systems. Finally, an example is also given to illustrate the applications of the main results. MSC:26A33, 49J15, 49K27, 93B05, 93C25.
Highlights
During the past few decades, fractional differential equations have proved to be valuable tools in the modeling of many phenomena in viscoelasticity, electrochemistry, control, porous media, and electromagnetism, etc
Due to its tremendous scopes and applications, several monographs have been devoted to the study of fractional differential equations; see the monographs [ – ]
Since approximately controllable systems are considered to be more prevalent and very often approximate controllability is completely adequate in applications, a considerable interest has been shown in approximate controllability of control systems consisting of a linear and a nonlinear part [ – ]
Summary
During the past few decades, fractional differential equations have proved to be valuable tools in the modeling of many phenomena in viscoelasticity, electrochemistry, control, porous media, and electromagnetism, etc. Approximate controllability for one order nonlinear evolution equations with monotone operators was attained in [ ]. The control function u is given in L ([ , b], U), U is a Hilbert space, B is a bounded linear operator from U into Xα.
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