Relative Controllability of Langevin Delayed Fractional System with Multiple Delays in Control

Abstract

A Langevin delayed fractional system with multiple delays in control, is a delayed fractional system that includes delay parameters in both state and control, is first introduced. This paper is devoted to investigating the relative controllability of the Langevin delayed fractional system with multiple delays in control. For linear systems to be relatively controllable, necessary and sufficient circumstances are identified by introducing and employing the Gramian matrix. The sufficient conditions for the relative controllability of semilinear systems are offered based on Schauder's fixed point theorem. As an unusual approach, the controllability results of the delayed system are built for the first time on the exact solution produced by the Mittag-Leffler type function although controllability ones in the literature are built on the Volterra integral equations or the mild solutions produced by resolvent families.