Controllability of Mild Solution of Nonlocal Conformable Fractional Differential Equations

Abstract

In many research works Bouaouid et al. have proved the existence of mild solutions of an abstract class of nonlocal conformable fractional Cauchy problem of the form: $d^{\alpha }x\left(t\right)/d{t}^{\alpha }=Ax\left(t\right)+f\left(t,x\left(t\right)\right),x\left(0\right)=x_0+g\left(x\right),t\in \left[0,\tau \right].$ The present paper is a continuation of these works in order to study the controllability of mild solution of the above Cauchy problem. Precisely, we shall be concerned with the controllability of mild solution of the following Cauchy problem $d^{\alpha }x\left(t\right)/d{t}^{\alpha }=Ax\left(t\right)+f\left(t,x\left(t\right)\right)+Bu\left(t\right),x\left(0\right)=x_0+g\left(x\right),t\in \left[0,\tau \right],$ where $d^{\alpha }\left(.\right)/d{t}^{\alpha }$ is the vectorial conformable fractional derivative of order $\left.\alpha \in \right]\left.0,1\right]$ in a Banach space $X$ and $A$ is the infinitesimal generator of a semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ on $X$ . The element $x_0$ is a fixed vector in $X$ and $f$ , $g$ are given functions. The control function $u$ is an element of $L^2\left(\left[0,\tau \right],U\right)$ with $U$ is a Banach space and $B$ is a bounded linear operator from $U$ into $X$ .