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 α x t / d t α = A x t + f t , x t , x 0 = x 0 + g x , t ∈ 0 , τ . 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 α x t / d t α = A x t + f t , x t + B u t , x 0 = x 0 + g x , t ∈ 0 , τ , where d α . / d t α is the vectorial conformable fractional derivative of order α ∈ 0 , 1 in a Banach space X and A is the infinitesimal generator of a semigroup T t t ≥ 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 0 , τ , U with U is a Banach space and B is a bounded linear operator from U into X .

A class of nonlocal impulsive differential equations with conformable fractional derivative

“…Indeed, for example, in the work [15] the authors gave the solution of conformable-fractional telegraph equations in terms of the classical exponential function, however for the Caputo-fractional telegraph equations considered in the very good papers [19,20,36], the fundamental solution cannot be given in terms of the exponential function as in the conformable-fractional case, and therefore the authors have been introduced the so-called Mittag-Leffler function. Another comparison, we notice that the constants of increases of the norms of the control bounded operators W and W −1 in the application of the work [27] are given directly in a simple way in terms of the exponential function, contrary, for the Caputo fractional derivative in the application of the nice work [51] these constants are given in terms of the so-called Mittag-Leffler function. For more details and conclusions concerning the uses and applications of conformable fractional calculus, we refer to the works [2,4,5,7,8,10,11,12,13,14,16,17,22,23,24,25,28,29,42,49].…”