Approximate controllability of Ψ‐Caputo fractional differential equation

Abstract

In this paper, we explain how ‐Caputo fractional differential equations are approximately controllable. The outcome is demonstrated using the infinitesimal operator, fractional calculus, semigroup theory, and Krasnoselskii fixed point theorem. To begin, we emphasize the presence of the mild solution and show that the ‐Caputo fractional system is approximately controllable. Additionally, an example is provided to illustrate the idea.

Approximate Controllability of Neutral Differential Systems with Fractional Deformable Derivatives

“…Moreover, assumptions (M3)-(M4) hold with I A 2 = e −β 1 + e β and C = 1 4 . According to Theorem 3.12 [29], the linear system corresponding to (27) with conditions (23) and (24)…”

“…ω(0, z) = ω 0 (z), 0 ≤ z ≤ 1; (24) where X = L 2 [0, 1], ω 0 (z) ∈ X. So, assumptions (M1)(i)(ii) and (M2)(i)(ii) hold with I A 2 = 1 36 , I A 1 = 1 64 , I A 2 = 1 4 , I A 1 = 1 4 , I A 2 = 1 36 , I A 1 = 1 64 .…”

“…It is common knowledge that engineering and mathematical control theory both strongly depend on the idea of controllability. As a result, several researchers have thoroughly examined the controllability of various nonlinear systems in recent years, for example [19][20][21][22][23][24][25], and the references therein. Finding an appropriate control function that will allow researchers to move the dynamical system's state towards the targeted final state is the controllability problem.…”

Finite-Approximate Controllability of ν-Caputo Fractional Systems

“…Thereafter, several researchers, Bora et al [8], Kavitha et al [9], Haq et al [10], Aimene [11], Bedi [12], Matar [13], Ge et al [14], Grudzka et al [15], Ke et al [16], Kumar et al [17,18], Liu et al [19], Sakthivel et al [20], Wang et al [21], Yan [22], Yang et al [23], Rykaczewski [24] have used different methods to study approximate controllability for several fractional differential and integro-differential systems. • Thereafter, several researchers, Vijayakumar et al [25], Ding et al [26], Bose et al [27] studied the approximate reachability for different kind of ν-fractional systems.…”