Exact Controllability for Hilfer Fractional Differential Inclusions Involving Nonlocal Initial Conditions

Abstract

The exact controllability results for Hilfer fractional differential inclusions involving nonlocal initial conditions are presented and proved. By means of the multivalued analysis, measure of noncompactness method, fractional calculus combined with the generalized Monch fixed point theorem, we derive some sufficient conditions to ensure the controllability for the nonlocal Hilfer fractional differential system. The results are new and generalize the existing results. Finally, we talk about an example to interpret the applications of our abstract results.

Solutions to Riemann–Liouville fractional integrodifferential equations via fractional resolvents

“…On the other hand, some new general fractional derivatives are introduced and studied, such as some types of extended Riemann-Liouville fractional derivative and the fractional derivative without singular kernel of exponential function [37][38][39][40][41]. Especially, the Hilfer fractional derivative is often used as a generalized Riemann-Liouville fractional derivative, which includes Riemann-Liouville and Caputo derivatives; see [4,9,42]. To the best of our knowledge, most of the existence and controllability results on the Hilfer fractional differential system are studied under the frame that A generates a strongly continuous semigroup and the solution is given by semigroup and probability density functions.…”

“…In fact, they can be regarded as alternative models to nonlinear differential equations and many physical phenomena with memory characteristics can be described by fractional differential equations; see, for instance, [1][2][3][4][5][6][7]. Recently, the theories of fractional differential equations with classical Caputo and Riemann-Liouville derivative have been developed and some basic properties are obtained including existence and controllability, see [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Among them, the differential equations with Caputo fractional derivative are studied extensively.…”

Results on the existence of Hilfer fractional neutral evolution equations with infinite delay via measures of noncompactness

“…Recently, many researchers show great interest in the existence and controllability of Hilfer fractional differential systems with or without delay, the authors can refer the previous articles 33–44 . In a previous work, 40 Harrat et al studied the solvability and optimal controls of an impulsive nonlinear Hilfer fractional delay evolution inclusion in Banach spaces by using fractional calculus, fixed‐point technique, semigroup theory, and multivalued analysis.…”

Approximate Controllability for a Kind of Fractional Neutral Differential Equations with Damping

“…In practice, FDEs are deemed to optimize the traditional differential equation model. About the elementary theory of fractional differential and evolution systems, one can refer to Podlubny [1], Kilbas et al [2], Zhou [3,4], and [5][6][7][8][9][10] and the references cited therein.…”

“…It lies in the fact that it is bound up with quadratic optimal control, observer design, and pole assignment. For this reason, the controllability has been actively investigated by many investigators, and an impressive progress has been made in recent years [7][8][9][10][11][12][14][15][16][17][18][19][20]. Controllability of the deterministic systems in infinite dimensional spaces has been broadly investigated.…”