Approximate Controllability of Hilfer Fractional Neutral Stochastic Differential Equations

Abstract

In this paper, we investigate the approximate controllability of Hilfer fractional neutral stochastic differential equations. Firstly, the existence and uniqueness of mild solutions for these equations are obtained by means of the Banach contraction mapping principle. Then, combining the techniques of stochastic analysis theory, fractional calculations and operator semigroup theory, a new set of sufficient conditions for approximate controllability of these equations is formulated. At last, an example is presented to illustrate the obtained results.

New results concerning to approximate controllability of Hilfer fractional neutral stochastic delay integro‐differential systems

“…Furthermore, stochastic agitation is inevitable and therefore it is essential and needed to assume stochastic impact into the inquiry of fractional systems [14, 15, 17, 26, 27, 32, 35, 42–45]. Recently, many authors are focusing their interest on the existence and controllability of the Hilfer fractional systems [5, 6, 14, 16, 18, 28–30, 32, 45–51]. Regardless, most elevated, the approximate controllability of Hilfer fractional systems presented in our article yet to be examined, and which motivates this manuscript.…”

“…The controllability problem is finding an appropriate control function to such a degree that one can steer the considered dynamical system to a final state. Accordingly, over recent years, the discussion about the controllability of numerous nonlinear frameworks has been comprehensively investigated by many researchers and one can view the articles, for instance [15, 17, 19, 26, 27, 29–41]. Furthermore, stochastic agitation is inevitable and therefore it is essential and needed to assume stochastic impact into the inquiry of fractional systems [14, 15, 17, 26, 27, 32, 35, 42–45].…”

“…Moreover, one can be noticed that noise or stochastic distress cannot be avoided in nature, including artificial systems. For this reason, stochastic differential systems have fascinated considerable attraction because of their large utilization in illustrating a lot of refined dynamical systems in biological, physical, and medical fields; one can verify [15,17,[31][32][33][34].…”

“…Accordingly, over recent years, the discussion about the controllability of numerous nonlinear frameworks has been comprehensively investigated by many researchers and one can view the articles, for instance [15,17,19,26,27,[29][30][31][32][33][34][35][36][37][38][39][40][41]. Furthermore, stochastic agitation is inevitable and therefore it is essential and needed to assume stochastic impact into the inquiry of fractional systems [14,15,17,26,27,32,35,[42][43][44][45]. Recently, many authors are focusing their interest on the existence and controllability of the Hilfer fractional systems [5, 6, 14, 16, 18, 28-30, 32, 45-51].…”

Existence and approximate controllability of Hilfer fractional evolution equations with almost sectorial operators

“…Later on, Debbouche et al [14] and Du et al [17] studied the approximate controllability of Hilfer FDEs and semilinear Hilfer FDEs with impulsive control inclusions in Banach spaces, respectively. In 2018, Lv and Yang [34] investigated the approximate controllability of neutral Hilfer FDEs by applying the techniques of stochastic analysis theory and semigroup operator theory in a Hilbert space. In recent works, Lv and Yang [35], with the help of Banach contraction principle, discussed the approximate controllability for a class of Hilfer FDEs of order 1 < α < 2 and type 0 ≤ β ≤ 1.…”