Approximate Controllability of Sobolev Type Nonlocal Fractional Stochastic Dynamic Systems in Hilbert Spaces

Abstract

We study a class of fractional stochastic dynamic control systems of Sobolev type in Hilbert spaces. We use fixed point technique, fractional calculus, stochastic analysis, and methods adopted directly from deterministic control problems for the main results. A new set of sufficient conditions for approximate controllability is formulated and proved. An example is also given to provide the obtained theory.

Stochastic fractional perturbed control systems with fractional Brownian motion and Sobolev stochastic non local conditions

“…Kerboua et al [12] proved the approximate controllability of Sobolev type non-local fractional stochastic dynamic systems in Hilbert spaces by using fixed point technique, fractional calculus, stochastic analysis, and methods adopted directly from deterministic control problems. Kerboua et al [13] introduced a new notion called fractional stochastic nonlocal condition for establishing approximate controllability of class of fractional stochastic nonlinear differential equations of Sobolev type in Hilbert spaces using Hölder's inequality, fixed point technique, fractional calculus, stochastic analysis and methods adopted directly from deterministic control problems.…”

Exact and trajectory controllability of second‐order evolution systems with impulses and deviated arguments

“…Several authors have extended the concept of controllability to infinite‐dimensional systems and established sufficient conditions for the controllability of nonlinear systems in abstract spaces. Among the various approaches to the study of the controllability of nonlinear systems, fixed‐point techniques have been used effectively for these systems . In the fixed‐point method, the controllability problem is transformed into a fixed‐point problem for an appropriate nonlinear operator in a function space …”

“…Among the various approaches to the study of the controllability of nonlinear systems, fixed-point techniques have been used effectively for these systems. [15][16][17][18] In the fixed-point method, the controllability problem is transformed into a fixedpoint problem for an appropriate nonlinear operator in a function space. [19][20][21] Second-order differential equations serve as abstract mathematical formulations of many partial differential equations that arise in several problems connected with the transverse motion of an extensible beam, the vibration of hinged bars, and many other physical phenomena.…”

Approximate Controllability of Impulsive Stochastic Fractional Differential Equations With Nonlocal Conditions

“…For more details, see [15,19,22,29,39,43,44,45,47,48] and references cited therein. For the study of differential equations with nonlocal initial conditions, we refer to the papers [11,12,17,19,20,36,37,39,40,42,44,49,50,52].…”

“…The biggest difficulty is the analysis of a stochastic control system and stochastic calculations induced by the stochastic process. For more details, see [14,16,19,23,34,36,39,50,52].…”