Approximate controllability of Sobolev type fractional stochastic nonlocal nonlinear differential equations in Hilbert spaces

We introduce a new notion called fractional stochastic nonlocal condition, and then we study approximate controllability of class of fractional stochastic nonlinear differential equations of Sobolev type in Hilbert spaces. We use Hölder's inequality, 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 is formulated and proved for the fractional stochastic control system to be approximately controllable. An example is given to illustrate the abstract results

Analysis of Hilfer fractional integro-differential equations with almost sectorial operators

In this work, we investigate a class of nonlocal integro-differential equations involving Hilfer fractional derivatives and almost sectorial operators. We prove our results by applying Schauder’s fixed point technique. Moreover, we show the fundamental properties of the representation of the solution by discussing two cases related to the associated semigroup. For that, we consider compactness and noncompactness properties, respectively. Furthermore, an example is given to illustrate the obtained theory.publishe

Approximate Controllability of Delayed Fractional Stochastic Differential Systems with Mixed Noise and Impulsive Effects

We herein report a new class of impulsive fractional stochastic differential systems driven by mixed fractional Brownian motions with infinite delay and Hurst parameter $\hat{\cal H} \in ( 1/2, 1)$. Using fixed point techniques, a $q$-resolvent family, and fractional calculus, we discuss the existence of a piecewise continuous mild solution for the proposed system. Moreover, under appropriate conditions, we investigate the approximate controllability of the considered system. Finally, the main results are demonstrated with an illustrative example.Comment: Please cite this paper as follows: Hakkar, N.; Dhayal, R.; Debbouche, A.; Torres, D.F.M. Approximate Controllability of Delayed Fractional Stochastic Differential Systems with Mixed Noise and Impulsive Effects. Fractal Fract. 2023, 7, 104. https://doi.org/10.3390/fractalfract702010

Sobolev type fractional dynamic equations and Optimal multi-integral controls with fractional nonlocal conditions

We prove existence and uniqueness of mild solutions to Sobolev type fractional nonlocal dynamic equations in Banach spaces. The Sobolev nonlocal condition is considered in terms of a Riemann-Liouville fractional derivative. A Lagrange optimal control problem is considered, and existence of a multi-integral solution obtained. Main tools include fractional calculus, semigroup theory, fractional power of operators, a singular version of Gronwall's inequality, and Leray-Schauder fixed point theorem. An example illustrating the theory is given

Approximate controllability of impulsive non-local non-linear fractional dynamical systems and optimal control

We establish existence, approximate controllability and optimal control of a class of impulsive non-local non-linear fractional dynamical systems in Banach spaces. We use fractional calculus, sectorial operators and Krasnoselskii fixed point theorems for the main results. Approximate controllability results are discussed with respect to the inhomogeneous non-linear part. Moreover, we prove existence results of optimal pairs of corresponding fractional control systems with a Bolza cost functional.publishe

Exact Null Controllability for Fractional Nonlocal Integrodifferential Equations via Implicit Evolution System

We introduce a new concept called implicit evolution system to establish the existence results of mild and strong solutions of a class of fractional nonlocal nonlinear integrodifferential system, then we prove the exact null controllability result of a class of fractional evolution nonlocal integrodifferential control system in Banach space. As an application that illustrates the abstract results, two examples are provided

Relaxation in controlled systems described by fractional integro-differential equations with nonlocal control conditions

A control system described by fractional evolution integro-differential equations and fractional integral nonlocal control conditions is investigated. This posed system is subjected to mixed multivalued control constraints whose values are nonconvex closed sets. Along with the original system, we consider the system in which the constraints on the controls are the closed convex hulls of the original constraints. More precisely, existence results for the mentioned nonlocal control systems are proved. Furthermore, we study relations between the solution sets of both two systems