Existence and Optimal Controls for Fractional Stochastic Evolution Equations of Sobolev Type Via Fractional Resolvent Operators

Abstract

This paper is mainly concerned with controlled stochastic evolution equations of Sobolev type for the Caputo and Riemann-Liouville fractional derivatives. Some sufficient conditions are established for the existence of mild solutions and optimal state-control pairs of the limited Lagrange optimal systems. The main results are investigated by compactness of fractional resolvent operator family, and the optimal control results are derived without uniqueness of solutions for controlled evolution equations.

Optimal Control for a Class of Riemann-Liouville Fractional Evolution Inclusions

“…In the present work, by using the fixed-point theorems of multivalued mapping, the existence theorem on mild solutions as well as optimal controls are investigated for (4) under the assumption that g is completely continuous or Lipschitz-continuous. The obtained results are natural improvements of [9,14].…”

Existence and optimal controls for nonlocal fractional evolution equations of order (1,2) in Banach spaces

“…Via fractional resolvent operator family and approximating minimizing sequences of suitable functions twice, the authors of [25] proposed the existence and optimal control of a class of Sobolev-type time fractional differential equations in the Caputo and Riemann-Liouville sense, respectively. The authors of [26] derived some results about the mild solutions and optimal controls for a class of Sobolevtype fractional stochastic evolution equations of order [1,2] via some compactness results of the corresponding fractional operators. Yan and Jia [27] studied optimal control for fractional stochastic functional differential equations of order (1, 2) in a Hilbert space via the fixed point theorem, approximation technique, and properties of the solution operator.…”

Optimal controls for fractional stochastic differential systems driven by Rosenblatt process with impulses

“…For recent studies on the optimal control problem, see References 31‐36 and the references therein. Recently, Chang et al 37 analyzed the existence of optimal controls for fractional stochastic evolution equations. Sathiyaraj et al 38 studied the controllability and optimal control for a class of time‐delayed fractional stochastic integro‐differential systems.…”