Chapter 7 - Controllability of Single-valued and Multivalued Fractional Stochastic Differential Equations

Abstract

In this chapter, approximate controllability, the solvability, and existence of optimal controls of different kinds of single-valued and multivalued differential equations are studied by transforming the considered problem into the fixed point problem of some nonlinear operator in appropriate function space. The new sets of sufficient conditions are formulated for the approximate controllability of some class of fractional stochastic differential equations/inclusions (FSDEs/FSDIs) under the assumption that the corresponding linear systems is approximately controllable. Further, sufficient conditions are formulated for the existence of mild solution and then existence of optimal control is investigated for the corresponding Lagrange optimal control problem of different classes of FSDEs/FSDIs by using fixed point techniques. The main results are obtained in various aspects under various hypotheses and assumptions by using fractional calculus, (a,k)-regularized families of bounded linear operators, resolvent operators, and various well-known fixed point theorems. Because of the fact that the Riemann–Liouville and Caputo fractional operators possess neither semigroup nor commutative properties, which are inherent to the derivatives of integer order, the well-developed theory of resolvent operators for integral equations and (a,k)-regularized families of bounded linear operators are employed to treat the abstract differential equations of fractional order.