Page 198 - Mathematical Techniques of Fractional Order Systems

Controllability of Single-valued Chapter | 7 187

and Zhou (2011a) studied the existence of mild solutions and the existence of
optimal pairs for semilinear fractional evolution equations in α-norm by
means of singular version Gronwall inequality and Leray Schauder fixed
point theorem. Fan and Mophou (2014) established the existence of optimal
control for semilinear composite fractional relaxation equations under
suitable conditions. Liu et al. (2013) studied the solvability and optimal con-
trols for some fractional impulsive differential equations by using fractional
calculus, Gronwall inequality, and Leray-Schauder fixed point theorem.
Highly inspired by the above research findings, this chapter deals with
the approximate controllability, solvability, and existence of optimal control
for some classes of single-valued and multivalued fractional stochastic differ-
ential equation (FSDEs). It is mandatory to mention that, except for standard
notation all other notations are uniquely defined for each subsection.
This chapter is organized as follows. In section 7.2, the approximate con-
trollability of multivalued fractional stochastic integro-differential equation,
solvability and optimal controls for FSDEs of order 1 , α , 2 are studied in
Hilbert space by using ða; kÞ-regularized families of bounded linear opera-
tors. Section 7.3 investigates the solvability and optimal controls for frac-
tional stochastic integro-differential equations with infinite delay in Hilbert
space by using analytic resolvent operators. Finally, section 7.4 deals with
some conclusions and future directions of these theoretical results.

7.2 CONTROLLABILITY RESULTS OF SINGLE-VALUED AND
MULTIVALUED FSDEs BY USING ða; kÞ-REGULARIZED
FAMILIES OF BOUNDED LINEAR OPERATORS

This section deals with controllability results of single-valued and multiva-
lued FDEs by using ða; kÞ-regularized families of bounded linear operators.
A new set of sufficient conditions is formulated for the approximate control-
lability of a class of multivalued fractional stochastic integro-differential
equation of order 1 , α , 2 in Hilbert space by using Bohnenblust Karlin’s
fixed point theorem . Further, the solvability and optimal control results are
investigated for FSDEs of order 1 , α , 2 in Hilbert space by using the clas-
sical Banach contraction mapping principle.

7.2.1 Approximate Controllability of Multivalued Fractional
Stochastic Integro-differential Equation

This subsection concerns the approximate controllability of a class of multi-
valued fractional stochastic integro-differential equations of the form
ð t ð t
c α c α21
D xðtÞAAxðtÞ1 D BuðtÞ1Ft;x t ; gðt;s;x s Þds 1 σðt;s;x s ÞdWðsÞ ;tAJ
t t
0 0
ð7:1Þ