Page 241 - Mathematical Techniques of Fractional Order Systems

230 Mathematical Techniques of Fractional Order Systems

upper semicontinuous of the right-hand side, but the class of continuously
differentiable functions is not large enough to guarantee the existence of
solution. Thus, the fractional differential inclusions described by
q
D xðtÞAfðxðtÞÞ
are required.
On the other hand, the theory of differential inclusions has become an
active area of investigation due to its applications in various fields such as
mechanics, electrical engineering, medicine biology, ecology, and so on
(Abbas and Mouffak, 2013; Balasubramaniam, 2002). There are many
applied problems in mathematics that induce the reader to the study of
dynamical systems having velocities nonuniquely determined by the state of
the system but depending loosely upon it.
Filippov (1988) systemized the theory of differential inclusions and intro-
duced the main properties of differential inclusions. As a matter of fact, there
exist extensive literature concerning differential and integral inclusions in
deterministic cases (see Aubin and Cellina, 1984; Hu and Papageorgiou,
1997; Deimling, 1992). Kree (1982) introduced stochastic differential inclu-
sions as a separate theory. Consequently there have been substantial works,
which have dealt with stochastic differential inclusions with different right-
hand sides and provided sufficient conditions for the existence of solutions
of corresponding stochastic differential inclusions.
However, there are only limited works considering the existence of solu-
tions, and controllability results of integer order stochastic differential inclu-
sions in finite and infinite dimensional space (Balasubramaniam and
Ntouyas, 2006; Balasubramaniam, 2002). Existence and controllability
results for fractional semilinear differential inclusions have been proposed by
Wang and Zhou (2011). Sakthivel et al. (2013) formulated a new set of suffi-
cient conditions for the approximate controllability of fractional nonlinear
differential inclusions. Fractional order Riemann Liouville integral inclu-
sions with two independent variables and multiple delays have been illus-
trated by Abbas and Mouffak (2013). So far no work has been reported in
the literature about the complete controllability of fractional higher order sto-
chastic integrodifferential inclusions.
Inspired by the abovementioned works, the aim of this chapter is to fill
this gap. The purpose of this chapter is to show the controllability of frac-
tional order stochastic integrodifferential inclusions. The sufficient condi-
tions are acquired via fixed point theorems, namely the Bohnenblust Karlin
fixed point theorem for the convex case and the Covitz Nadler fixed point
theorem for the nonconvex case where the controllability Grammian matrix
is defined by using Mittag Leffler matrix function. In particular, the com-
plete controllability results are investigated for the following nonlinear frac-
tional higher order stochastic integrodifferential inclusions.