Page 240 - Mathematical Techniques of Fractional Order Systems

Chapter 8

Controllability of Fractional

Higher Order Stochastic
Integrodifferential Inclusions

T. Sathiyaraj and Pagavathigounder Balasubramaniam
Gandhigram Rural Institute (Deemed to be University), Dindigul, Tamil Nadu, India

8.1 INTRODUCTION
Fractional differential equations serve as an appropriate phenomenon such that
it can even describe the real-world problems which are impossible to do using
classical integer order differential equations. Over the past decades, the theory
of fractional differential equation has gained more attention, and has obtained
a prior position in the field of physics, signal processing, fluid mechanics, vis-
coelasticity, mathematical biology, electro chemistry, and many other science
and engineering fields (for detail one may refer to the books and monographs:
Azar et al., 2017; Miller and Ross, 1993; Oldham and Spanier, 1974;
Podlubny, 1998; Samko et al., 1993; Sabatier et al., 2007).
Controllability of dynamical systems is one of the fundamental notions of
modern control theory. Generally speaking, controllability enables one to
steer the control system from an arbitrary initial state to an arbitrary final
state using the set of admissible controls. This concept leads to some impor-
tant conclusions regarding the behavior of linear and nonlinear dynamical
systems. Controllability of fractional order deterministic and stochastic dif-
ferential equations and inclusions in finite dimensional space have been stud-
ied by Balachandran and Kokila (2012) and Sathiyaraj and Balasubramaniam
(2016). Balasubramaniam and Ntouyas (2006) studied controllability for neu-
tral stochastic functional differential inclusions with infinite delay in abstract
space. Controllability of linear stochastic systems has been investigated by
Mahmudov and Denker (2000).
It is natural to define a solution as continuously differentiable functions
q
satisfying the fractional differential equations D xðtÞ 5 fðxðtÞÞ in all points of
some interval with continuous function in the right-hand side. In many
applied problems, there is a need to consider differential equations with an

Mathematical Techniques of Fractional Order Systems. DOI: https://doi.org/10.1016/B978-0-12-813592-1.00008-8
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