Page 196 - Mathematical Techniques of Fractional Order Systems

Chapter 7

Controllability of Single-valued

and Multivalued Fractional
Stochastic Differential Equations

1
Peachimuthu Tamilalagan and Pagavathigounder Balasubramaniam 2
1
Amrita Vishwa Vidyapeetham (Deemed to be University), Coimbatore, Tamil Nadu, India,
2
Gandhigram Rural Institute (Deemed to be University), Dindigul, Tamil Nadu, India
7.1 INTRODUCTION

The fractional calculus instills new dimensions to understand or describe
basic nature of the real life phenomena arising in science and engineering in
a better way. Hence, it serves as an eminent tool for providing more accurate
and successful results than integer order differential equations in the model-
ing of several real-life phenomena. Accordingly, the fractional abstract
Cauchy problems attain its necessity of studying the existence, controlla-
bility, and other qualitative and quantitative properties (see Azar et al., 2017;
Bajlekova, 2001; Srivastava et al., 2006; Podlubny, 1998).
It is well known that, random differential and integral equations play an
important role in characterizing many social, physical, biological, and engi-
neering problems. The modeling of natural phenomena by stochastic differ-
ential equations (SDEs) plays an important role, wherever we encounter the
fluctuations in nature (see Da Prato and Zabczyk, 2014; Mao, 2007). Hence,
it is of great significant to study fractional differential equations (FDEs) with
stochastic effects.
Further, the notion of multimaps arises in many branches of mathematics,
namely mathematical economics, theory of games, convex analysis, etc. The
multimaps play a significant role in the description of process in control
theory, since the presence of control provides an intrinsic multivalence in the
evolution of the system (Kamenskii et al., 2001). Differential inclusions have
wide applications in economics, engineering, and so on. The theory of differ-
ential inclusions has been developed accordingly in the past three decades
(see Aubin and Cellina, 1984; Balasubramaniam and Ntouyas, 2006;
Balasubramaniam et al., 2005; Deimling, 1992; Hu, 1997). The extension of

Mathematical Techniques of Fractional Order Systems. DOI: https://doi.org/10.1016/B978-0-12-813592-1.00007-6
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