Page 237 - Mathematical Techniques of Fractional Order Systems

226 Mathematical Techniques of Fractional Order Systems

1α , 2 have been studied in Hilbert space by using ða; kÞ-regularized families
of bounded linear operators. The peculiarity of the obtained theoretical
results lies in the fact that ða; kÞ-regularized families of bounded linear opera-
tors cover the theories of C 0 -semigroups and cosine families as particular
cases. The second part of this chapter dealt with the solvability and existence
of optimal control for a class of fractional stochastic integro-differential
equations with infinite delay in Hilbert space by employing the well devel-
oped theory of resolvent operators for integral equations.
Our future investigations are mainly focused on stability analysis of new
classes of single and multivalued FSDEs in the sense of Caputo Fabrizio’s
or Atangana Baleanu’s fractional derivative by employing fixed point tech-
nique. Since, the fractional derivatives are nonlocal and have weakly singular
kernels, hence the stability analysis of FDEs is more complex than that of
classical integer order differential equations. However, the new definitions of
fractional derivative, namely Caputo Fabrizio’s and Atangana Baleanu’s
fractional derivative can be employed to study the stability analysis, since
they are defined without singular kernel. Caputo Fabrizio’s fractional deriv-
ative is based on exponential kernel and Atangana Baleanu’s fractional
derivative used Mittag Leffler function as nonlocal kernels.

REFERENCES

Agrawal, O.P., 2004. A general formulation and solution scheme for fractional optimal control
problems. Nonlinear Dynam. 38 (1-4), 323 337.
Agrawal, O.P., Defterli, O., Baleanu, D., 2010. Fractional optimal control problems with several
state and control variables. J. Vibrat. Control 16 (13), 1967 1976.
Aubin, J., Cellina, A., 1984. Differential Inclusions. Heidelberg, Berlin.
Azar, A.T., Vaidyanathan, S., Ouannas, A., 2017. Fractional Order Control and Synchronization
of Chaotic Systems, Vol. 688. Springer, New York.
Bajlekova, E.G., 2001. Fractional evolution equations in banach spaces. Ph.D. thesis, University
Press Facilities, Eindhoven University of Technology Eindhoven, The Netherlands.
Balachandran, K., Kiruthika, S., 2011. Existence results for fractional integrodifferential equa-
tions with nonlocal condition via resolvent operators. Comput. Math. Applicat. 62 (3),
1350 1358.
Balasubramaniam, P., Ntouyas, S., 2006. Controllability for neutral stochastic functional differ-
ential inclusions with infinite delay in abstract space. J. Math. Anal. Applicat. 324 (1),
161 176.
Balasubramaniam, P., Tamilalagan, P., 2015. Approximate controllability of a class of fractional
neutral stochastic integro-differential inclusions with infinite delay by using mainardis func-
tion. Appl. Math. Comput. 256, 232 246.
Balasubramaniam, P., Tamilalagan, P., 2016. The solvability and optimal controls for impulsive
fractional stochastic integro-differential equations via resolvent operators. J. Optimizat.
Theory Applicat. 1 17.
Balasubramaniam, P., Ntouyas, S., Vinayagam, D., 2005. Existence of solutions of semilinear
stochastic delay evolution inclusions in a hilbert space. J. Math. Anal. Applicat. 305 (2),
438 451.