Page 258 - Mathematical Techniques of Fractional Order Systems
P. 258

Controllability of Fractional Chapter | 8  247


             8.5  CONCLUSION
             This chapter has promoted the complete controllability result of fractional
             higher order stochastic integrodifferential inclusions. Based on fractional cal-
             culus, stochastic analysis approach, and suitable fixed point theorems,
             namely the Bohnenblust Karlin fixed point theorem for the convex case and
             the Covitz Nadler for the nonconvex case, sufficient conditions have been
             derived for the controllability of fractional higher order stochastic integrodif-
             ferential inclusions in finite dimensional space. Many practical problems
             could be modeled into linear fractional stochastic differential inclusions,
             hence a suitable controllability Grammian matrix may be constructed to
             establish the result and an appropriate nonlinear model may also be formu-
             lated and one can use proposed technique to derive the controllability results.
             The result may be extended to infinite dimensional Hilbert space for approxi-
             mate controllability results by suitable hypotheses on nonlinear functions and
             appropriate fixed point theorems.



             ACKNOWLEDGMENTS
             The work of authors are supported by Council of Scientific and Industrial Research
             (CSIR), Extramural Research Division, Pusa, New Delhi, India under the grant No. 25/
             (0217)/13/EMR-II.

             REFERENCES
             Abbas, S., Mouffak, B., 2013. Fractional order Riemann-Liouville integral inclusions with two
                independent variables and multiple delay. Opuscula Math. 33 (2), 209 222.
             Aubin, J.B., Cellina, A., 1984. Differential Inclusions: Set-Valued Maps and Viability Theory.
                Springer-Verlag, New York.
             Azar, A.T., Vaidyanathan, S., Ouannas, A., 2017. Fractional Order Control and Synchronization
                of Chaotic Systems, Studies in Computational Intelligence. Springer-Verlag, Germany.
             Balachandran, K., Kokila, J., 2012. On the controllability of fractional dynamical systems. Int. J.
                Appl. Math. Comput. Sci. 22 (3), 523 531.
             Balasubramaniam, P., 2002. Existence of solutions of functional stochastic differential inclu-
                sions. Tamkang J. Math. 33 (1), 25 34.
             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.
             Bohnenblust, H.F., Karlin, S., 1950. On a Theorem of Ville, Contributions to the Theory of
                Games. Princeton University Press, Princeton.
             Castaing, C., Valadier, M., 2006. Convex Analysis and Measurable Multifunctions. Springer,
                New York.
             Covitz, H., Nadler, S.B., 1970. Multi-valued contraction mappings in generalized metric spaces.
                Israel J. Math. 8 (1), 5 11.
             Deimling, K., 1992. Multivalued Differential Equations. Walter de Gruyter, New York.
   253   254   255   256   257   258   259   260   261   262   263