Existence and approximate controllability for systems governed by fractional delay evolution inclusions

Abstract

We prove the existence of mild solutions for the control system governed by fractional delay evolution inclusion in a Banach space X:assuming that the operator A generates a strongly continuous semigroup and the nonlinear part F is a multi-valued function with convex and closed values for which is weakly upper semicontinuous for a.e. and has a -integrable selection for each . Moreover, in this setting, we obtain the approximate controllability for such system provided that the corresponding linear control system is approximately controllable. Finally, our results apply, in particular, to the control problem of fractional partial differential inclusion with time delay.

Keywords:

• fractional evolution inclusion
• delay
• mild solution
• approximate controllability

AMS Subject Classifications:

• Primary: 35R11
• Secondary: 34K09
• 93B05

Acknowledgements

We would like to express our thanks to the anonymous referees for their suggestions and comments that improved the quality of the paper.

Notes

This research was supported in part by NNSF of China [grant number 11101202] and the Guangdong Province Natural Science Foundation [grant number S2013010013050].