Optimal control of fractional non-autonomous evolution inclusions with Clarke subdifferential

Abstract

In this paper, the non-autonomous fractional evolution inclusions of Clarke subdifferential type in a separable reflexive Banach space are investigated. The mild solution of the non-autonomous fractional evolution inclusions of Clarke subdifferential type is defined by introducing the operators \(\psi (t,\tau )\) and \(\phi (t,\tau )\) and V(t), which are generated by the operator \(-\mathcal {A}(t)\) and probability density function. Combined the measure of non-compactness, some properties of the Clarke subdifferential with fixed point theorem of \(\kappa -\)condensing multi-valued maps, a new existence result of mild solution is established. Moreover, an existence result of optimal control pair for the Lagrange problem is also derived. The results obtained in this paper extend the study of fractional autonomous evolution equations to the non-autonomous fractional evolution inclusions. Finally, a fractional partial differential inclusion with control is provided to illustrate the applications of the obtained main results.