Solvability and optimal controls of impulsive Hilfer fractional delay evolution inclusions with Clarke subdifferential

We study the solvability and optimal controls of an impulsive nonlinear Hilfer fractional delay evolution inclusion in Banach spaces. For the main results, we use fractional calculus, fixed point technique, semigroup theory and multivalued analysis. We introduce the notion of Clarke delay subdifferential. To continue and extend previous works in the field, we prove an existence result of optimal control pair for Lagrange problem LP under appropriate set of sufficient conditions. Finally, as application, we give an impulsive control partial differential inclusion with Hilfer fractional derivative.