Necessary conditions of optimality for second-order nonlinear impulsive integro-differential equations on Banach spaces

We discuss a class of optimal control problems of systems governed by the second-order nonlinear impulsive integro-differential equations with time-varying generating operators in fractional power spaces. Introducing the reasonable PC–αPC–α-mild solution of the second-order nonlinear impulsive integro-differential equations we prove the existence of PC–αPC–α-mild solutions. Existence of optimal controls for a Lagrange problem of systems governed by the second-order nonlinear impulsive integro-equations is presented. We apply a direct approach to derive the maximum principle for the problem at hand.