An analysis on the optimal control results for second-order Sobolev-type delay differential inclusions of Clarke's subdifferential type

In this paper, we study the existence and optimal control results for second-order Sobolevtype delay systems with Clarke's subdifferential type. Initially, the existence of mild solution is established for the proposed second-order delay differential system with the novel ideas of Clarke's subdifferential. The fixed point theorem of condensing multivalued maps, the strongly continuous cosine family, and the properties of Clarke's subdifferential are used to establish the existence of mild solution. Moreover, the existence of optimal control pair that is governed by the presented system is verified through Balder's theorem. Finally, an example is provided to illustrate the main results.