Optimization of Lagrange problem with higher-order differential inclusion and special boundary-value conditions

This article concerns about optimality conditions for boundary-value problems related to differential inclusions (DFIs) of higher orders. We intend to attain optimality conditions when a general Lagrange functional takes place in the cost function. Moreover, it is intended that these conditions are applicable to the non-convex case as well. The notion of locally adjoint mapping for both convex and non-convex functions is used via Hamiltonian functions and arg-max sets of set-valued functions to obtain results. The presented main problem turns into a problem in the calculus of variations with some simplifications. It is noteworthy to see that the famous Euler-Poisson equation arises in this case. Furthermore, a higher-order semilinear optimal control problem is considered as an application, and its sufficient conditions, including Weierstrass-Pontryagin maximum principle, are derived. Then, the dual problems for the presented primal problems are established and their duality theorems are proved. Finally, the third-order polyhedral DFI with duality relations is considered.