Optimization of the Lagrange Problem with second order discrete and differential inclusions and spatial boundary conditions

In this paper, we derive optimality conditions for the Lagrange problem with second-order differential inclusions (DFIs) and spatial boundary conditions. The Lagrangian and the set-valued mapping here also depend on the derivative of the sought trajectory. The difficulties that arise here are related to the construction of the adjoint discrete and differential inclusions. The novelty here lies in using the discretized method to establish the optimality condition for both discrete and DFIs. Optimality conditions for the discrete problem are generated by applying the concept locally adjoint mapping(LAM). Equivalence theorems are used to obtain the so-called Mahmudov's adjoint conditions for the discrete-approximat problem. Moreover passing to the limit, we get sufficient optimality conditions for the continuous problem. Unlike the Euler-Lagrange DFI, which only provides first-order optimality conditions, Mahmudov's adjoint inclusion is a powerful tool for establishing optimality conditions for higher-order problems. The findings are reinforced with an example. We also obtain similar results for the non-convex problem by using the concept of local tents.