Optimization of semilinear higher-order delay differential inclusions

The paper investigates an optimal control problem described by higher-order semi-linear delay differential inclusions (DFIs). In terms of the Euler-Lagrange type adjoint DFIs and Hamiltonian, a sufficient optimality condition for higher-order DFIs is derived, which have different forms in different time intervals depending on the delay parameter, and in the problem without delay effect these two adjoint DFIs coincide. At the same time, when constructing the Euler-Lagrange type adjoint DFI, without using traditional approaches to constructing an adjoint inclusion based on a discrete-approximate method, the new method of adjoint DFI of Mahmudov for ‘higher-order problems’ is used. It is shown also that the adjoint DFI for the first order DFI coincides with the classical Euler-Lagrange inclusion, and the optimality conditions coincide with the results of Rockafellar on the Mayer problem with first order DFIs. At the end of the article, some problems closely related to the Pontryagin maximum principle are considered.