Optimization of fourth-order discrete-approximation inclusions

The paper concerns the necessary and sufficient conditions of optimality for Cauchy problem of fourth-order discrete (P_D) and discrete-approximate (P_DA) inclusions. The main problem is the formulation of the fourth-order adjoint discrete and discrete-approximate inclusions (DAIs) and transversality conditions, which are peculiar to problems including fourth-order derivatives and approximate derivatives. Thus, the necessary and sufficient conditions of optimality are proved, incorporating the Euler–Lagrange and Hamiltonian forms of inclusions. Derivation of optimality conditions is based on the apparatus of locally adjoint mapping (LAM) and equivalence of LAMs theorems. Moreover, in the application of these results the fourth-order linear optimal control problems with linear discrete and discrete-approximate inclusions are considered. An approach to obtain approximate numerical solutions of linear discrete inclusions is presented.