Optimization of first-order Nicoletti boundary value problem with discrete and differential inclusions and duality

This paper is devoted to optimal control of the first-order Nicoletti boundary value problem (BVP) with discrete and differential inclusions (DFIs) and duality. First, we define Nicoletti-type problem with discrete inclusions, formulate optimality conditions for it and, based on the concept of infimal convolution, dual problems. Then, using the auxiliary discrete-approximate problem, we construct dual problems for Nicoletti DFIs and prove the duality theorems. Here, for the transition to the continuous problem, some results on the equivalence of locally adjoint mappings and support functions to the graph of set-valued mapping are proved. It turns out that the Euler–Lagrange type inclusions are ‘duality relations’ for both primal and dual problems, which means that a pair consisting of solutions to the primal and dual problems satisfies this extremal relation and vice versa. Finally, as an appication of the results obtained, we consider the first-order Nicoletti BVP with polyhedral DFIs.