On Duality in Second-Order Discrete and Differential Inclusions with Delay

The present paper studies the duality theory for the Mayer problem with second-order evolution differential inclusions with delay and state constraints. Although all the proofs in the paper relating to dual problems are carried out in the case of delay, these results are new for problems without delay, too. To this end, first we use an auxiliary problem with second-order discrete- and discrete-approximate inclusions. Second, applying infimal convolution concept of convex functions, step-by-step, we construct the dual problems for discrete, discrete–approximate, and differential inclusions, and prove duality results, where the Euler–Lagrange type inclusions are “duality relations” for both primary and dual problems and that the dual problem for discrete–approximate problem make a bridge between the dual problems of discrete and continuous problems. Thus, proceeding to the limit procedure, we establish the dual problem for the continuous problem. In addition, semilinear problems with discrete and differential inclusions of second order with delay are also considered. These problems show that supremum in the dual problems are realized over the set of solutions of the Euler–Lagrange-type discrete/differential inclusions, respectively.