Infimal Convolution and Duality in Problems with Third-Order Discrete and Differential Inclusions

Abstract

This paper is concerned with the Mayer problem for third-order evolution differential inclusions; to this end, first we use auxiliary problems with third-order discrete and discrete-approximate inclusions. In the form of Euler–Lagrange-type inclusions and transversality conditions, necessary and sufficient optimality conditions are derived. The principal idea of obtaining optimal conditions is locally adjoint mappings. Then, applying infimal convolution concept of convex functions, step by step, we construct the dual problems for third-order discrete, discrete-approximate and differential inclusions and prove duality results. It appears that the Euler–Lagrange-type inclusions are duality relations for both primary and dual problems and that the dual problem for discrete-approximate problem makes a bridge between the dual problems of discrete and continuous problems. As a result, the passage to the limit in the dual problem with discrete approximations plays a substantial role in the next investigations, without which it is hardly ever possible to establish any duality to continuous problem. In this way, relying to the described method for computation of the conjugate and support functions of discrete-approximate problems, a Pascal triangle with binomial coefficients can be successfully used for any “higher order” calculations. Thus, to demonstrate this approach, some semilinear problems with discrete and differential inclusions of third order are considered. These problems show that maximization in the dual problems is realized over the set of solutions of the Euler–Lagrange-type discrete/differential inclusions.