Optimization of first-order impulsive differential inclusions and duality

The paper studies optimization problem described by first order evolution impulsive differential inclusions (DFIs); in terms of locally adjoint mappings in framework of convex and nonsmooth analysis we formulate sufficient conditions of optimality. Then we construct the dual problems for impulsive DFIs and prove duality results. It turns out that the Euler-Lagrange inclusions are "duality relations" for both the primal and dual problems, that is, if some pair of functions satisfies this relation, then one of them is a solution to the primal problem, and the other is a solution to a dual problem. At the end of the paper duality in optimal control problems with first order linear and polyhedral DFIs are considered, where the supremum is taken over the class of non-negative absolutely continuous functions.