The article is devoted to the optimization of first-order evolution inclusions (DFI) with undivided conditions. Optimality conditions are formulated in terms of locally adjoint mappings (LAMs). The construction of \duality relations" is an indispensable approach for the differential inclusions. In this case, the presence of discrete-approximate problems is a bridge between discrete and continuous problems. At the end of the article, as an example, we consider duality in optimization problems with linear discrete and first-order polyhedral DFIs.