Existence of solutions in dynamic optimization

A new existence theory is presented for a standard problem in optimum control consisting of minimizing the cost integral over the processes satisfying given dynamics and prescribed conditions on the values of the state. The theory presented proceeds in an indirect way by invoking necessary conditions at a certain intermediate point, thereby generating a minimizing sequence with special properties. The comparison with the existing literature, as well as the presentation of the main idea, is presented in terms of the classical variational framework. The proof, which is explained in full, constitutes the first application to existence theory of the technique known as proximal analysis of value functions.>