Alexander Aurell: Does your control problem lack an optimal solution? Just relax...

Techniques such as Pontryagin's Maximum Principle and Dynamic Programming characterize optimal controls under the assumption that they exist and are elements of the set of admissible controls. If there is no optimal control among the admissible controls, is it still possible to determine the minimal value of the performance functional?

In this talk, I will answer this question by introducing relaxed controls. The admissible controls will be embedded into a larger set of controls, the set of probability measures on the set of control values. This set has a richer topological structure that allows for existence of a minimizer.