A first-order numerical approach to switched-mode systems optimization

This paper studies optimal control processes governed by switched-mode systems. We consider Optimal Control Problems (OCPs) with smooth cost functionals and apply a newly elaborated abstraction for the system dynamics under consideration. The control design we finally obtain includes an optimal switching times selection (”timing”) as well as an optimal modes sequence scheduling (”sequencing”). For purpose of numerical treatment of the initially given OCP we use a newly elaborated relaxation concept and analyse the resulting ”weakly relaxed” optimization problems. In contrast to the conventional relaxations our approach is based on the infimal prox convolution technique and does not use the celebrated Chattering Lemma. This fact causes a lower relaxation gap. Our aim is to propose a gradient-based computational algorithms for the OCPs with switched-mode dynamics. In particular, we deal with the celebrated Armijo-type gradient methods and establish the corresponding convergence properties. The numerical consistency (numerical stability) analysis makes it possible to apply a class of relative simple first-order numerical procedures to a sophisticated initial OCP involved in specific switched-mode dynamics.