On the Projected Gradient Methods for Switched - Mode Systems Optimization

This paper is devoted to optimal control problems (OCPs) associated with a class of switched - mode systems. We propose a new formalism for the autonomous variant of these dynamic systems and study an OCP with a smooth cost functional. The control design we finally propose includes an optimal switching times selection (”timing”) as well as an optimal modes sequence scheduling (”sequencing”). The initial nonlinear optimization problem is replaced by a suitable relaxed OCP with some specific control restrictions. We consider a fully convexified OCP as well as a ”weakly” relaxed optimization problems. Our aim is to analyse a family of numerically tractable control algorithms, namely, the projected gradient methods involved Armijo line search and to establish the corresponding convergence properties. The abstract consistency analysis we implement makes it possible to apply relative simple first - order numerical procedures to a sophisticated initial OCP involved an autonomous switched - mode dynamics.