# Nonsmooth Optimization for Discretized Switching Time Problems

Kathrin Flaßkamp^{*}

In this contribution, optimization problems for discretized switched dynamical systems are introduced. A switched system is defined by a family of vector fields together with a switching law which chooses the active vector field at any time. Switching time optimization focuses on the optimization of the switching times in order to govern the system evolution to a desired behavior described by some cost function.\newline It turns out that in contrast to the original, continuous time problem, the discretized problem loses differentiability with respect to the optimization variables. Thus, this class of problems has to be addressed by nonsmooth optimization techniques. \newline In this talk, we study the structure of nondifferentiability for discretized switching time problems and illustrate the analytic results by means of several examples, e.g. a hybrid double pendulum or a switched linear system with nonsmooth optimality conditions. Finally, a subgradient descent algorithm is applied to solve these problems numerically.\newline This is joint work with Todd Murphey (Northwestern University, USA) and Sina Ober-Blöbaum (University of Paderborn, Germany).

Mathematics Subject Classification: 49K15 93C30

Keywords: Switched Systems, Nonsmooth Optimization

Minisymposion: Nonsmooth Optimization