Discretized switching time optimization problems

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. Thus, the switching law encoding the switching times and the sequence of modes may serve as a design parameter. Switching time optimization (STO) focuses on the optimization of the switching times in order to govern the system evolution to a desired behavior described by some cost function. However, it is rare that a STO problem can be solved analytically leading to the use of numerical approximation methods. In this contribution, we directly start with applying integration schemes to approximate the system´s state and adjoint trajectories and study the effect of this discretization. It turns out that in contrast to the continuous time problem, the discretized problem loses differentiability with respect to the optimization variables. The isolated nondifferentiable points can be precisely identified though. Nevertheless, to solve the STO problem, nonsmooth optimization techniques have to be applied which we illustrate using a hybrid double pendulum.