Regularity and directional differentiability with respect to parameters of solutions to nonlinear switching systems

Many physical systems and processes encompass several modes of operation with a different dynamical behavior in each mode. Switching systems provide a suitable mathematical model for such processes. It is well known that the continuous states of switching systems are often non-differentiable with respect to parameters at the switching instants. However, states at these instants are often involved in optimal control of switching systems, resulting in nonsmooth dynamic programming problems. Directional derivative plays an important role in optimization and control theory. In this paper, we explore the directional differentiability of solutions to a class of switching systems with respect to parameters. We present some basic properties for the switching systems, including non-Zenoness, existence and uniqueness of solution, and local Lipschitzean of solution with respect to parameters. On this basis, we assert the directional differentiability of the solution with respect to parameters and give the formula of the directional derivative.