Bounds on the number of switchings for trajectories of piecewise analytic vector fields

Abstract

We study the trajectories of systems x ̇ = X(x) , where X is a continuous “extendably piecewise analytic” vector field, i.e., a continuous vector field X such that the domain of ƒ admits a locally finite partition I into sets such that for each A ∈ I there is a vector field X A which is analytic on a neighborhood of the closure of A and whose restriction to A coincides with that of X. We prove that the trajectories are piecewise analytic, with a priori bounds on the number of switchings for all trajectories that stay in a fixed compact set and whose duration does not exceed a fixed number T. This result implies the existence of a regular synthesis for optimal control problems with a strictly convex Lagrangian, and a linear dynamics with polyhedral constraints on the controls.