Linearization and Boundary Trajectories of Nonsmooth Control Systems

Abstract

This paper deals with boundary trajectories of non-smooth control systems and differential inclusions.Consider a control system(1.1)and denote by R(t) its reachable set at time t. Let (z, u*) be a trajectory-control pair. If for every t from the time interval [0, 1], z(t) lies on the boundary of R(t) then z is called a boundary trajectory. It is known that for systems with Lipschitzian in x right-hand side, z is a boundary trajectory if and only if z(1) belongs to the boundary of the set R(1). If z is not a boundary trajectory, that is, z(1) ∊ Int R(1) then the system is said to be locally controllable around z at time 1.A first-order necessary condition for boundary trajectories of smooth systems comes from the Pontriagin maximum principle, (see e.g. [12]).