ABSTRACT

The subsequent discussion is confined to control systems described by a differential equation dxdt = F(x, u) in which the control functions u(t) = (u1(t), . . . , um(t)) take values in a fixed set U in Rm. Most of the theory presented in this chapter is extracted through F and its derivatives, and for that reason it is expedient to assume that the state variable x(t) belongs to an analytic manifold M and that F(x, u) is an analytic vector field for each u in U . The reader not familiar with these notions may assume at the beginning that M is a finite dimensional vector space and that for each u ∈ U , F(x, u) can be represented by its Taylor series at each point x in M. The extensions to arbitrary manifolds will be defined as needed.