Linearization at a Singular Point

Abstract

In this chapter we consider nonlinear control systems near a singular point, i.e., a common fixed point of the drift vector field and the control vector fields. Linearization at this point yields a bilinear system in ℝ^d; hence the linearized system is a special case of the general model considered in the preceding chapter. However, here we use an additional structure: The systems group is a Lie group and the usual Lie algebra rank condition for local accessibility implies that the systems group acts transitively on the projective space ℙ^d-1. Thus the projective space is a homogeneous space for the systems group. As observed in Proposition 4.5.21, this implies that the inner pair condition for a piecewise constant periodic control is satisfied everywhere on projective space, if and only if the corresponding element of the systems semigroup lies in the interior of this semigroup. This allows us to construct the control sets with nonvoid interior directly (without variation ρ of the control range and use of a ρ-inner pair condition). Furthermore, the relation between the control sets and the chain control sets in projective space can be made much more precise than in the general situation.