On the existence of stabilising feedback controls for real analytic small-time locally controllable systems

Abstract

It is shown that, for real analytic control systems of the form f : M × Ω (q, u) → f (q, u) ∈ T q M, where M is a real analytic manifold and Ω is a separable metric space, small-time local controllability from an equilibrium p ∈ M implies the existence of a piecewise analytic feedback control that locally stabilises f at p. The proof is similar in spirit to an earlier analogous result for globally controllable systems; however, it resolves several technical obstructions that emerge when the assumption of small-time local controllability is substituted for that of global controllability. In the light of a recent characterisation of small-time local controllability for homogeneous control systems, the main result of the paper implies that, for a large class of control systems that appear in applications and the literature, there is a computable sufficient condition for stabilisability by means of a piecewise analytic feedback control.