On the existence of stabilising feedback controls for real analytic small-time locally controllable systems. Issue 4 (December 2015) - Isaiah, Pantelis
Abstract It is shown that, for real analytic control systems of the form $$f:M× \varOmega \ni (q, u)\mapsto f(q, u)\in {T}_qM$$ f : M × Ω ∋ ( q, u ) ↦ f ( q, u ) ∈ T q M, whereM is a real analytic manifold and $$\varOmega $$ Ω is a separable metric space, small-time local controllability from an equilibrium $$p\in {M}$$ p ∈ M implies the existence of a piecewise analytic feedback control that locally stabilisesf atp . 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.