On the existence of stabilising feedback controls for real analytic small-time locally controllable systems. Issue 4 (December 2015)

• Record Type: Journal Article
• Authors: Isaiah, Pantelis
• Is Part Of: Mathematics of control, signals, and systems. Volume 27:Issue 4(2015)
• Page Start: 467
• Page End: 492
• Publication Date: 2015-12
• Subjects: Small-time local controllability -- State-feedback stabilisation -- Piecewise analytic feedback control -- Subanalytic sets; Control theory -- Periodicals; Signal processing -- Periodicals; System analysis -- Periodicals; 621.38223
• DOI: 10.1007/s00498-015-0148-z
• Languages: English
• ISSNs: 0932-4194

Abstract

Abstract It is shown that, for real analytic control systems of the form $$f:{M}\times \varOmega \ni (q, u)\mapsto f(q, u)\in {T}_q{M}$$ 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.