On the existence of stabilising feedback controls for real analytic small-time locally controllable systems. (English) Zbl 1327.93315
Summary: It is shown that, for real analytic control systems of the form \(f:M\times\Omega\ni (q,u)\mapsto f(q,u)\in T_qM\), where \(M\) is a real analytic manifold and \(\Omega\) is a separable metric space, small-time local controllability from an equilibrium \(p\in M\) 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.
MSC:
| 93D15 | Stabilization of systems by feedback |
| 93B05 | Controllability |
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 93C10 | Nonlinear systems in control theory |
Keywords:
small-time local controllability; state-feedback stabilisation; piecewise analytic feedback control; subanalytic setsReferences:
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