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A necessary condition for feedback stabilization. (English) Zbl 0699.93075
Summary: We give a new necessary condition for the existence of a - static or dynamic - continuous asymptotically stabilizing feedback control for the system $$\dot x=f(x,u)$$. We apply our condition to give an example of a system in $${\mathbb{R}}^ 3$$ which is small time locally controllable, satisfies ‘Brockett’s second and third condition’ [see R. W. Brockett, Differential geometric control theory, Proc. Conf., Mich. Technol. Univ. 1982, Prog. Math. 27, 181-191 (1983; Zbl 0528.93051)] and cannot be locally asymptotically stabilized by employing a - static or dynamic - continuous feedback law.

##### MSC:
 93D15 Stabilization of systems by feedback 93C10 Nonlinear systems in control theory 93C15 Control/observation systems governed by ordinary differential equations 93D20 Asymptotic stability in control theory
Zbl 0528.93051
Full Text:
##### References:
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