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Smooth stabilization implies coprime factorization. (English) Zbl 0682.93045
The main theorem states that if a nonlinear system $$\dot x=f(x)+G(x)u$$ can be made globally asymptotically stable by a smooth feedback $$u=K(x)$$, then there also exists a smooth feedback $$u=K'(x)+v$$ such that the feedback modified system is input-to-state stable. The construction of $$K'(x)$$ is given explicitly for feedback linearizable systems, which are trivially smoothly stabilizable. Based upon this main result it is also shown that smoothly stabilizable systems admit coprime factorizations. Finally some results about input-to-output stability are given.
Reviewer: A.van der Schaft

##### MSC:
 93D15 Stabilization of systems by feedback 93B28 Operator-theoretic methods 47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators 93C10 Nonlinear systems in control theory 93D25 Input-output approaches in control theory
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