Artstein's theorem
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Short description: Theorem in control theory
Artstein's theorem states that a nonlinear dynamical system in the control-affine form
[math]\displaystyle{ \dot{\mathbf{x}} = \mathbf{f(x)} + \sum_{i=1}^m \mathbf{g}_i(\mathbf{x})u_i }[/math]
has a differentiable control-Lyapunov function if and only if it admits a regular stabilizing feedback u(x), that is a locally Lipschitz function on Rn\{0}.[1]
The original 1983 proof by Zvi Artstein proceeds by a nonconstructive argument. In 1989 Eduardo D. Sontag provided a constructive version of this theorem explicitly exhibiting the feedback.[2][3]
See also
References
- ↑ Artstein, Zvi (1983). "Stabilization with relaxed controls" (in en). Nonlinear Analysis: Theory, Methods & Applications 7 (11): 1163–1173. doi:10.1016/0362-546X(83)90049-4.
- ↑ Sontag, Eduardo D. A Universal Construction Of Artstein's Theorem On Nonlinear Stabilization
- ↑ Sontag, Eduardo D. (1999), "Stability and stabilization: discontinuities and the effect of disturbances", in Clarke, F. H.; Stern, R. J.; Sabidussi, G., Nonlinear Analysis, Differential Equations and Control, Springer Netherlands, pp. 551–598, doi:10.1007/978-94-011-4560-2_10, ISBN 9780792356660
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