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Artstein's theorem states that a nonlinear dynamical system in the control-affine form

${\dot {\mathbf {x} }}=\mathbf {f(x)} +\sum _{i=1}^{m}\mathbf {g} _{i}(\mathbf {x} )u_{i}$

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]

References

^ Artstein, Zvi (1983). "Stabilization with relaxed controls". 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. (eds.), Nonlinear Analysis, Differential Equations and Control, Springer Netherlands, pp. 551–598, arXiv:math/9902026, doi:10.1007/978-94-011-4560-2_10, ISBN 9780792356660, S2CID 13439

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[1] Artstein, Zvi (1983). "Stabilization with relaxed controls". Nonlinear Analysis: Theory, Methods & Applications. 7 (11): 1163–1173. doi:10.1016/0362-546X(83)90049-4.

[2] Sontag, Eduardo D. A Universal Construction Of Artstein's Theorem On Nonlinear Stabilization

[3] Sontag, Eduardo D. (1999), "Stability and stabilization: discontinuities and the effect of disturbances", in Clarke, F. H.; Stern, R. J.; Sabidussi, G. (eds.), Nonlinear Analysis, Differential Equations and Control, Springer Netherlands, pp. 551–598, arXiv:math/9902026, doi:10.1007/978-94-011-4560-2_10, ISBN 9780792356660, S2CID 13439

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Artstein's theorem

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