Artstein's theorem

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

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

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}.

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.

See also

• Analysis and control of nonlinear systems
• Control-Lyapunov function

References

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