Artstein's theorem

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

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

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]

See also

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