Artstein's theorem explained

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

• x = f(x) + \sum ^{m} _{i =1} 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\.^[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]

See also

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

Notes and References

1. Artstein. Zvi. 1983. Stabilization with relaxed controls. Nonlinear Analysis: Theory, Methods & Applications. en. 7. 11. 1163–1173. 10.1016/0362-546X(83)90049-4.
2. Sontag, Eduardo D. A Universal Construction Of Artstein's Theorem On Nonlinear Stabilization