Artstein's theorem states that a
nonlinear dynamical system in the control-affine form
has a differentiable
control-Lyapunov function if and only if it admits a regular stabilizing feedback ''u''(''x''), that is a
locally Lipschitz
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exis ...
function on R
n\.
The original 1983 proof by
Zvi Artstein Zvi ( he, צְבִי and , ''Tzvi'', Ṣvi, "gazelle") is a Jewish masculine given name. Notable people with this name include:
* Zvi Aharoni (1921–2012), Israeli Mossad agent
* Zvi Arad (1942–2018), Israeli mathematician, acting president of B ...
proceeds by a nonconstructive argument. In 1989
Eduardo D. Sontag
Eduardo Daniel Sontag (born April 16, 1951, in Buenos Aires, Argentina) is an Argentine-American mathematician, and distinguished university professor at Northeastern University, who works in the fields control theory, dynamical systems, syste ...
provided a constructive version of this theorem explicitly exhibiting the feedback.
See also
*
Analysis and control of nonlinear systems
*
Control-Lyapunov function
References
Control theory
Theorems in dynamical systems
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