Control-Lyapunov function: Difference between revisions

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|chapter-url=https://link.springer.com/chapter/10.1007/978-0-8176-4759-9_3

|publisher=Birkhäuser

|year=2008|pages=33–63
|doi=10.1007/978-0-8176-4759-9_3

|edition=illustrated, reprint

|isbn=978-0-8176-4758-2|

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==Theorems==

E.[[ D.Eduardo Sontag]] showed that for a given control system, there exists a continuous CLF if and only if the origin is asymptotic stabilizable.<ref>{{cite journal |first=E.D. |last=Sontag |title=A Lyapunov-like characterization of asymptotic controllability|journal=SIAM J. Control Optim.|volume=21 |issue=3 |year=1983 |pages=462–471|doi=10.1137/0321028 |s2cid=450209 }}</ref> It was later shown by [[Francis Clarke (mathematician)|Francis H. Clarke]], Yuri Ledyaev, [[Eduardo Sontag]], and A.I. Subbotin that every [[Controllability|asymptotically controllable]] system can be stabilized by a (generally discontinuous) feedback.<ref>{{cite journal |first1=F.H.|last1=Clarke |first2=Y.S.|last2=Ledyaev |first3=E.D.|last3=Sontag |first4=A.I.|last4=Subbotin |title=Asymptotic controllability implies feedback stabilization |journal=IEEE Trans. Autom. Control|volume=42 |issue=10 |year=1997 |pages=1394–1407|doi=10.1109/9.633828 }}</ref>

Artstein proved that the dynamical system ({{EquationNote|2}}) has a differentiable control-Lyapunov function if and only if there exists a regular stabilizing feedback ''u''(''x'').

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A Control-Lyapunov candidate is then

:<math>

r \mapsto V(r) :=\frac{1}{2}r^2

</math>

which is positive definite for all <math> r \ne 0</math>.

Now taking the time derivative of <math>V</math>