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'').
Line 72 ⟶ 74:
A Control-Lyapunov candidate is then
:<math>
r \mapsto V(r) :=\frac{1}{2}r^2
</math>
which is positive definite for all <math> q \ne 0</math>, <math>\dot{q}r \ne 0</math>.
