THEOREM

If, in a ball B, there exists a scalar function V(x) with continuous first partial derivatives such that :

- V(x) is positive definite (locally in B),

- V'(x) = dV(x)/dt is negative semi-definite (locally in B)

Then the equilibrium point 0 is stable. if, actually, the derivative V'(x) is locally negative definite in B then the stability is asymptotic.