LYAPUNOV THEOREM FOR LOCAL STABILITY




 
 

        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.