On the stability analysis of nonlinear systems using polynomial Lyapunov functions

In the stability study of nonlinear systems, not to found feasible solution for the LMI problem associated with a quadratic Lyapunov function shows that it doesn’t exist positive definite quadratic Lyapunov function that proves stability of the system, but doesn’t show that the system isn’t stable. So, we must search for other Lyapunov functions. That's why, in the present paper, the construction of polynomial Lyapunov candidate functions is investigated and sufficient conditions for global asymptotic stability of analytical nonlinear systems are proposed to allow computational implementation.