Nontangency-based Lyapunov tests for stability and convergence

We give new results for Lyapunov and asymptotic stability of nonlinear systems. In addition, we also give results for convergence and semistability. Convergence is the property whereby every trajectory of a system converges to a limit point that may depend upon the initial condition. Semistability is the additional requirement that the limit points of the trajectories of a convergent system also be Lyapunov stable. Our results do not make assumptions of sign definiteness on the Lyapunov function. Instead, our results use a novel nontangency condition between the system dynamics and the level sets of the Lyapunov function or its derivative. Using this nontangency condition, we extend previously known Lyapunov stability and asymptotic stability results involving semidefinite Lyapunov functions