Lyapunov functions and stability criteria for nonlinear systems with multiple critical eigenvalues

Lyapunov functions are explicitly constructed for nonlinear systems of ordinary differential equations whose linearizations possess multiple critically stable eigenvalues. The construction yields efficient criteria imposed upon a nonlinear stability matrix constructed from the system dynamics for local asymptotic stability inference. Direct formulae for the entries of the nonlinear stability matrix are derived. A less restrictive notion of definiteness for symmetric real matrices, referred to as the relaxed definiteness, is presented. It is shown that in the presence of multiple critical modes, the equilibrium point is locally asymptotically stable if the so-constructed nonlinear stability matrix is relaxed negative definite. The results not only facilitate stabilizing feedback synthesis and stabilizability analysis, but also provide additional design options for stabilization