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

Abstract

Efficient criteria are derived via explicit construction of Liapunov functions for local asymptotic stability inference of nonlinear systems, whose linearizations possessc ≥ 2 critical modes at an equilibrium point. The stability criteria are obtained in the context of two novel notions, relaxed definiteness and relaxed stability. A real symmetricc×c matrixQ isrelaxed negative definite ifw ^TQw<0 for any 0≠wεℜ ^c₊ , ℜ₊=[0, ∞); a matrixR isrelaxed stable if there is aP>0 such thatPR+R ^TP is relaxed negative definite. The construction leads to some characterizations of the nonlinear system's local structure,in the sense of Liapunov, and the so-called stability characteristic matrices and tensors. It is shown that a nonlinear system with multiple critical modes is locally asymptotically stable generically if the stability characteristic matrix is relaxed stable and less generically if the stability characteristic tensor is trivial or degenerate in certain way and the perturbed stability characteristic matrix is relaxed stable.