LIAPUNOV-FUNCTIONS AND STABILITY-CRITERIA FOR NONLINEAR-SYSTEMS WITH MULTIPLE CRITICAL EIGENVALUES

Efficient criteria are derived via explicit construction of Liapunov f unctions for local asymptotic stability inference of nonlinear systems , whose linearizations possess c greater-than-or-equal-to 2 critical m odes at an equilibrium point. The stability criteria are obtained in t he context of two novel notions, relaxed definiteness and relaxed stab ility. A real symmetric c x c matrix Q is relaxed negative definite if w(T)Qw < 0 for any 0 not-equal w is-an-element-of R+c, R+ = [0, infin ity); a matrix R is relaxed stable if there is a P > 0 such that PR R(T)P is relaxed negative definite. The construction leads to some cha racterizations 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 mo des is locally asymptotically stable generically if the stability char acteristic matrix is relaxed stable and less generically if the stabil ity characteristic tensor is trivial or degenerate in certain way and the perturbed stability characteristic matrix is relaxed stable.