Generalized absolute stability using Lyapunov functions with relaxed positivity conditions

Conditions are given for verifying stability and computing upper bounds on the induced (regional) L2 gain for systems defined by vector fields which are, along with their Jacobian, rational in the states and sector bounded nonlinearities. A class of candidate Lyapunov functions is considered that are polynomial on the states and the nonlinearities and have a polynomial scaled Lurie-Postnikov term. The main result of this letter is a set of conditions that relax the requirement on the candidate Lyapunov function from being sum-of-squares with respect to the nonlinearities and the Lurie-Postnikov terms from being non-negative.