Generalized Absolute Stability Using Lyapunov Functions With Relaxed Positivity Conditions

Conditions are given for verifying stability and computing upper bounds on the induced (regional) <inline-formula> <tex-math notation="LaTeX">$\mathcal {L}_{2}$ </tex-math></inline-formula> 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.