AbstractIn this paper, a new test for the absolute stability of nonlinear systems with state‐dependent nonlinearities is developed. Scalar nonlinearities are assumed to lie in sectors. Using a Lur'e function as a Lyapunov function, a linear matrix inequalities (LMI) stability condition is derived. The new condition lets one go from a pure integral (Persidskii) to a pure quadratic Lyapunov function in an unified framework. Several results available in the literature are generated as particular cases of the new test. An example shows that the proposed condition can be much less conservative than available diagonal stability and passivity based methods, as the circle and Popov criteria. Tests for infinite as well as finite nonlinearity sectors can be easily generated, since the parameters of the nonlinearity sectors appear in the LMI condition in a very convenient way. This feature can also provide optimization of the absolute stability sector through convex programming techniques. Copyright © 2002 John Wiley & Sons, Ltd.