About the necessity of Popov criterion for a special Lyapunov function existence for the systems with multiple nonlinearities

Necessary and sufficient conditions for existence of Lyapunov function from the class ‘quadratic form plus integral of nonlinearity’ (Lyapunov‐Lurie function) for systems with several nonlinearities are considered. It is assumed that the nonlinearity graphs belong to the infinite sectors, i.e., belong to the union of the first and third quadrants in the plane. It is proven that Popov criterion is necessary and sufficient for existence of Lyapunov‐Lurie function if the relative degree of the linear part is greater than one. The proof is based on the result concerning losslessness of the S‐procedure for several respective quadratic constraints.