The circle criterion and Tsypkin's criterion for systems with several nonlinearities without the use of the \(S\)-procedure (Q6569581)

This paper presents results related to a theorem of Pyatnitskii. The author discusses two formulations of the theorem and, separately, an important particular case.\N\NThen, using this theorem, the author presents new proofs of two classical results: the circle criterion and Tsypkin's criterion for systems with several nonlinearities. Secondly, more efficient conditions for the existence of quadratic Lyapunov functions are obtained for a wide class of Lurie systems and switched systems.\N\NThe results can also be used to reduce the dimension of arbitrary systems of linear matrix inequalities.