The circle criterion and Tsypkin's criterion for systems with several nonlinearities without the use of the $S$-procedure

The circle criterion (for continuous-time systems) and Tsypkin's criterion (for discrete-time systems) of absolute stability for Lurie systems with several nonlinearities are obtained with the use of the convolution theorem and without use of the $S$-procedure. On the basis of the convolution theorem, two theorems are proved which lead to a substantial reduction in the dimension of connected systems of linear matrix inequalities. Bibliography: 19 titles.