Matrix inequalities in the stability theory: new results based on the convolution theorem

Using Pyatnitskiy's convolution theorem, the circle criterion of absolute stability for Lurie systems with several nonlinearities is obtained without use of the S-lemma. For connected systems with switching between three linear subsystems, a new criterion for the existence of a quadratic Lyapunov function is proposed. On the basis of the convolution theorem, two theorems are proved which lead to a substantial reduction in the dimensionality of connected systems of linear matrix inequalities. Issues of improving the circle criterion for Lurie systems with two nonlinearities are also discussed.