Uniform ultimate boundedness of Lur'e systems with switchings and delays

The paper investigates a hybrid system consisting of Lur'e subsystems with constant delays and time-dependent switching. It is assumed that nonlinearities from the right side of the systems have degrees less than unity. An analysis of such a property of the system as the uniform ultimate boundedness of all its solutions is conducted. The linear part of the system is supposed to be asymptotically stable. As is known, this means that there is a correspondent homogeneous Lyapunov function. Using this function, a common Lyapunov–Krasovskii functional is constructed which makes it possible to find sufficient conditions for the uniform ultimate boundedness with arbitrary choices of positive delays and switching laws. Moreover, delays can occur during switching, for example, when generating feedback. The derived conditions are found to be less conservative in the case of asynchronous switching compared to synchronous ones. The validity of the theoretical results is confirmed through numerical modeling.