On cycle slipping in infinite-dimensional control systems with periodic nonlinearities

In this paper we consider control systems with periodic nonlinearities characterized by countable sets of equilib- ria, both Lyapunov stable and unstable. The simplest ex- ample of such a system is the mathematical pendulum; therefore, these systems are often called “pendulum- like” systems. In pendulum-like systems, the very con- cept of stability differs from that in systems with a unique equilibrium. Stability is defined as the conver- gence of any solution to a certain equilibrium. For sta- ble pendulum-like systems, the problem of cycle slip- ping arises. In the case of a mathematical pendulum, the number of slipped cycles corresponds to the num- ber of rotations of the pendulum around its suspension point. In general, it represents the distance between the initial value of the input and its limit value. In this pa- per, we obtain frequency-domain estimates for the num- ber of slipped cycles in infinite-dimensional systems us- ing the Popov method of a priori integral indices. These estimates are tighter than those established in previous works. The paper presents an expanded version of the talk delivered at the International Conference on Physics and Control (PhysCon 2024).