Recurrence of nonlinear control systems: Entropy, bit rates, and finite alphabet controllers

In this paper, we introduce the notion of recurrence entropy in the context of nonlinear control systems. A set is said to be (τ-)recurrent if every trajectory that starts in the set returns to it (within at most τ units of time). The recurrence entropy of a control system quantifies the complexity of making a set τ-recurrent measured by the average rate of growth, as time increases, of the number of control signals required to achieve this goal. Our analysis reveals that, compared to invariance, recurrence is quantitatively less complex, meaning that the recurrence entropy of a set is no larger than, and often strictly smaller than, the invariance entropy. We provide upper and lower bounds on recurrence entropy and show that they converge to the bounds on invariance entropy as τ decreases to zero. Further, our results show that recurrence entropy lower bounds the minimum data rate between the sensor and controller required for achieving recurrence. We present an algorithm according to which the sensor can send state estimates to the controller over a limited-bandwidth channel to achieve recurrence asymptotically at an exponential rate. Finally, we show that, under mild stricter conditions on the set and dynamics, the control signals that enforce the τ-recurrence of a set can be generated by a finite alphabet of control signals of durations of at most τ units of time, which allows us to store them for quick online execution.