Data-driven Practical Stabilization of Nonlinear Systems via Chain Policies: Sample Complexity and Incremental Learning

Abstract: We propose a method for data-driven practical stabilization of nonlinear systems with provable guarantees, based on the concept of Nonparametric Chain Policies (NCPs). The approach employs a normalized nearest-neighbor rule to assign, at each state, a finite-duration control signal derived from stored data, after which the process repeats. Unlike recent works that model the system as linear, polynomial, or polynomial fraction, we only assume the system to be locally Lipschitz. Our analysis builds on the framework of Recurrent Lyapunov Functions (RLFs), which enable data-driven certification of practical stability using standard norm functions instead of requiring the explicit construction of a classical Lyapunov function. To extend this framework, we introduce the concept of Recurrent Control Lyapunov Functions (R-CLFs), which can certify the existence of an NCP that practically stabilizes an arbitrarily small c-neighborhood of an equilibrium point. We also provide an explicit sample complexity guarantee of O((3/rho)^d log(R/c)) number of trajectories, where R is the domain radius, d the state dimension, and rho a system-dependent constant. The proposed Chain Policies are nonparametric, thus allowing new verified data to be readily incorporated into the policy to either improve convergence rate or enlarge the certified region. Numerical experiments illustrate and validate these properties.