Data-Driven optimal control via Koopman operators and Hamilton-Jacobi-Bellman equations
This paper presents a data-driven stable manifold (DD-SM) method, which integrates Koopman operator representation learning with the geometric stable manifold approach to Hamilton-Jacobi-Bellman (HJB) equations, enabling end-to-end optimal feedback control synthesis from raw trajectory data without prior knowledge of system dynamics. We construct an augmented control system under a unified symmetric subspace decomposition (SSD) and extended dynamic mode decomposition (EDMD) framework for joint approximation of the drift field, control matrix and their spatial derivatives, and derive probabilistic finite-sample error bounds for invariant and non-invariant dictionary spaces to yield a provably accurate approximate characteristic system of HJB equation. Via Lyapunov-Perron operator and ODE perturbation analysis, we prove the data-driven stable manifold achieves monotonically decreasing semi-global error with growing training data. We further establish closed-loop exponential stability and quantify the optimality gap, both tightenable by refining model accuracy. An efficient algorithm pipeline with adaptive data generation and deep neural approximation is developed, outputting control signals within 1 millisecond. Experiments on a modified van der Pol oscillator verify the effectiveness of our method.