Constructive Finite-Rank Koopman Control via ESL Theory: Real-Time Nonlinear Control with Provable Stability Guarantees

We present a single-loop, real-time Koopman control framework where the model and controller are updated continuously from streaming data. Leveraging the ESL hierarchy, we show that Koopman-learnable systems admit streamable finite-rank surrogates and we instantiate them via Oja/PAST updates with TSQR orthonormalization. At each sample, a warm-started Kleinman/Riccati step updates the gain, and a lightweight safety projection enforces a contraction margin in a Lyapunov metric. We provide Bauer–Fike stability transfer and ISS bounds under model mismatch, yielding explicit tube radii rt+1 =√εt/(1−γ). The per-step complexity is O(d r2) with rd. This eliminates train→deploy→retrain phases and offers a constructive path from ESL guarantees to real-time nonlinear control.