Sharp Sample Complexity for Lyapunov-Based Control of Unknown Systems

Learning control from offline demonstrations requires transferring certificate guarantees to a replacement controller's trajectories. We study control-affine systems using one fixed batch of $N$ independent expert episodes, including recorded controls. The method jointly learns a certificate and a drift predictor, then synthesizes constrained feedback through a soft control-Lyapunov quadratic program, without further plant interaction. A bounded, one-sided warning loss measures expert-margin deficits and harmful directional model errors. Under explicit encounter coverage, its population value bounds the probability of any certificate-decrease violation over a prescribed deployment horizon. Soundness requires no expert stability premise; useful guarantees additionally require an in-class decrease witness and directional approximation. We give same-batch bounds for parametric and smooth reproducing-kernel classes. For a fixed library of $K$ certificates and an affine predictor with $p$ coefficients, a convex scenario formulation yields a report of order O\(\frac{p+\log(K/\delta)}{\nu_{enc} N}) at zero optimal warning slack. We quantify the probability of passing a requested risk tolerance. Separately, spatial logging coverage gives practical stability outside a ball whose radius shrinks with the loss bound. A realizable control family establishes matching sample, dimension, and coverage dependence within prescribed classes and synthesis architecture, despite nonvanishing minimax uniform drift-estimation error. Experiments illustrate directional learning and the distinct roles of source confidence and deployment coverage.