  In this letter, we discuss the problem of optimal control for affine systems
in the context of data-driven linear programming. First, we introduce a unified
framework for the fixed point characterization of the value function,
Q-function and relaxed Bellman operators. Then, in a model-free setting, we
show how to synthesize and estimate Bellman inequalities from a small but
sufficiently rich dataset. To guarantee exploration richness, we complete the
extension of Willem's fundamental lemma to affine systems.
