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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, <inline-formula> <tex-math notation="LaTeX">${Q}$ </tex-math></inline-formula>-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 Willems&#x2019; fundamental lemma to affine systems.

Data-Driven Optimal Control of Affine Systems: A Linear Programming Perspective