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We propose a Model Predictive Control (MPC) with a single-step predictionhorizon to solve infinite horizon optimal control problems with the expectedsum of convex stage costs for constrained linear uncertain systems. Theproposed method relies on two techniques. First, we estimate the expectedvalues of the convex costs using a computationally tractable approximation,achieved by sampling across the space of disturbances. Second, we implement adata-driven approach to approximate the optimal value function and itscorresponding domain, through systematic exploration of the system's statespace. These estimates are subsequently used as the terminal cost and terminalset within the proposed MPC. We prove recursive feasibility, robust constraintsatisfaction, and convergence in probability to the target set. Furthermore, weprove that the estimated value function converges to the optimal value functionin a local region. The effectiveness of the proposed MPC is illustrated withdetailed numerical simulations and comparisons with a value iteration methodand a Learning MPC that minimizes a certainty equivalent cost.

Approximate solution of stochastic infinite horizon optimal control problems for constrained linear uncertain systems