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This paper presents a novel synthesis method for designing an optimal and robust guidance law for a non-throttleable upper stage of a launch vehicle, using a convex approach. In the unperturbed scenario, a combination of lossless and successive convexification techniques is employed to formulate the guidance problem as a sequence of convex problems that yields the optimal trajectory, to be used as a reference for the design of a feedback controller, with little computational effort. Then, based on the reference state and control, a stochastic optimal control problem is defined to find a closed-loop control law that rejects random in-flight disturbance. The control is parameterized as a multiplicative feedback law; thus, only the control direction is regulated, while the magnitude corresponds to the nominal one, enabling its use for solid rocket motors. The objective of the optimization is to minimize the splash-down dispersion to ensure that the spent stage falls as close as possible to the nominal point. Thanks to an original convexification strategy, the stochastic optimal control problem can be solved in polynomial time since it reduces to a semidefinite programming problem. Numerical results assess the robustness of the stochastic controller and compare its performance with a model predictive control algorithm via extensive Monte Carlo campaigns.