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We study the regularity properties of the value function associated with anaffine optimal control problem with quadratic cost plus a potential, for afixed final time and initial point. Without assuming any condition on singularminimizers, we prove that the value function is continuous on an open and densesubset of the interior of the attainable set. As a byproduct we obtain that itis actually smooth on a possibly smaller set, still open and dense.

On the set of points of smoothness for the value function for affine optimal control problems