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The present article is a study of an optimal control problem having a non-differentiable, but Lipschitz, cost function. It is inspired by the minimisation of the energy consumption of a car-like vehicle or robot along a road which profile is known. This problem is stated by means of a simple model of the longitudinal dynamics and a running cost that comprises both an absolute value function and a function that accounts for the efficiency of the energy conversion process. A regularity result that excludes chattering phenomena from the set of solutions is proven. It is valid for the class of control affine systems, which includes the considered problem. Three case studies are detailed and analysed. The optimal trajectories are shown to be made of bang-bang, inactivated, singular and backward arcs.

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