When does relaxation reduce the minimum cost of an optimal control problem?

Michele Palladino (1, 2) , Richard Vinter (1, 2)

1 DEEE - Department of Electrical and Electronic Engineering [London]
2 Control and Power Research Group

Relaxation is a regularization procedure used in optimal control, involving the replacement of velocity sets by their convex hulls, to ensure the existence of a minimizer. It can be an important step in the construction of sub-optimal controls for the original, unrelaxed, optimal control problem (which may not have a minimizer), based on obtaining a minimizer for the relaxed problem and approximating it. In some cases the infimum cost of the unrelaxed problem is strictly greater than the infimum cost over relaxed state trajectories; there is a need to identify such situations because then the above procedure fails. Following on from earlier work by Warga, we explore the relation between, on the one hand, non-coincidence of the minimum cost of the optimal control and its relaxation and, on the other, abnormality of necessary conditions (in the sense that they take a degenerate form in which the cost multiplier set to zero).

Domaines: Optimisation et contrôle [math.OC]

Michele Palladino, Richard Vinter. When does relaxation reduce the minimum cost of an optimal control problem?. 52nd IEEE Control and Decision Conference (CDC), 2013, Florence, Italy. pp.526-531, ⟨10.1109/CDC.2013.6759935⟩. ⟨hal-01067548⟩