On classical envelopes in optimal control theory

The method of characteristics is utilized to investigate the value function of an optimal control problem with a smooth control near singularities of the corresponding flow of extremals. A direct generalization of the classical concept of envelopes from the calculus of variations to the optimal control problem is formulated. As a simple illustration it is shown that the local geometric properties of the flow of extremals near a fold singularity are identical with those of the field of catenaries in the classical example of minimum surfaces. The geometry of trajectories near a simple cusp singularity is described as well.