Sub-Finslerian metric associated to an optimal control system

The problem of minimizing the cost functional of an optimal control system through the use of constrained variational calculus is a generalization of the geodetic problem in Riemannian geometry. In the framework of a geometric formulation of optimal control, we de ne a metric structure associated to the optimal control system on the enlarged space of state and time variables, such that the minimal length curves of the metric are the optimal solutions of the system. A twofold generalization of metric structure is applied, considering Finslerian-type metrics as well as allowed and forbidden directions (like in sub-Riemannian geometry). Free (null Hamiltonian) or fixed final parameter problems are identified with constant energy leaves, and the restriction of the metric to these leaves gives way to a family of metric structures on the usual state manifold.