A local field of extremals for single-input systems with state space constraints

For optimal control problems with state space constraints given by control-invariant manifolds of relative degree 1 we construct a local embedding of a boundary arc into a field of extremals providing sufficient conditions for strong local optimality of the reference trajectory. Because of the presence of the constraint this local field is made of portions of trajectories which are only partially defined since the full trajectories are not close to the reference trajectory.