A Variational Approach to Perturbation Feedback Control for Optimal Control Problems with Terminal Constraints and Free Terminal Time

Using the method of characteristics, for an optimal control problem with terminal constraints and free terminal time, we construct a family of extremals along a non-singular reference extremal and analyze when this family defines an embedding, i.e., a local field of extremals. An essential aspect of this construction is the desingularization of the transversality conditions near the terminal point. The construction leads to an extension of what are classical sufficient conditions for a strong local minimum of a reference trajectory in the engineering literature. These are expressed in terms of the existence of a solution to a matrix Riccati differential equation and the non-singularity of certain matrices associated with this solution. The respective matrices then can be used to set up a perturbation feedback control law around the reference extremal. Mathematically, these conditions represent the strengthened Jacobi condition and our constructions clarify the geometric meaning of the terms arising in the perturbation feedback control law.