A direct second-variational method for unconstrained optimal control problems

A direct second-variational algorithm is proposed for the iterative solution of optimal control problems in continuous, as well as discrete, systems. While the classical method using the Riccati transformation is shown to follow from the direct method, the latter has certain promising features in terms of structural simplicity, ease of programming, and substantial reduction in computation time. The process involves a predictor-corrector algorithm for a forward sweep in the state equations, followed by a backwards sweep in the adjoint equations. Applied to discrete systems, this method functions under conditions less stringent than in the classical scheme. Linearized quasioptimal feedback control is examined from the point of view of the direct method and a scheme for successive improvement in quasi-optimal policy is proposed for systems that suffer small perturbations from a given optimal trajectory.