A Fixed-Point Characterization of the Optimal Costate in Finite-Horizon Optimal Control Problems

A fixed-point characterization of the optimal costate in finite-horizon optimal control problems for nonlinear systems is presented. It is shown that the optimal initial condition of the costate variable must be a fixed-point, for any time, of the composition of the forward and backward flows of the underlying Hamiltonian dynamics. Such an abstract property is then translated into a constructive condition by relying on a sequence of repeated Lie brackets involving the Hamiltonian dynamics and evaluated at a single point in the state space. This leads to a system of algebraic equations in the unknown initial optimal costate that allows achieving a desired degree of accuracy of the approximation while always consisting of a number of equations equal to the dimension of the state of the underlying system, regardless of the achieved accuracy. A dual characterization of the optimal terminal value of the state is also discussed, together with a few computational aspects of the proposed strategy.