Manifold-Following Approximate Solution of Completely Hypersensitive Optimal Control Problems

Abstract

The solution to a completely hypersensitive optimal control problem may shadow trajectories on the stable and unstable manifolds of an equilibrium point in the state---costate phase space. If such shadowing occurs, the solution of the Hamiltonian boundary value problem, that constitutes the first-order necessary conditions for optimality, can be approximated as a composite of an initial segment on the stable manifold leading to the equilibrium point and a final segment on the unstable manifold departing from the equilibrium point. Using a dichotomic basis, the Hamiltonian vector field can be decomposed into stable and unstable components, and the unspecified boundary conditions for the initial and terminal segments can be determined such that the initial and final conditions are, respectively, on the stable and unstable manifolds of the equilibrium point. In this paper, we propose and justify the use of finite-time Lyapunov vectors to construct an approximate dichotomic basis and develop a corresponding manifold-following solution approximation method. The method is illustrated on two examples and shown to be more accurate than a similar method that uses eigenvectors of the frozen-time linearized dynamics.