Solving Partially Hyper-Sensitive Optimal Control Problems Using Manifold Structure

Hyper-sensitivity to unknown boundary conditions plagues indirect methods of solving optimal control problems as a Hamiltonian boundary-value problem for both state and costate. Yet the hyper-sensitivity may imply manifold structure in the Hamiltonian flow, knowledge of which would yield insight regarding the optimal solutions and suggest a solution approximation strategy that circumvents the hyper-sensitivity. Finite-time Lyapunov exponents and vectors provide a means of diagnosing hypersensitivity and determining the associated manifold structure. A solution approximation approach is described that requires determining the unknown boundary conditions, such that the solution end points lie on certain invariant manifolds, and matching of forward and backward segments. The approach is applied to the optimal control of a nonlinear spring-mass-damper system. The approximate solution is shown to be accurate by comparison with a solution obtained by a collocation method.