Approximate solution of hyper-sensitive optimal control problems using finite-time Lyapunov analysis

Solving optimal control problems by an indirect method is often abandoned in favor of a direct method due to hyper-sensitivity with respect to unknown boundary conditions for the Hamiltonian boundary-value problem that represents the first-order necessary conditions. Yet the hyper-sensitivity may imply a manifold structure for the flow in the Hamiltonian phase space, structure that provides insight regarding the optimal solutions and suggests a solution approximation strategy that avoids the hyper-sensitivity. This paper concerns the development of a solution approximation method based on finite-time Lyapunov exponents and vectors. The focus is on determining the unknown boundary conditions such that the solution end points lie on certain invariant manifolds. Using kinematic eigenvalues, a systematic approach to determine the appropriate finite-time for the Lyapunov analysis is presented. A simple example is used to illustrate the approximation method and its implementation.