Riccati Dichotomic Basis Method for Solving Hypersensitive Optimal Control Problems

Abstract

The solution of a completely hyper-sensitive Hamiltonian boundary-value problem (HBVP) arising in optimal control can be accurately approximated by concatenating an initial boundary-layer segment, an equilibrium segment, and a terminal boundary-layer segment. The approximations in the boundary-layers are essentially similar; only the direction of time is different. Consequently, it is sufficient to focus on the initial boundary-layer. In this paper it is shown that by using a dichotomic basis, the Hamiltonian vector field can be decomposed into its stable and unstable components. Furthermore, it is shown that a dichotomic basis can be constructed from by solving a Riccati differential equation. Using this result, a successive approximation procedure arises from which from which the approximate solution in the initial boundary-layer can be computed. The successive approximation procedure is illustrated on a problem in supersonic aircraft trajectory optimization and its range of applicability is discussed.