Eigenvector approximate dichotomic basis method for solving hyper-sensitive optimal control problems

The dichotomic basis method is further developed for solving completely hyper-sensitive Hamiltonian boundary value problems arising in optimal control. For this class of problems, the solution can be accurately approximated by concatenating an initial boundary-layer segment, an equilibrium segment, and a terminal boundary-layer segment. Constructing the solution in this composite manner alleviates the sensitivity. The method uses a dichotomic basis to decompose the Hamiltonian vector field into its stable and unstable components, thus allowing the missing initial conditions needed to specify the initial and terminal boundary-layer segments to be determined from partial equilibrium conditions. The dichotomic basis reveals the phase-space manifold structure in the neighbourhood of the optimal solution. The challenge is to determine a sufficiently accurate approximation to a dichotomic basis. In this paper we use an approximate dichotomic basis derived from local eigenvectors. An iterative scheme is proposed to handle the approximate nature of the basis. The method is illustrated on an example problem and its general applicability is assessed.