An Initial Examination of Using Pseudospectral Methods for Timescale and Differential Geometric Analysis of Nonlinear Optimal Control Problems

An initial examination of the development of a framework for analyzing the time-scale and differential geometric structure of nonlinear optimal control problems is considered. The framework is synthesizedby combining a recently developeddirect collocationmethod called the Gauss pseudospectral method (GPM) with concepts from differentialgeometry. In particular, the GPM is known to provide optimal state and costate information, thus enabling the computation of accurate Hamiltonian phase space trajectories. Using the optimal Hamiltonian phase space trajectories from the GPM, it is possible to analyze the timescale and differential geometric structure by computing the finite-time Lyapunov exponents and Lyapunov vectors. The Lyapunov exponents provide information about both the stable/unstableand slow/fast behavior of the Hamiltonian system along the optimal trajectory. Furthermore, the directions in the phase space along which these different behaviors act are isolated by decomposing the tangent space of the Hamiltonian system using the finitetime Lyapunov vectors. The approach is demonstrated successfully on two examples. The main contribution of this paper is to demonstrate the effectiveness of combining the Gauss pseudospectral method with results from differential geometry to assess the structure of optimally controlled systems, but without having to solve the Hamiltonian boundary-value problem that arises from the calculus of variations.