High-Precision Algorithms for Constructing Optimal Trajectories via Solving Hamiltonian Systems

Abstract

The paper deals with analysis of optimal control problems arising in models of economic growth. The Pontryagin maximum principle is applied for analysis of the optimal investment problem. Specifically, the research is based on existence results and necessary conditions of optimality in problems with infinite horizon. Properties of Hamiltonian systems are examined for different regimes of optimal control. The existence and uniqueness result is proved for a steady state of the Hamiltonian system. Analysis of properties of eigenvalues and eigenvectors is completed for the linearized system in a neighborhood of the steady state. Description of behavior of the nonlinear Hamiltonian system is provided on the basis of results of the qualitative theory of differential equations. This analysis allows us to outline proportions of the main economic factors and trends of optimal growth in the model. A numerical algorithm for construction of optimal trajectories of economic growth is elaborated on the basis of constructions of backward procedures and conjugation of an approximation linear dynamics with the nonlinear Hamiltonian dynamics. High order precision estimates are obtained for the proposed algorithm. These estimates establish connection between precision parameters in the phase space and precision parameters for functional indices.