Optimal trajectory construction by integration of Hamiltonian dynamics in models of economic growth under resource constraints

The paper is devoted to an optimal control problem based on a model of optimization of natural resource productivity. The analysis of the problem is conducted with the use of Pontryagin's maximum principle adjusted to problems with infinite time horizon. Properties of the Hamiltonian function are investigated. Based on methods for resolving singularities, a special change of variables is suggested, which allows to simplify essentially the solution of the problem by means of analyzing steady states and corresponding Jacobian matrices of the Hamiltonian system. An important property of the change of variables is the possibility of an adequate and meaningful economic interpretation of the new variables. The existence of steady states of the Hamiltonian dynamics in the domain of transient control regime is studied, and a stable manifold is constructed for finding boundary conditions of integration of the Hamiltonian system in backward time. On the basis of the implemented analysis, an algorithm is proposed for constructing optimal trajectories under resource constraints. The analysis of the algorithm provides estimates for its convergence time and for its error with respect to the utility functional of the control problem based on the properties of the Hamiltonian system and constraints of the model. The asymptotic behavior of optimal trajectories is studied with the use of the implemented research. The operation of the algorithm is illustrated by graphical results.