Hamiltonian Trajectories in a Heterogeneous Economic Growth Model for Optimization Resource Productivity*

In the paper, a dynamic mechanism for optimization of resource productivity is analyzed within the economic growth model framework. The research deals with the problem of shortage of natural resources, security of supply of energy and materials, and environmental impact of uncontrollable resource consumption. The model is constructed within the framework of the mathematical theory of optimal control for problems with infinite horizon. The main element of the model is phase constraints on the resource consumption. The natural value of resources is introduced in the model construction as a penalty for the intensive resource consumption. The natural value of resources is balanced with the consumption index in the utility function. The optimal control problem is posed to optimize investment in capital, as a basic factor of production, and investment in technology for raising resource productivity. These two different types of investments generate heterogeneous character of the optimal control problem. The optimal steady trajectories satisfying phase constraints of the resource consumption are constructed within the Pontryagin maximum principle in the model. It is shown that under certain conditions for parameters the model admits trajectories with growing trends of production approaching steady states. A structure of nonlinear stabilizers is proposed for designing a feedback mechanism leading the economic system along trajectories of sustainable growth to steady states. Numerical algorithms are elaborated for constructing balanced investment levels in capital and in technology of resource productivity and reaching trajectories of sustainable growth.