Cyclic Behavior of Optimal Trajectories in Growth Models

Abstract

Abstract: This paper estimates the accuracy of an algorithm for constructing optimal solutions of optimization resource productivity problem within the framework of economic growth modeling. The problem is investigated using the generalized Pontryagin maximum principle for infinite time interval problems. Qualitative analysis of the Hamiltonian system reveals that optimal trajectories have a cyclic behavior at a steady state neighborhood. This means that the Jacobian evaluated at a steady state has complex eigenvalues. A nonlinear stabilizer, constructed for saddle steady state, does not exist in this case. In the paper, we generalize the structure of a nonlinear stabilizer for the case of a focal steady state. Applying the stabilizer to the Hamiltonian system obtained from the Pontryagin maximum principle in the case of complex eigenvalues, one can derive a closed-loop system which has spiral-formed solutions converging to the steady state over time. Optimal trajectories in a steady state vicinity have similar behavior and infinitesimal closeness of high order to trajectories generated by the proposed nonlinear stabilizer. This observation provides the basis of the algorithm for constructing solutions of the considered class of optimal control problems. Estimation of the algorithm accuracy and performance time is provided in terms of the utility function of the optimal control problem.