Estimate of a Smooth Approximation to the Production Function for Integrating Hamiltonian Systems

Abstract

In many applied control problems in economics, ecology, demography, and other areas, the relationship between dependent and independent main variables is determined statistically, which does not guarantee the smoothness of the model functional dependence. Particularly, in economic growth models, the production function describing the dependence of the output on the production factors is commonly supposed to be everywhere smooth; however, because of this constraint, qualitative parameters affecting the output cannot be included in the model. We propose an approach overcoming the requirement for the production function to be everywhere differentiable. The method is based on a smooth approximation to the production function, which is constructed simultaneously with the integration of the Hamiltonian system. A differentiable approximation to the production function is derived by constructing an asymptotic state observer for an auxiliary system. It should be noted that the standard approach to the approximation of nonsmooth components of the model on a finite time interval may not work here, which necessitates stabilizing the Hamiltonian system on an infinite time interval. The theoretical results are supported by numerical experiments for the one-sector economic growth model.