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.

中文翻译：

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对集成哈密顿系统的生产函数的平滑逼近估计

摘要

在经济学、生态学、人口学等领域的许多应用控制问题中，主要因变量和自变量的关系是统计确定的，不能保证模型函数依赖的平滑性。特别是在经济增长模型中，描述产出对生产要素依赖性的生产函数通常被认为是处处光滑的；然而，由于这个限制，影响输出的定性参数不能包含在模型中。我们提出了一种方法来克服生产函数处处可微的要求。该方法基于对生产函数的平滑逼近，该函数与哈密顿系统的积分同时构建。通过为辅助系统构建渐近状态观测器，推导出生产函数的可微近似。应该注意的是，在有限时间间隔上逼近模型的非光滑分量的标准方法在这里可能不起作用，这需要在无限时间间隔上稳定哈密顿系统。理论结果得到了单一部门经济增长模型的数值实验的支持。