On group analysis of optimal control problems in economic growth models

Gülden Gün Polat; Teoman Özer

doi:10.3934/dcdss.2020215

Article Contents

2020, Volume 13, Issue 10: 2853-2876. Doi: 10.3934/dcdss.2020215

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On group analysis of optimal control problems in economic growth models

Gülden Gün Polat^1, and
Teoman Özer^2, ,

1.
İstanbul Kültür University, Faculty of Science and Letters, Department of Mathematics and Computer Science, 34156 Bakırköy, İstanbul, Turkey
2.
İstanbul Technical University, Faculty of Civil Engineering, Division of Mechanics, 34469 Maslak, İstanbul, Turkey

^* Corresponding author: Teoman Özer
^* Corresponding author: Teoman Özer

Received: January 2019

Revised: June 2019

Early access: December 2019

Published: July 2020

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Abstract

The optimal control problems in economic growth theory are analyzed by considering the Pontryagin's maximum principle for both current and present value Hamiltonian functions based on the theory of Lie groups. As a result of these necessary conditions, two coupled first-order differential equations are obtained for two different economic growth models. The first integrals and the analytical solutions (closed-form solutions) of two different economic growth models are analyzed via the group theory including Lie point symmetries, Jacobi last multiplier, Prelle-Singer method, $ \lambda $-symmetry and the mathematical relations among them.

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Mathematics Subject Classification: 70G65, 65K10, 34A05.

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Table 1. Null forms, $ \lambda $-functions and $ \lambda $-symmetries of the system (82)

Null Forms	$ \lambda $-functions and $ \lambda $-symmetries
$ S_{21}=\frac{2(-\sqrt{k}+pk)}{p^2} $,	$ \lambda_{21}=\frac{1}{2pk}-\frac{1}{\sqrt{k}}-r $,
	$ V_{21}=\frac{\partial}{\partial p}-(\frac{2(-\sqrt{k}+pk)}{p^2} ) \frac{\partial}{\partial k} $,
$ S_{22}=\frac{2\sqrt{k}(-1+p(\sqrt{k}+2kr))}{p^2(1+2\sqrt{k}r)} $	$ \lambda_{22}=-\frac{1}{\sqrt{k}}+\frac{1}{2pk+4pr k^{3/2}-r} $,
	$ V_{22}=\frac{\partial}{\partial p}-(\frac{2\sqrt{k}(-1+p(\sqrt{k}+2kr))}{p^2(1+2\sqrt{k}r)} ) \frac{\partial}{\partial k} $,
$ S_{23}=-\frac{2 \sqrt{k}}{p^2} $	$ \lambda_{23}=\frac{1}{2} (\frac{1}{pk}-\frac{1}{\sqrt{k}}-2r) $,
	$ V_{23}=\frac{\partial}{\partial p}+\left(\frac{2 \sqrt{k}}{p^2} \right) \frac{\partial}{\partial k} . $

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Table 2. Null forms, $ \bar{\lambda} $-functions and $ \bar{\lambda} $-symmetries of the system (97)

Null Forms	$ \lambda $-functions and $ \lambda $-symmetries
$ \bar{S}_{21}=-\frac{2(e^{-rt} \sqrt{\bar{k}}-\bar{k}\bar{p})}{\bar{p}^2} $	$ \bar{\lambda}_{21}=\frac{e^{-rt} }{2\bar{k}\bar{p}}-\frac{1}{\sqrt{\bar{k}}}-2r $,
	$ \bar{V}_{21}=\frac{\partial}{\partial p}+(\frac{2(e^{-rt} \sqrt{\bar{k}}-\bar{k}\bar{p})}{\bar{p}^2} ) \frac{\partial}{\partial k} $
$ \bar{S}_{22}=-\frac{2(e^{-rt} \sqrt{\bar{k}}-2\bar{k}\bar{p}(1+2r \sqrt{\bar{k}})}{\bar{p}^2(1+2r \sqrt{\bar{k}})} $	$ \bar{\lambda}_{22}=-\frac{1}{\sqrt{\bar{k}}}-2r+\frac{e^{-rt} }{2\bar{k}\bar{p}+4r \bar{p} \bar{k}^{3/2} } $,
	$ \bar{V}_{22}=\frac{\partial}{\partial p}+(\frac{2(e^{-rt} \sqrt{\bar{k}}-2\bar{k}\bar{p}(1+2r \sqrt{\bar{k}})}{\bar{p}^2(1+2r \sqrt{\bar{k}})} ) \frac{\partial}{\partial k} $
$ \bar{S}_{23}= -\frac{2 e^{-rt} \sqrt{\bar{k}}}{\bar{p}^2} $	$ \bar{\lambda}_{23}=\frac{1}{2}(\frac{e^{-rt} }{\bar{k}\bar{p}}-\frac{1}{\sqrt{\bar{k}}}-4r) $,
	$ \bar{V}_{23}=\frac{\partial}{\partial p}+(\frac{2 e^{-rt} \sqrt{\bar{k}}}{\bar{p}^2} ) \frac{\partial}{\partial k} $

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