doi: 10.3934/dcdss.2020215

On group analysis of optimal control problems in economic growth models

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

Received  January 2019 Revised  June 2019 Published  December 2019

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.

Citation: Gülden Gün Polat, Teoman Özer. On group analysis of optimal control problems in economic growth models. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020215
References:
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D. Acemoglu, Introduction to Modern Economic Growth (Levine's Bibliography), Department of Economics, UCLA, 2007. Google Scholar

[2]

S. C. Anco and G. Bluman, Integrating factors and first integrals for ordinary differential equation, European J. Appl. Math., 9 (1998), 245-259.  doi: 10.1017/S0956792598003477.  Google Scholar

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G. Bauman, Symmetry analysis of differential equations using MathLie, J. Math. Sci. (New York), 108 (2002), 1052-1069.  doi: 10.1023/A:1013548607060.  Google Scholar

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G. W. Bluman and S. C. Anco, Symmetries and Integration Methods for Differential Equations, Applied Mathematical Sciences, 154. Springer-Verlag, New York, 2002.  Google Scholar

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V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, Extended Prelle-Singer method and integrability/solvability of a class of nonlinear $n$th order ordinary differential equations, J. Nonlinear Math. Phys., 12 (2005) 184–201. doi: 10.2991/jnmp.2005.12.s1.16.  Google Scholar

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A. Cheviakov, GeM software package for computation of symmetries and conservation laws of differential equations, Comput. Phys. Comm., 176 (2007), 48-61.  doi: 10.1016/j.cpc.2006.08.001.  Google Scholar

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G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré, Bulletin des Sciences Mathématiques et Astronomiques, 2 (1878), 151-200.   Google Scholar

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F. DieleC. Marangi and S. Ragni, Exponential Lawson integration for nearly Hamiltonian systems arising in optimal control, Math. Comput. Simulation, 81 (2011), 1057-1067.  doi: 10.1016/j.matcom.2010.10.010.  Google Scholar

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B. U. Haq and I. Naeem, First integrals and analytical solutions of some dynamical systems, Nonlinear Dynamics, 95 (2019), 1747-1765.  doi: 10.1007/s11071-018-4657-4.  Google Scholar

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B. U. Haq and I. Naeem, First integrals and exact solutions of some compartmental disease models, Zeitschrift für Naturforschung A, 74 (2019). doi: 10.1515/zna-2018-0450.  Google Scholar

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K. S. Mahomed and R. J. Moitsheki, First integrals of generalized Ermakov systems via the Hamiltonian formulation, Internat. J. Modern Phys. B, 30 (2016), 12 pp. doi: 10.1142/S0217979216400191.  Google Scholar

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R. MohanasubhaM. Senthilvelan and M. Lakshmanan, On the interconnections between various analytic approaches in coupled first-order nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simul., 62 (2018), 213-228.  doi: 10.1016/j.cnsns.2018.02.021.  Google Scholar

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C. Muriel and J. L. Romero, First integrals, integrating factors and symmetries of second order differential equations, J. Phys. A, 42 (2009), 17 pp. doi: 10.1088/1751-8113/42/36/365207.  Google Scholar

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C. Muriel and J. L. Romero, $C^{\infty}$ symmetries and reduction of equations without Lie point symmetries, J. Lie Theory, 13 (2003), 167-188.   Google Scholar

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R. Naz and A. Chaudhry, Comparison of Closed-Form Solutions for the Lucas-Uzawa Model via the Partial Hamiltonian Approach and the Classical Approach, Math. Model. Anal., 22 (2017), 464-483.  doi: 10.3846/13926292.2017.1323035.  Google Scholar

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R. NazF. M. Mahomed and A. Chaudhry, A partial Lagrangian method for dynamical systems, Nonlinear Dynamics, 84 (2016), 1783-1794.  doi: 10.1007/s11071-016-2605-8.  Google Scholar

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R. Naz, The applications of the partial Hamiltonian approach to mechanics and other areas, International Journal of Non-Linear Mechanics, 86 (2016), 1-6.  doi: 10.1016/j.ijnonlinmec.2016.07.009.  Google Scholar

[25]

R. NazF. M. Mahomed and A. Chaudhry, A partial Hamiltonian approach for current value Hamiltonian systems, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 3600-3610.  doi: 10.1016/j.cnsns.2014.03.023.  Google Scholar

[26]

M. C. Nucci and G. Sanchini, Symmetries, Lagrangians and conservation laws of an Easter island population model, Symmetry, 7 (2015), 1613-1632.  doi: 10.3390/sym7031613.  Google Scholar

[27]

M. C. Nucci, Jacobi last multiplier and Lie symmetries: A novel application of an old relationship, J. Nonlinear Math. Phys., 12 (2005), 284-304.  doi: 10.2991/jnmp.2005.12.2.9.  Google Scholar

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M. K. Nucci, Seeking (and Finding) Lagrangians, Theoret. and Math. Phys., 160 (2009), 1014-1021.  doi: 10.1007/s11232-009-0092-5.  Google Scholar

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M. C. Nucci and P. G. L. Leach, An old method of Jacobi to find Lagrangians, J. Nonlinear Math. Phys., 16 (2009), 431-441.  doi: 10.1142/S1402925109000467.  Google Scholar

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P. J. Olver, Applications of Lie Groups to Differential Equations, Second edition. Graduate Texts in Mathematics, 107. Springer-Verlag, New York, 1993.  Google Scholar

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F. Ramsey, A Mathematical theory of saving, Economic Journal, 38 (1928), 543-559.  doi: 10.2307/2224098.  Google Scholar

show all references

References:
[1]

D. Acemoglu, Introduction to Modern Economic Growth (Levine's Bibliography), Department of Economics, UCLA, 2007. Google Scholar

[2]

S. C. Anco and G. Bluman, Integrating factors and first integrals for ordinary differential equation, European J. Appl. Math., 9 (1998), 245-259.  doi: 10.1017/S0956792598003477.  Google Scholar

[3] R. J. Barro and X. Sala-i-Martin, Economic Growth, Cambridge, The MIT press, 2004.   Google Scholar
[4]

G. Bauman, Symmetry analysis of differential equations using MathLie, J. Math. Sci. (New York), 108 (2002), 1052-1069.  doi: 10.1023/A:1013548607060.  Google Scholar

[5] R. Bellman, Dynamic Programming, Princeton University Press, Princeton, N. J., 1957.   Google Scholar
[6]

G. W. Bluman and S. C. Anco, Symmetries and Integration Methods for Differential Equations, Applied Mathematical Sciences, 154. Springer-Verlag, New York, 2002.  Google Scholar

[7]

V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, Extended Prelle-Singer method and integrability/solvability of a class of nonlinear $n$th order ordinary differential equations, J. Nonlinear Math. Phys., 12 (2005) 184–201. doi: 10.2991/jnmp.2005.12.s1.16.  Google Scholar

[8]

A. Cheviakov, GeM software package for computation of symmetries and conservation laws of differential equations, Comput. Phys. Comm., 176 (2007), 48-61.  doi: 10.1016/j.cpc.2006.08.001.  Google Scholar

[9]

A. C. Chiang, Elements of Dynamic Optimization, Illinois, Waveland Press Inc, 2000. Google Scholar

[10]

A. C. Chiang and K. Wainwright, Fundamental methods of Mathematical Economics, McGraw Hill 4th Edition, 2005. Google Scholar

[11]

G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré, Bulletin des Sciences Mathématiques et Astronomiques, 2 (1878), 151-200.   Google Scholar

[12]

F. DieleC. Marangi and S. Ragni, Exponential Lawson integration for nearly Hamiltonian systems arising in optimal control, Math. Comput. Simulation, 81 (2011), 1057-1067.  doi: 10.1016/j.matcom.2010.10.010.  Google Scholar

[13]

B. U. Haq and I. Naeem, First integrals and analytical solutions of some dynamical systems, Nonlinear Dynamics, 95 (2019), 1747-1765.  doi: 10.1007/s11071-018-4657-4.  Google Scholar

[14]

B. U. Haq and I. Naeem, First integrals and exact solutions of some compartmental disease models, Zeitschrift für Naturforschung A, 74 (2019). doi: 10.1515/zna-2018-0450.  Google Scholar

[15]

C. G. J. Jacobi, Sul principio dellultimo moltiplicatore, e suo come nuovo principio generale di meccanica, Giornale Arcadico di Scienze Lettere ed Arti, 99 (1844), 129-146.   Google Scholar

[16]

K. S. Mahomed and R. J. Moitsheki, First integrals of generalized Ermakov systems via the Hamiltonian formulation, Internat. J. Modern Phys. B, 30 (2016), 12 pp. doi: 10.1142/S0217979216400191.  Google Scholar

[17]

R. MohanasubhaM. Senthilvelan and M. Lakshmanan, On the interconnections between various analytic approaches in coupled first-order nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simul., 62 (2018), 213-228.  doi: 10.1016/j.cnsns.2018.02.021.  Google Scholar

[18]

C. Muriel and J. L. Romero, New methods of reduction for ordinary differential equations, IMA J. Appl. Math., 66 (2001), 111-125.  doi: 10.1093/imamat/66.2.111.  Google Scholar

[19]

C. Muriel and J. L. Romero, First integrals, integrating factors and symmetries of second order differential equations, J. Phys. A, 42 (2009), 17 pp. doi: 10.1088/1751-8113/42/36/365207.  Google Scholar

[20]

C. Muriel and J. L. Romero, $C^{\infty}$ symmetries and reduction of equations without Lie point symmetries, J. Lie Theory, 13 (2003), 167-188.   Google Scholar

[21]

R. Naz and A. Chaudhry, Comparison of Closed-Form Solutions for the Lucas-Uzawa Model via the Partial Hamiltonian Approach and the Classical Approach, Math. Model. Anal., 22 (2017), 464-483.  doi: 10.3846/13926292.2017.1323035.  Google Scholar

[22]

R. Naz and A. Chaudhry, Closed-form solutions of Lucas-Uzawa model with externalities via partial Hamiltonian approach, Comput. Appl. Math., 37 (2018), 5146-5161.  doi: 10.1007/s40314-018-0622-6.  Google Scholar

[23]

R. NazF. M. Mahomed and A. Chaudhry, A partial Lagrangian method for dynamical systems, Nonlinear Dynamics, 84 (2016), 1783-1794.  doi: 10.1007/s11071-016-2605-8.  Google Scholar

[24]

R. Naz, The applications of the partial Hamiltonian approach to mechanics and other areas, International Journal of Non-Linear Mechanics, 86 (2016), 1-6.  doi: 10.1016/j.ijnonlinmec.2016.07.009.  Google Scholar

[25]

R. NazF. M. Mahomed and A. Chaudhry, A partial Hamiltonian approach for current value Hamiltonian systems, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 3600-3610.  doi: 10.1016/j.cnsns.2014.03.023.  Google Scholar

[26]

M. C. Nucci and G. Sanchini, Symmetries, Lagrangians and conservation laws of an Easter island population model, Symmetry, 7 (2015), 1613-1632.  doi: 10.3390/sym7031613.  Google Scholar

[27]

M. C. Nucci, Jacobi last multiplier and Lie symmetries: A novel application of an old relationship, J. Nonlinear Math. Phys., 12 (2005), 284-304.  doi: 10.2991/jnmp.2005.12.2.9.  Google Scholar

[28]

M. K. Nucci, Seeking (and Finding) Lagrangians, Theoret. and Math. Phys., 160 (2009), 1014-1021.  doi: 10.1007/s11232-009-0092-5.  Google Scholar

[29]

M. C. Nucci and P. G. L. Leach, An old method of Jacobi to find Lagrangians, J. Nonlinear Math. Phys., 16 (2009), 431-441.  doi: 10.1142/S1402925109000467.  Google Scholar

[30]

P. J. Olver, Applications of Lie Groups to Differential Equations, Second edition. Graduate Texts in Mathematics, 107. Springer-Verlag, New York, 1993.  Google Scholar

[31]

F. Ramsey, A Mathematical theory of saving, Economic Journal, 38 (1928), 543-559.  doi: 10.2307/2224098.  Google Scholar

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} . $
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} . $
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} $
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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