October  2020, 13(10): 2853-2876. 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  October 2020 Early access  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 and Continuous Dynamical Systems - S, 2020, 13 (10) : 2853-2876. doi: 10.3934/dcdss.2020215
References:
[1]

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

[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.

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

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

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

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

[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.

[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.

[9]

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

[10]

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

[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. 

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[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.

[30]

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

[31]

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

show all references

References:
[1]

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

[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.

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

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

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

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

[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.

[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.

[9]

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

[10]

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

[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. 

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[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.

[30]

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

[31]

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

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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