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On group analysis of optimal control problems in economic growth models

  • * Corresponding author: Teoman Özer

    * Corresponding author: Teoman Özer
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  • 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.

    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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