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On the Optimal Control of Some Parabolic Partial Differential Equations Arising in Economics

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  • Raouf Boucekkine

    (GREQAM - Groupement de Recherche en Économie Quantitative d'Aix-Marseille - EHESS - École des hautes études en sciences sociales - AMU - Aix Marseille Université - ECM - École Centrale de Marseille - CNRS - Centre National de la Recherche Scientifique, IRES-CORE - UCL - Université Catholique de Louvain = Catholic University of Louvain)

  • Carmen Camacho

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

  • Giorgio Fabbri

    (EPEE - Centre d'Etudes des Politiques Economiques - UEVE - Université d'Évry-Val-d'Essonne)

Abstract

We review an emerging application field to parabolic partial differential equations (PDEs), that's economic growth theory. After a short presentation of concrete applications, we highlight the peculiarities of optimal control problems of parabolic PDEs with infinite time horizons. In particular, the heuristic application of the maximum principle to the latter leads to single out a serious ill-posedness problem, which is, in our view, a barrier to the use of parabolic PDEs in economic growth studies as the latter are interested in long-run asymptotic solutions, thus requiring the solution to infinite time horizon optimal control problems. Adapted dynamic programming methods are used to dig deeper into the identified ill-posedness issue.

Suggested Citation

  • Raouf Boucekkine & Carmen Camacho & Giorgio Fabbri, 2013. "On the Optimal Control of Some Parabolic Partial Differential Equations Arising in Economics," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00831042, HAL.
    • Handle: RePEc:hal:cesptp:halshs-00831042
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-00831042
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    References listed on IDEAS

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    1. Boucekkine, R. & Camacho, C. & Fabbri, G., 2013. "Spatial dynamics and convergence: The spatial AK model," Journal of Economic Theory, Elsevier, vol. 148(6), pages 2719-2736.
    2. Paulo B. Brito, 2022. "The dynamics of growth and distribution in a spatially heterogeneous world," Portuguese Economic Journal, Springer;Instituto Superior de Economia e Gestao, vol. 21(3), pages 311-350, September.
    3. Carmen Camacho, 2013. "Spatial migration," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00801109, HAL.
    4. Brock, William & Xepapadeas, Anastasios, 2008. "Diffusion-induced instability and pattern formation in infinite horizon recursive optimal control," Journal of Economic Dynamics and Control, Elsevier, vol. 32(9), pages 2745-2787, September.
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    Cited by:

    1. Fabbri, Giorgio, 2017. "International borrowing without commitment and informational lags: Choice under uncertainty," Journal of Mathematical Economics, Elsevier, vol. 68(C), pages 103-114.
    2. La Torre, Davide & Liuzzi, Danilo & Marsiglio, Simone, 2015. "Pollution diffusion and abatement activities across space and over time," Mathematical Social Sciences, Elsevier, vol. 78(C), pages 48-63.
    3. Herb Kunze & Davide La Torre & Simone Marsiglio, 2019. "A Multicriteria Macroeconomic Model with Intertemporal Equity and Spatial Spillovers," Papers 1911.08247, arXiv.org.
    4. Camacho, Carmen & Pérez-Barahona, Agustín, 2015. "Land use dynamics and the environment," Journal of Economic Dynamics and Control, Elsevier, vol. 52(C), pages 96-118.
    5. Brock, William A. & Xepapadeas, Anastasios & Yannacopoulos, Athanasios N., 2014. "Spatial externalities and agglomeration in a competitive industry," Journal of Economic Dynamics and Control, Elsevier, vol. 42(C), pages 143-174.
    6. Raouf Boucekkine & Giorgio Fabbri & Patrick A. Pintus, 2018. "Short-run pain, long-run gain: the conditional welfare gains from international financial integration," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 65(2), pages 329-360, March.
    7. Addie, Ron & Taranto, Aldo, 2024. "Economic Similarities and their Application to Inflation," EconStor Preprints 283286, ZBW - Leibniz Information Centre for Economics.
    8. de Frutos, Javier & Martín-Herrán, Guiomar, 2019. "Spatial vs. non-spatial transboundary pollution control in a class of cooperative and non-cooperative dynamic games," European Journal of Operational Research, Elsevier, vol. 276(1), pages 379-394.
    9. W.A. Brock & A. Xepapadeas & A.N. Yannacopoulos, 2014. "Optimal Control in Space and Time and the Management of Environmental Resources," Annual Review of Resource Economics, Annual Reviews, vol. 6(1), pages 33-68, October.
    10. Carmen Camacho & Agustín Pérez-Barahona, 2017. "The diffusion of economic activity across space: a new approach," Working Papers halshs-01670532, HAL.
    11. Emmanuelle Augeraud-Véron & Raouf Boucekkine & Vladimir Veliov, 2019. "Distributed Optimal Control Models in Environmental Economics: A Review," AMSE Working Papers 1902, Aix-Marseille School of Economics, France.
    12. Javier de Frutos & Guiomar Martín-Herrán, 2016. "Pollution control in a multiregional setting: a differential game with spatially distributed controls," Gecomplexity Discussion Paper Series 201601, Action IS1104 "The EU in the new complex geography of economic systems: models, tools and policy evaluation", revised Jan 2016.
    13. Carmen Camacho & Agustín Pérez-Barahona, 2017. "The diffusion of economic activity across space: a new approach," PSE Working Papers halshs-01670532, HAL.
    14. Xepapadeas, Anastasios & Yannacopoulos, Athanasios N., 2023. "Spatial growth theory: Optimality and spatial heterogeneity," Journal of Economic Dynamics and Control, Elsevier, vol. 146(C).
    15. Torre, Davide La & Liuzzi, Danilo & Marsiglio, Simone, 2021. "Transboundary pollution externalities: Think globally, act locally?," Journal of Mathematical Economics, Elsevier, vol. 96(C).
    16. Xepapadeas, A. & Yannacopoulos, A.N., 2016. "Spatial growth with exogenous saving rates," Journal of Mathematical Economics, Elsevier, vol. 67(C), pages 125-137.
    17. Ballestra, Luca Vincenzo, 2016. "The spatial AK model and the Pontryagin maximum principle," Journal of Mathematical Economics, Elsevier, vol. 67(C), pages 87-94.
    18. Brock, William A. & Xepapadeas, Anastasios & Yannacopoulos, Athanasios N., 2014. "Optimal agglomerations in dynamic economics," Journal of Mathematical Economics, Elsevier, vol. 53(C), pages 1-15.
    19. Herb Kunze & Davide Torre & Simone Marsiglio, 2022. "Sustainability and spatial spillovers in a multicriteria macroeconomic model," Annals of Operations Research, Springer, vol. 311(2), pages 1067-1084, April.
    20. Berenguer, M.I. & Gámez, D. & Kunze, H. & La Torre, D. & Ruiz Galán, M., 2024. "Solving direct and inverse problems for Fredholm-type integro-differential equations with application to pollution diffusion modeling," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 223(C), pages 394-404.
    21. Anastasios Xepapadeas & Athanasios Yannacopoulos & Andreas Ioannidis, 2014. "Spatial Growth: The Distribution of Capital across Locations when Saving Rates are Exogenous," DEOS Working Papers 1412, Athens University of Economics and Business.
    22. Raouf Boucekkine & Giorgio Fabbri & Patrick Pintus, 2012. "Short-Run Pain, Long-Run Gain: The Conditional Welfare Gains from International Financial Integration The Conditional Welfare Gains from International Financial Integration," AMSE Working Papers 1202, Aix-Marseille School of Economics, France, revised 27 Jun 2016.
    23. Vahdani, Behnam & Mohammadi, Mehrdad & Thevenin, Simon & Meyer, Patrick & Dolgui, Alexandre, 2023. "Production-sharing of critical resources with dynamic demand under pandemic situation: The COVID-19 pandemic," Omega, Elsevier, vol. 120(C).

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