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



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.

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  • Raouf Boucekkine & Carmen Camacho & Giorgio Fabbri, 2013. "On the Optimal Control of Some Parabolic Partial Differential Equations Arising in Economics," AMSE Working Papers 1334, Aix-Marseille School of Economics, Marseille, France, revised 05 Jun 2013.
  • Handle: RePEc:aim:wpaimx:1334

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    References listed on IDEAS

    1. Carmen Camacho, 2013. "Spatial migration," Documents de travail du Centre d'Economie de la Sorbonne 13017, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    2. 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.
    3. Paulo Brito, 2004. "The Dynamics of Growth and Distribution in a Spatially Heterogeneous World," Working Papers Department of Economics 2004/14, ISEG - Lisbon School of Economics and Management, Department of Economics, Universidade de Lisboa.
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    Cited by:

    1. Xepapadeas, A. & Yannacopoulos, A.N., 2016. "Spatial growth with exogenous saving rates," Journal of Mathematical Economics, Elsevier, vol. 67(C), pages 125-137.
    2. 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.
    3. repec:spr:joecth:v:65:y:2018:i:2:d:10.1007_s00199-016-1019-7 is not listed on IDEAS
    4. 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.
    5. 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.
    6. 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.
    7. Ballestra, Luca Vincenzo, 2016. "The spatial AK model and the Pontryagin maximum principle," Journal of Mathematical Economics, Elsevier, vol. 67(C), pages 87-94.
    8. 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.
    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. 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.
    11. 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.

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    Parabolic partial differential equations; optimal control; infinite dimensional problems; infinite time horizons; ill-posedness; dynamic programming;

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