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Optimal control in infinite horizon problems: a Sobolev space approach

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  • Cuong Le Van

    (CERMSEM - CEntre de Recherche en Mathématiques, Statistique et Économie Mathématique - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, CORE - Center of Operation Research and Econometrics [Louvain] - UCL - Université Catholique de Louvain = Catholic University of Louvain)

  • Raouf Boucekkine

    (CORE - Center of Operation Research and Econometrics [Louvain] - UCL - Université Catholique de Louvain = Catholic University of Louvain)

  • Cagri Saglam

    (Bilkent University [Ankara])

Abstract

In this paper, we make use of the Sobolev space W1,1(R+, Rn) to derive at once the Pontryagin conditions for the standard optimal growth model in continuous time, including a necessary and sufficient transversality condition. An application to the Ramsey model is given. We use an order ideal argument to solve the problem inherent to the fact that L1 spaces have natural positive cones with no interior points.

Suggested Citation

  • Cuong Le Van & Raouf Boucekkine & Cagri Saglam, 2005. "Optimal control in infinite horizon problems: a Sobolev space approach," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00197546, HAL.
    • Handle: RePEc:hal:cesptp:halshs-00197546
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-00197546
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    References listed on IDEAS

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    1. Askenazy, Philippe & Le Van, Cuong, 1999. "A Model of Optimal Growth Strategy," Journal of Economic Theory, Elsevier, vol. 85(1), pages 24-51, March.
    2. Dana, R.A. & Le Van, C. & Magnien, F., 1994. "General Equilibrium in Asset Markets with or without Short-Selling," Papers 9492, Tilburg - Center for Economic Research.
    3. Le Van, Cuong, 1996. "Complete Characterization of Yannelis-Zame and Chichilnisky-Kalman-Mas-Colell Properness Conditions on Preferences for Separable Concave Functions Defined in L[superscript p subscript +] and L[supersc," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 8(1), pages 155-166, June.
    4. Chichilnisky, Graciela, 1977. "Nonlinear functional analysis and optimal economic growth," MPRA Paper 7990, University Library of Munich, Germany.
    5. Ngo Van Long & Koji Shimomura, 2003. "A Note on Transversality Conditions," Discussion Paper Series 144, Research Institute for Economics & Business Administration, Kobe University.
    6. Bonnisseau, Jean-Marc & Le Van, Cuong, 1996. "On the subdifferential of the value function in economic optimization problems," Journal of Mathematical Economics, Elsevier, vol. 25(1), pages 55-73.
    7. Cuong Le Van, 1996. "Complete characterization of Yannelis-Zame and Chichilnisky-Kalman-Mas-Colell properness conditions on preferences for separable concave functions defined in $L^{p}_{+}.$ and Lp (*)," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 8(1), pages 155-166.
    8. Michel, Philippe, 1982. "On the Transversality Condition in Infinite Horizon Optimal Problems," Econometrica, Econometric Society, vol. 50(4), pages 975-985, July.
    9. Paulo Monteiro, 2005. "General equilibrium in Rio," Economics Bulletin, AccessEcon, vol. 28(35), pages 1.
    10. Benveniste, L. M. & Scheinkman, J. A., 1982. "Duality theory for dynamic optimization models of economics: The continuous time case," Journal of Economic Theory, Elsevier, vol. 27(1), pages 1-19, June.
    11. Kamihigashi, Takashi, 2001. "Necessity of Transversality Conditions for Infinite Horizon Problems," Econometrica, Econometric Society, vol. 69(4), pages 995-1012, July.
    12. Halkin, Hubert, 1974. "Necessary Conditions for Optimal Control Problems with Infinite Horizons," Econometrica, Econometric Society, vol. 42(2), pages 267-272, March.
    13. Mas-Colell, Andreu & Zame, William R., 1991. "Equilibrium theory in infinite dimensional spaces," Handbook of Mathematical Economics, in: W. Hildenbrand & H. Sonnenschein (ed.), Handbook of Mathematical Economics, edition 1, volume 4, chapter 34, pages 1835-1898, Elsevier.
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    Cited by:

    1. Skerdilajda Zanaj & Patrice Pieretti & Benteng Zou, 2021. "On the long run sustainability of small jurisdictions," Economia e Politica Industriale: Journal of Industrial and Business Economics, Springer;Associazione Amici di Economia e Politica Industriale, vol. 48(1), pages 15-35, March.
    2. Dogan, Erol & Le Van, Cuong & Saglam, Cagri, 2011. "Optimal timing of regime switching in optimal growth models: A Sobolev space approach," Mathematical Social Sciences, Elsevier, vol. 61(2), pages 97-103, March.
    3. Xepapadeas, Anastasios & Yannacopoulos, Athanasios N., 2023. "Spatial growth theory: Optimality and spatial heterogeneity," Journal of Economic Dynamics and Control, Elsevier, vol. 146(C).
    4. Jean-Michel Grandmont, 2013. "Tribute to Cuong Le Van," International Journal of Economic Theory, The International Society for Economic Theory, vol. 9(1), pages 5-10, March.
    5. Greiner, Alfred & Bondarev, Anton, 2017. "Optimal R&D investment with learning-by-doing: Multiple steady-states and thresholds," Working papers 2017/06, Faculty of Business and Economics - University of Basel.
    6. Mahdi Ebrahimi Kahou & James Yu & Jesse Perla & Geoff Pleiss, 2024. "How Inductive Bias in Machine Learning Aligns with Optimality in Economic Dynamics," Papers 2406.01898, arXiv.org, revised Jun 2024.
    7. Goenka, Aditya & Liu, Lin & Nguyen, Manh-Hung, 2014. "Infectious diseases and economic growth," Journal of Mathematical Economics, Elsevier, vol. 50(C), pages 34-53.
    8. Patrice Pieretti & Skerdilajda Zanaj & Benteng Zou, 2012. "On the long run economic performance of small economies," DEM Discussion Paper Series 12-14, Department of Economics at the University of Luxembourg.

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    More about this item

    Keywords

    Optimal control; Sobolev spaces; transversality conditions; order ideal; condition de transversalité; l'idéal pour l'ordre; espaces de Sobolev; Contrôle optimal;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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