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Optimal control in infinite horizon problems: A Sobolev space approach

Author

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  • LE VAN, Cuong
  • BOUCEKKINE, Raouf
  • SAGLAM, Cagri
Abstract
In this paper, we make use of the Sobolev space W exp.1,1 (R+, R exp.n) 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 L exp.1 spaces have natural positive cones with no interior points.

Suggested Citation

  • LE VAN, Cuong & BOUCEKKINE, Raouf & SAGLAM, Cagri, 2004. "Optimal control in infinite horizon problems: A Sobolev space approach," LIDAM Discussion Papers CORE 2004089, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:2004089
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    References listed on IDEAS

    as
    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. Halkin, Hubert, 1974. "Necessary Conditions for Optimal Control Problems with Infinite Horizons," Econometrica, Econometric Society, vol. 42(2), pages 267-272, March.
    3. 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.
    4. 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.
    5. 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.
    6. Michel, Philippe, 1982. "On the Transversality Condition in Infinite Horizon Optimal Problems," Econometrica, Econometric Society, vol. 50(4), pages 975-985, July.
    7. Chichilnisky, Graciela, 1977. "Nonlinear functional analysis and optimal economic growth," MPRA Paper 7990, University Library of Munich, Germany.
    8. Paulo Monteiro, 2005. "General equilibrium in Rio," Economics Bulletin, AccessEcon, vol. 28(35), pages 1.
    9. 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.
    10. 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.
    11. 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.
    12. Kamihigashi, Takashi, 2001. "Necessity of Transversality Conditions for Infinite Horizon Problems," Econometrica, Econometric Society, vol. 69(4), pages 995-1012, July.
    13. Ngo Van Long & Koji Shimomura, 2003. "A Note on Transversality Conditions," Discussion Paper Series 144, Research Institute for Economics & Business Administration, Kobe University.
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    Cited by:

    1. 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.
    2. Goenka, Aditya & Liu, Lin & Nguyen, Manh-Hung, 2014. "Infectious diseases and economic growth," Journal of Mathematical Economics, Elsevier, vol. 50(C), pages 34-53.
    3. 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.
    4. 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.
    5. 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.
    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. Xepapadeas, Anastasios & Yannacopoulos, Athanasios N., 2023. "Spatial growth theory: Optimality and spatial heterogeneity," Journal of Economic Dynamics and Control, Elsevier, vol. 146(C).
    8. Yoichi Otsubo & Theoharry Grammatikos & Thorsten Lehnert, 2012. "Market Perceptions of US and European Policy Actions Around the Subprime Crisis," 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;
    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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