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Optimal Control in Infinite Dimensional Spaces and Economic Modeling: State of the Art and Perspectives

Author

Listed:
  • Giorgio Fabbri
  • Silvia Faggian
  • Salvatore Federico
  • Fausto Gozzi

Abstract

This survey collects, within a unified framework, various results (primarily by the authors themselves) on the use of Deterministic Infinite-Dimensional Optimal Control Theory to address applied economic models. The main aim is to illustrate, through several examples, the typical features of such models (including state constraints, non-Lipschitz data, and non-regularizing differential operators) and the corresponding methods needed to handle them. This necessitates developing aspects of the existing Deterministic Infinite-Dimensional Optimal Control Theory (see, e.g., the book by 108) in specific and often nontrivial directions. Given the breadth of this area, we emphasize the Dynamic Programming Approach and its application to problems where explicit or quasi-explicit solutions of the associated Hamilton–Jacobi–Bellman (HJB) equations can be obtained. We also provide insights and references for cases where such explicit solutions are not available.

Suggested Citation

  • Giorgio Fabbri & Silvia Faggian & Salvatore Federico & Fausto Gozzi, 2025. "Optimal Control in Infinite Dimensional Spaces and Economic Modeling: State of the Art and Perspectives," Working Papers 2025-06, Grenoble Applied Economics Laboratory (GAEL).
    • Handle: RePEc:gbl:wpaper:2025-06
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    References listed on IDEAS

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    1. Emmanuelle Augeraud-Veron & Mauro Bambi & Fausto Gozzi, 2017. "Solving Internal Habit Formation Models Through Dynamic Programming in Infinite Dimension," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 584-611, May.
    2. Asea, Patrick K. & Zak, Paul J., 1999. "Time-to-build and cycles," Journal of Economic Dynamics and Control, Elsevier, vol. 23(8), pages 1155-1175, August.
    3. Xepapadeas, A. & Yannacopoulos, A.N., 2016. "Spatial growth with exogenous saving rates," Journal of Mathematical Economics, Elsevier, vol. 67(C), pages 125-137.
    4. Albeverio, Sergio & Mastrogiacomo, Elisa, 2022. "Large deviation principle for spatial economic growth model on networks," Journal of Mathematical Economics, Elsevier, vol. 103(C).
    5. Ballestra, Luca Vincenzo, 2016. "The spatial AK model and the Pontryagin maximum principle," Journal of Mathematical Economics, Elsevier, vol. 67(C), pages 87-94.
    6. Smith, Martin D. & Sanchirico, James N. & Wilen, James E., 2009. "The economics of spatial-dynamic processes: Applications to renewable resources," Journal of Environmental Economics and Management, Elsevier, vol. 57(1), pages 104-121, January.
    7. M. Bambi & G. Fabbri & F. Gozzi, 2012. "Optimal policy and consumption smoothing effects in the time-to-build AK model," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 50(3), pages 635-669, August.
    8. Xepapadeas, Anastasios & Yannacopoulos, Athanasios N., 2023. "Spatial growth theory: Optimality and spatial heterogeneity," Journal of Economic Dynamics and Control, Elsevier, vol. 146(C).
    9. Chunsheng Zhou, 2000. "Time-to-Build and Investment," The Review of Economics and Statistics, MIT Press, vol. 82(2), pages 273-282, May.
    10. Mauro Bambi & Cristina Girolami & Salvatore Federico & Fausto Gozzi, 2017. "Generically distributed investments on flexible projects and endogenous growth," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 63(2), pages 521-558, February.
    11. Bambi, Mauro & Gozzi, Fausto, 2020. "Internal habits formation and optimality," Journal of Mathematical Economics, Elsevier, vol. 91(C), pages 165-172.
    12. Augeraud-Veron, Emmanuelle & Bambi, Mauro, 2015. "Endogenous growth with addictive habits," Journal of Mathematical Economics, Elsevier, vol. 56(C), pages 15-25.
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    Keywords

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    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • E12 - Macroeconomics and Monetary Economics - - General Aggregative Models - - - Keynes; Keynesian; Post-Keynesian; Modern Monetary Theory
    • E22 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - Investment; Capital; Intangible Capital; Capacity

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