IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v205y2025i2d10.1007_s10957-025-02657-w.html
   My bibliography  Save this article

Linear–State Control Problems and Differential Games: Deterministic and Stochastic Systems

Author

Listed:
  • José Márquez–Prado

    (CINVESTAV–IPN)

  • Onésimo Hernández–Lerma

    (CINVESTAV–IPN)

Abstract

This paper concerns a class of linear-state optimal control problems and noncooperative differential games. Deterministic and stochastic systems are considered, as well as finite- and infinite-horizon problems. We give conditions under which these systems have degenerate feedback optimal controls so that the optimal control actions $$a(t,x) \equiv a(t)$$ a ( t , x ) ≡ a ( t ) are independent of the state variable x. As a consequence, open-loop and feedback (or Markov) optimal controls coincide, the value (or optimal objective) function is linear in the state x, and the certainty equivalence principle is satisfied.

Suggested Citation

  • José Márquez–Prado & Onésimo Hernández–Lerma, 2025. "Linear–State Control Problems and Differential Games: Deterministic and Stochastic Systems," Journal of Optimization Theory and Applications, Springer, vol. 205(2), pages 1-37, May.
  • Handle: RePEc:spr:joptap:v:205:y:2025:i:2:d:10.1007_s10957-025-02657-w
    DOI: 10.1007/s10957-025-02657-w
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-025-02657-w
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10957-025-02657-w?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Alejandra Fonseca-Morales & Onésimo Hernández-Lerma, 2018. "Potential Differential Games," Dynamic Games and Applications, Springer, vol. 8(2), pages 254-279, June.
    2. S. Jørgensen & G. Martín-Herrán & G. Zaccour, 2003. "Agreeability and Time Consistency in Linear-State Differential Games," Journal of Optimization Theory and Applications, Springer, vol. 119(1), pages 49-63, October.
    3. Ngo Long, 2011. "Dynamic Games in the Economics of Natural Resources: A Survey," Dynamic Games and Applications, Springer, vol. 1(1), pages 115-148, March.
    4. Matthew J. Sobel, 1990. "Myopic Solutions of Affine Dynamic Models," Operations Research, INFORMS, vol. 38(5), pages 847-853, October.
    5. Nico Tauchnitz, 2015. "The Pontryagin Maximum Principle for Nonlinear Optimal Control Problems with Infinite Horizon," Journal of Optimization Theory and Applications, Springer, vol. 167(1), pages 27-48, October.
    6. E. Bacchiega & L. Lambertini & A. Palestini, 2010. "On the Time Consistency of Equilibria in a Class of Additively separable Differential Games," Journal of Optimization Theory and Applications, Springer, vol. 145(3), pages 415-427, June.
    7. Peng, Shige & Shi, Yufeng, 2000. "Infinite horizon forward-backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 85(1), pages 75-92, January.
    8. Dockner,Engelbert J. & Jorgensen,Steffen & Long,Ngo Van & Sorger,Gerhard, 2000. "Differential Games in Economics and Management Science," Cambridge Books, Cambridge University Press, number 9780521637329, September.
    9. Ricardo Josa-Fombellida & Juan Pablo Rincón-Zapatero, 2018. "Stochastic Differential Games for Which the Open-Loop Equilibrium is Subgame Perfect," Dynamic Games and Applications, Springer, vol. 8(2), pages 379-400, June.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Caputo, Michael R. & Ling, Chen, 2013. "The intrinsic comparative dynamics of locally differentiable feedback Nash equilibria of autonomous and exponentially discounted infinite horizon differential games," Journal of Economic Dynamics and Control, Elsevier, vol. 37(10), pages 1982-1994.
    2. Alejandra Fonseca-Morales & Onésimo Hernández-Lerma, 2018. "Potential Differential Games," Dynamic Games and Applications, Springer, vol. 8(2), pages 254-279, June.
    3. Ekaterina Gromova & Anastasiya Malakhova & Arsen Palestini, 2018. "Payoff Distribution in a Multi-Company Extraction Game with Uncertain Duration," Mathematics, MDPI, vol. 6(9), pages 1-17, September.
    4. Açıkgöz, Ömer T. & Benchekroun, Hassan, 2017. "Anticipated international environmental agreements," European Economic Review, Elsevier, vol. 92(C), pages 306-336.
    5. Hassan Benchekroun & Farnaz Taherkhani, 2014. "Adaptation and the Allocation of Pollution Reduction Costs," Dynamic Games and Applications, Springer, vol. 4(1), pages 32-57, March.
    6. N. Baris Vardar & Georges Zaccour, 2020. "Exploitation of a Productive Asset in the Presence of Strategic Behavior and Pollution Externalities," Mathematics, MDPI, vol. 8(10), pages 1-28, October.
    7. Jaakkola, Niko, 2019. "Carbon taxation, OPEC and the end of oil," Journal of Environmental Economics and Management, Elsevier, vol. 94(C), pages 101-117.
    8. José Daniel López-Barrientos & Ekaterina Viktorovna Gromova & Ekaterina Sergeevna Miroshnichenko, 2020. "Resource Exploitation in a Stochastic Horizon under Two Parametric Interpretations," Mathematics, MDPI, vol. 8(7), pages 1-29, July.
    9. Ricardo Josa-Fombellida & Juan Rincón-Zapatero, 2015. "Euler–Lagrange equations of stochastic differential games: application to a game of a productive asset," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 59(1), pages 61-108, May.
    10. Massol, Olivier & Rifaat, Omer, 2018. "Phasing out the U.S. Federal Helium Reserve: Policy insights from a world helium model," Resource and Energy Economics, Elsevier, vol. 54(C), pages 186-211.
    11. Fabbri, Giorgio & Faggian, Silvia & Freni, Giuseppe, 2024. "On competition for spatially distributed resources in networks," Theoretical Economics, Econometric Society, vol. 19(2), May.
    12. Jiequn Han & Ruimeng Hu & Jihao Long, 2020. "Convergence of Deep Fictitious Play for Stochastic Differential Games," Papers 2008.05519, arXiv.org, revised Mar 2021.
    13. Michael Caputo & Chen Ling, 2015. "Intrinsic Comparative Dynamics of Locally Differentiable Feedback Stackelberg Equilibria," Dynamic Games and Applications, Springer, vol. 5(1), pages 1-25, March.
    14. Emre M. Demirezen & Subodha Kumar & Bala Shetty, 2020. "Two Is Better Than One: A Dynamic Analysis of Value Co‐Creation," Production and Operations Management, Production and Operations Management Society, vol. 29(9), pages 2057-2076, September.
    15. Ricardo Josa-Fombellida & Juan Pablo Rincón-Zapatero, 2018. "Stochastic Differential Games for Which the Open-Loop Equilibrium is Subgame Perfect," Dynamic Games and Applications, Springer, vol. 8(2), pages 379-400, June.
    16. Engwerda, J.C., 2012. "Prospects of Tools from Differential Games in the Study Of Macroeconomics of Climate Change," Other publications TiSEM cac36d07-227b-4cf2-83cb-7, Tilburg University, School of Economics and Management.
    17. Christos Koulovatianos, 2024. "Rational Noncooperative Strategic Exploitation of Species in a Predator–Prey Ecosystem with Random Disturbances," Dynamic Games and Applications, Springer, vol. 14(1), pages 57-77, March.
    18. Sébastien Rouillon, 2014. "Do Social Status Seeking Behaviors Worsen the Tragedy of the Commons?," Dynamic Games and Applications, Springer, vol. 4(1), pages 73-94, March.
    19. Giorgio Fabbri & Silvia Faggian & Giuseppe Freni, 2022. "On competition for spatially distributed resources in networks: an extended version," Working Papers 2022:03, Department of Economics, University of Venice "Ca' Foscari".
    20. Colombo, Luca & Labrecciosa, Paola, 2018. "Consumer surplus-enhancing cooperation in a natural resource oligopoly," Journal of Environmental Economics and Management, Elsevier, vol. 92(C), pages 185-193.

    More about this item

    Keywords

    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:205:y:2025:i:2:d:10.1007_s10957-025-02657-w. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.