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

Author

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  • 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
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    References listed on IDEAS

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    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.
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    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, Enero-Abr.
    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.
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