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Soft-Constrained Output Feedback Guaranteed Cost Equilibria in Infinite-Horizon Uncertain Linear-Quadratic Differential Games

Author

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  • Aniruddha Roy

    (Indian Institute of Technology–Madras)

  • Puduru Viswanadha Reddy

    (Indian Institute of Technology–Madras)

Abstract

In this paper, we study infinite-horizon linear-quadratic uncertain differential games with an output feedback information structure. We assume linear time-invariant nominal dynamics influenced by deterministic external disturbances, and players’ risk preferences are expressed by a soft-constrained quadratic cost criterion over an infinite horizon. We demonstrate that the conditions available in the literature for the existence of a soft-constrained output feedback Nash equilibrium (SCONE) are too stringent to satisfy, even in low-dimensional games. To address this issue, using ideas from suboptimal control, we introduce the concept of a soft-constrained output feedback guaranteed cost equilibrium (SCOGCE). At an SCOGCE, the players’ worst-case costs are upper-bounded by a specified cost profile while maintaining an equilibrium property. We show that SCOGCE strategies form a larger class of equilibrium strategies; that is, whenever an SCONE exists, it is also an SCOGCE. We demonstrate that sufficient conditions for the existence of SCOGCE are related to the solvability of a set of coupled bi-linear matrix inequalities. Using semi-definite programming relaxations, we provide linear matrix inequality-based iterative algorithms for the synthesis of SCOGCE strategies. Finally, we illustrate the performance of SCOGCE controllers with numerical examples.

Suggested Citation

  • Aniruddha Roy & Puduru Viswanadha Reddy, 2025. "Soft-Constrained Output Feedback Guaranteed Cost Equilibria in Infinite-Horizon Uncertain Linear-Quadratic Differential Games," Journal of Optimization Theory and Applications, Springer, vol. 204(2), pages 1-33, February.
    • Handle: RePEc:spr:joptap:v:204:y:2025:i:2:d:10.1007_s10957-024-02575-3
    DOI: 10.1007/s10957-024-02575-3
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    References listed on IDEAS

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    1. Jacob Engwerda, 2007. "Algorithms for computing Nash equilibria in deterministic LQ games," Computational Management Science, Springer, vol. 4(2), pages 113-140, April.
    2. Engwerda, J.C. & Weeren, A.J.T.M., 2006. "A Result on Output Feedback Linear Quadratic Control," Other publications TiSEM a11b5333-1364-4a3c-a637-4, Tilburg University, School of Economics and Management.
    3. van den Broek, W.A., 2001. "Uncertainty in differential games," Other publications TiSEM 195bcb68-8943-49c1-8acb-0, Tilburg University, School of Economics and Management.
    4. Tamer Başar & Quanyan Zhu, 2011. "Prices of Anarchy, Information, and Cooperation in Differential Games," Dynamic Games and Applications, Springer, vol. 1(1), pages 50-73, March.
    5. 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, January.
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