TY - JOUR
T1 - Feedback Nash Equilibrium for Randomly Switching Differential-Algebraic Games
AU - Tanwanimember, Aneel
AU - Zhu, Quanyan Zhu
N1 - Funding Information:
Manuscript received May 10, 2019; accepted September 9, 2019. Date of publication September 25, 2019; date of current version July 28, 2020. This work was supported in part by the ANR project CONVAN with grant number ANR-17-CE40-0019-01, in part by NSF grants ECCS-1847056, CNS-1544782, and SES-1541164, and in part by ARO grant W911NF1910041. Recommended by Associate Editor R. P. Malhame. (Corresponding author: Aneel Tanwani.) A. Tanwani is with the LAAS–CNRS, University of Toulouse, 31400 Toulouse, France (e-mail: aneel.tanwani@laas.fr).
Publisher Copyright:
© 1963-2012 IEEE.
PY - 2020/8
Y1 - 2020/8
N2 - As a subclass of stochastic differential games with algebraic constraints, this article studies dynamic noncooperative games where the dynamics are described by Markov jump differential-algebraic equations (DAEs). Theoretical tools, which require computing the extended generator and deriving Hamilton-Jacobi-Bellman equation for Markov jump DAEs, are developed. These fundamental results lead to pure feedback optimal strategies to compute the Nash equilibrium in noncooperative setting. In case of quadratic cost and linear dynamics, these strategies are obtained by solving coupled Riccati-like differential equations. Under an appropriate stabilizability assumption on system dynamics, these differential equations reduce to coupled algebraic Riccati equations when the cost functionals are considered over infinite horizon. As a first case-study, the application of our results is studied in the context of an economic system where different suppliers aim to maximize their profits subject to the market demands and fluctuations in operating conditions. The second case-study refers to the conventional problem of robust control for randomly switching linear DAEs, which can be formulated as a two-player zero sum game and is solved using the results developed in this article.
AB - As a subclass of stochastic differential games with algebraic constraints, this article studies dynamic noncooperative games where the dynamics are described by Markov jump differential-algebraic equations (DAEs). Theoretical tools, which require computing the extended generator and deriving Hamilton-Jacobi-Bellman equation for Markov jump DAEs, are developed. These fundamental results lead to pure feedback optimal strategies to compute the Nash equilibrium in noncooperative setting. In case of quadratic cost and linear dynamics, these strategies are obtained by solving coupled Riccati-like differential equations. Under an appropriate stabilizability assumption on system dynamics, these differential equations reduce to coupled algebraic Riccati equations when the cost functionals are considered over infinite horizon. As a first case-study, the application of our results is studied in the context of an economic system where different suppliers aim to maximize their profits subject to the market demands and fluctuations in operating conditions. The second case-study refers to the conventional problem of robust control for randomly switching linear DAEs, which can be formulated as a two-player zero sum game and is solved using the results developed in this article.
KW - Coupled Riccati equations
KW - Leontief input-output (IO) models
KW - differential-algebraic dynamical systems
KW - generalized Hamilton-Jacobi-Bellman (HJB) equation
KW - infinitesimal generators
KW - minimax robust control
KW - noncooperative games
KW - piecewise deterministic Markov processes
UR - http://www.scopus.com/inward/record.url?scp=85088897934&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85088897934&partnerID=8YFLogxK
U2 - 10.1109/TAC.2019.2943577
DO - 10.1109/TAC.2019.2943577
M3 - Article
AN - SCOPUS:85088897934
VL - 65
SP - 3286
EP - 3301
JO - IRE Transactions on Automatic Control
JF - IRE Transactions on Automatic Control
SN - 0018-9286
IS - 8
M1 - 8848414
ER -