Necessary and Sufficient Conditions for Feedback Nash Equilibria for the Affine-Quadratic Differential Game

In this note, we consider the non-cooperative linear feedback Nash quadratic differential game with an infinite planning horizon. The performance function is assumed to be indefinite and the underlying system affine. We derive both necessary and sufficient conditions under which this game has a Nash equilibrium. As a special case, we derive existence conditions for the multi-player zero-sum game.

[1]  Jacob C. Engwerda,et al.  Solving the scalar feedback Nash algebraic Riccati equations: an eigenvector approach , 2003, IEEE Trans. Autom. Control..

[2]  T. Başar,et al.  Dynamic Noncooperative Game Theory , 1982 .

[3]  Hisham Abou-Kandil,et al.  On global existence of solutions to coupled matrix Riccati equations in closed-loop Nash games , 1996, IEEE Trans. Autom. Control..

[4]  Georges Zaccour,et al.  Differential Games in Marketing , 2003 .

[5]  Jacob Engwerda,et al.  LQ Dynamic Optimization and Differential Games , 2005 .

[6]  M. Vidyasagar,et al.  Control for Nonlinear Descriptor Systems , 2002 .

[7]  J. Caulkins,et al.  Optimal Control of Nonlinear Processes: With Applications in Drugs, Corruption, and Terror , 2008 .

[8]  A. Weeren,et al.  Asymptotic Analysis of Linear Feedback Nash Equilibria in Nonzero-Sum Linear-Quadratic Differential Games , 1999 .

[9]  Gábor Kun,et al.  Stabilizability, controllability and optimal strategies of linear and nonlinear dynamical games , 2000 .

[10]  B. Anderson,et al.  A Nash game approach to mixed H/sub 2//H/sub /spl infin// control , 1994 .

[11]  E. Dockner,et al.  Differential Games in Economics and Management Science: Basic concepts of game theory , 2000 .

[12]  S. Liberty,et al.  Linear Systems , 2010, Scientific Parallel Computing.

[13]  G. Olsder,et al.  On the linear-quadratic, closed-loop, no-memory Nash game , 1984 .

[14]  Hiroaki Mukaidani,et al.  Soft-constrained stochastic Nash games for weakly coupled large-scale systems , 2009, Autom..

[15]  Y. Ho,et al.  Nonzero-sum differential games , 1969 .

[16]  I. E. Wijayanti,et al.  The (multi-player) optimal linear quadratic feedback state regulator problem for index one descriptor systems , 2009 .

[17]  Ngo Van Long,et al.  Differential Games in Economics and Management Science: Answers and hints for exercises , 2000 .

[18]  Y. Ho.  On the Linear-Quadratic , Closed-Loop , No-Memory Nash Game , 2004 .

[19]  Jacob Engwerda,et al.  Algorithms for computing Nash equilibria in deterministic LQ games , 2006, Comput. Manag. Sci..

[20]  Dongbing Gu,et al.  A Differential Game Approach to Formation Control , 2008, IEEE Transactions on Control Systems Technology.

[21]  W. Hackbusch.  Integral Equations: Theory and Numerical Treatment , 1995 .

[22]  J. B. CruzJr.,et al.  On the existence of Nash strategies and solutions to coupled riccati equations in linear-quadratic games , 1979 .

[23]  Tamer Başar,et al.  H1-Optimal Control and Related Minimax Design Problems , 1995 .

[24]  D. Lukes.  Equilibrium Feedback Control in Linear Games with Quadratic Costs , 1971 .

[25]  Xian-Wei Zhou,et al.  Noncooperative Differential Game Based Efficiency-aware Traffic Assignment for Multipath Routing in CRAHN , 2012, Wirel. Pers. Commun..

[26]  Fan-Ren Chang,et al.  H-infinity Control for Nonlinear Descriptor Systems (Lecture Notes in Control and Information Sciences) , 2006 .

[27]  Tomasz P. Michalak,et al.  Dynamic Modeling of Monetary and Fiscal Cooperation Among Nations , 2006 .

[28]  J. Schumacher,et al.  Robust Equilibria in Indefinite Linear-Quadratic Differential Games , 2003 .

[29]  Jacob Engwerda,et al.  The (multi-player) linear quadratic state feedback control problem for index one descriptor systems , 2009, 2009 European Control Conference (ECC).

[30]  Jacob Engwerda,et al.  A Result on Output Feedback Linear Quadratic Control , 2006, Autom..

[31]  Fan-Ren Chang,et al.  H∞ control for nonlinear descriptor systems , 2006, IEEE Trans. Autom. Control..

[32]  Hiroaki Mukaidani.  Robust guaranteed cost control for uncertain stochastic systems with multiple decision makers , 2009, Autom..

[33]  J. B. Cruz,et al.  Series Nash solution of two-person, nonzero-sum, linear-quadratic differential games , 1971 .

[34]  H. Abou-Kandil,et al.  Kronecker products and coupled matrix Riccati differential systems , 1989 .