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Series Nash solution of two-person, nonzero-sum, linear-quadratic differential games

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It is well-known that the Nash equilibrium solution of a two-person, nonzero-sum, linear differential game with a quadratic cost function can be expressed in terms of the solution of coupled generalized Riccati-type matrix differential equations. For high-order games, the numerical determination of the solution of the nonlinear coupled equations may be difficult or even impossible when the application dictates the use of small-memory computers. In this paper, a series solution is suggested by means of a parameter imbedding method. Instead of solving a high-order matrix-Riccati equation, a lower-order matrix-Riccati equation corresponding to a zero-sum game is solved. In addition, lower-order linear equations have to be solved. These solutions to lower-order equations are the coefficients of the series solution for the nonzero-sum game. Cost functions corresponding to truncated solutions are compared with those for exact Nash equilibrium solutions.

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This research was supported in part by the National Science Foundation under Grant No. GK-3893, in part by the Air Force under Grant No. AFOSR-68-1579B, and in part by the Joint Services Electronics Program under Contract No. DAAB-07-67-C-0199 with the Coordinated Science Laboratory, University of Illinois, Urbana, Illinois.

Communicated by Y. C. Ho

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Cruz, J.B., Chen, C.I. Series Nash solution of two-person, nonzero-sum, linear-quadratic differential games. J Optim Theory Appl 7, 240–257 (1971).

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  • Cost Function
  • Nash
  • Series Solution
  • Differential Game
  • Numerical Determination