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Lyapunov Iterations for Solving Coupled Algebraic Riccati Equations of Nash Differential Games and Algebraic Riccati Equations of Zero-Sum Games

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New Trends in Dynamic Games and Applications

Part of the book series: Annals of the International Society of Dynamic Games ((AISDG,volume 3))

Abstract

In this paper we study the symmetric coupled algebraic Riccati equations corresponding to the steady state Nash strategies. Under control-oriented assumptions, imposed on the problem matrices, the Lyapunov iterations are constructed such that the proposed algorithm converges to the nonnegative (positive) definite stabilizing solution of the coupled algebraic Riccati equations. In addition, the problem order reduction is achieved since the obtained Lyapunov equations are of the reduced-order and can be solved independently. As a matter of fact a parallel synchronous algorithm is obtained. A high-order numerical example is included in order to demonstrate the efficiency of the proposed algorithm. In the second part of this paper we have proposed an algorithm, in terms of the Lyapunov iterations, for finding the positive semidefinite stabilizing solution of the algebraic Riccati equation of the zero-sum differential games. The similar algebraic Riccati type equations appear in the H ∞ optimal control and related problems.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Michigan State University, E. Lansing, MI, 48824, USA

    T-Y. Li

  2. Department of Electrical and Computer Engineering, Rutgers University, Piscataway, NJ, 08855-0909, USA

    Z. Gajic

Authors
  1. T-Y. Li
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  2. Z. Gajic
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Editor information

Editors and Affiliations

  1. Section of Applied Analysis, Delft University of Technology, Mekelweg 4, Delft, The Netherlands

    Geert Jan Olsder

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© 1995 Birkhäuser Boston

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Li, TY., Gajic, Z. (1995). Lyapunov Iterations for Solving Coupled Algebraic Riccati Equations of Nash Differential Games and Algebraic Riccati Equations of Zero-Sum Games. In: Olsder, G.J. (eds) New Trends in Dynamic Games and Applications. Annals of the International Society of Dynamic Games, vol 3. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4274-1_17

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  • DOI: https://doi.org/10.1007/978-1-4612-4274-1_17

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4612-8719-3

  • Online ISBN: 978-1-4612-4274-1

  • eBook Packages: Springer Book Archive

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