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Existence and Uniqueness of Disturbed Open-Loop Nash Equilibria for Affine-Quadratic Differential Games

  • Teresa-Paula Azevedo-Perdicoúlis
  • Gerhard Jank
Chapter
Part of the Annals of the International Society of Dynamic Games book series (AISDG, volume 11)

Abstract

In this note, we investigate the solution of a disturbed quadratic open loop (OL) Nash game, whose underlying system is an affine differential equation and with a finite time horizon. We derive necessary and sufficient conditions for the existence/uniqueness of the Nash/worst-case equilibrium. The solution is obtained either via solving initial/terminal value problems (IVP/TVP, respectively) in terms of Riccati differential equations or solving an associated boundary value problem (BVP). The motivation for studying the case of the affine dynamics comes from practical applications, namely the optimization of gas networks. As an illustration, we applied the results obtained to a scalar problem and compare the numerical effectiveness between the proposed approach and an usual Scilab BVP solver.

Keywords

Nash Equilibrium Boundary Value Problem Open Loop Differential Game Dynamic Game 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Abou-Kandil, H., Freiling, G., Ionescu, V., Jank, G.: Matrix Riccati Equations in Control and Systems Theory. Birkhäuser, Basel (2003)MATHGoogle Scholar
  2. 2.
    Azevedo-Perdicoúlis, T., Jank, G.: A 2–DAE system gas network model in view to optimization as a dynamic game. Proceedings of the 5th International Workshop in Multidimensional (nD) Systems 2007, held at Universidade do Aveiro, Aveiro, Portugal (2007)Google Scholar
  3. 3.
    Azevedo Perdicoúlis, T.-P., Jank, G.: A Disturbed Nash Game Approach for Gas Network Optimization. International Journal of Tomography and Statistics, Special Issue on: Control Applications of Optimisation – optimisation methods, differential games, time delay control systems, economics and management 6, 43–49 (2007)Google Scholar
  4. 4.
    Basar, T., Olsder, G.: Dynamic Noncooperative Game Theory. Second ed. SIAM, London (1999)MATHGoogle Scholar
  5. 5.
    Engwerda, J.C.: LQ Dynamic Optimization and Differential Games. Wiley, Chichester (2005)Google Scholar
  6. 6.
    Engwerda, J.C.: Uniqueness conditions for the affine open-loop linear quadratic differential game. Automatica 44, 504–511 (2008)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Engwerda, J.C.: Addendum: Errors in LQ Dynamic Optimization and Differential Games. http://center.uvt.nl/staff/engwerda/books/framefoutjes.pdf
  8. 8.
    Jank, G., Kun, G.: Optimal Control of Disturbed Linear-Quadratic Differential Games. European Journal of Control 8(2), 152–162 (2002)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Teresa-Paula Azevedo-Perdicoúlis
    • 1
  • Gerhard Jank
    • 2
  1. 1.ISR-pólo de Coimbra & Departamento MatemáticaUTADVila RealPortugal
  2. 2.Universidade de AveiroAveiroPortugal

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