Existence and Uniqueness of Disturbed Open-Loop Nash Equilibria for Affine-Quadratic Differential Games
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.
KeywordsNash Equilibrium Boundary Value Problem Open Loop Differential Game Dynamic Game
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- 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.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
- 5.Engwerda, J.C.: LQ Dynamic Optimization and Differential Games. Wiley, Chichester (2005)Google Scholar
- 7.Engwerda, J.C.: Addendum: Errors in LQ Dynamic Optimization and Differential Games. http://center.uvt.nl/staff/engwerda/books/framefoutjes.pdf