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A numerical solution of open-loop Nash equilibrium in nonlinear differential games based on Chebyshev pseudospectral method. (English) Zbl 1335.49051

Summary: In general, the applications of differential games for solving practical problems have been limited, because all calculations had to be done analytically. In this investigation, a simple and efficient numerical method for solving nonlinear nonzero-sum differential games with finite- and infinite-time horizon is presented. In both cases, derivation of open-loop Nash equilibria solutions usually leads to solving nonlinear boundary value problems for a system of ODEs. The proposed numerical method is based on a combination of the minimum principle of Pontryagin and expanding the required approximate solutions as the elements of Chebyshev polynomials. Applying the Chebyshev pseudospectral method, two-point boundary value problems in differential games are reduced to the solution of a system of algebraic equations. Finally, several examples are given to demonstrate the accuracy and efficiency of the proposed method and a comparison is made with the results obtained by the fourth order Runge-Kutta method.

MSC:

49M30 Other numerical methods in calculus of variations (MSC2010)
49N70 Differential games and control
49K15 Optimality conditions for problems involving ordinary differential equations
91A23 Differential games (aspects of game theory)

Software:

SOCS; Matlab; bvp4c
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Full Text: DOI

References:

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