Geodesic
Piecewise-linear price function of a differential game with simple dynamics and integral-terminal price functional
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Geometric Methods in Control Theory and Mathematical Physics" dedicated to the 70th anniversary of S.L. Atanasyan, the 70th anniversary of I.S. Krasilshchik, the 70th anniversary of A.V. Samokhin, and the 80th anniversary of V.T. Fomenko. S.A. Esenin Ryazan State University, Ryazan, September 25–28, 2018. Part 1, Tome 168 (2019), pp. 114-122.

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In this paper, we consider an antagonistic differential game of two persons with dynamics described by a differential equation with simple motions and an integral-terminal board functional. In this game, there exists a price function, which is a generalized (minimax or viscous) solution of the corresponding Hamilton–Jacobi equation. For the case where the terminal function and the Hamiltonian are piecewise linear and the dimension of the phase space is equal to $2$, we propose a finite algorithm for the exact construction of the price function. The algorithm is reduced to the sequential solution of elementary problems arising in a certain order. The piecewise linear price function of a differential game is constructed by gluing piecewise linear solutions of elementary problems. Structural matrices are a convenient tool of representing such functions.
Keywords: differential game, price function, Hamilton–Jacobi equation, generalized solution, minimax solution, algorithm.
Mots-clés : simple motion 
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     title = {Piecewise-linear price function of a differential game with simple dynamics and integral-terminal price functional},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
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L. G. Shagalova. Piecewise-linear price function of a differential game with simple dynamics and integral-terminal price functional. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Geometric Methods in Control Theory and Mathematical Physics" dedicated to the 70th anniversary of S.L. Atanasyan, the 70th anniversary of I.S. Krasilshchik, the 70th anniversary of A.V. Samokhin, and the 80th anniversary of V.T. Fomenko. S.A. Esenin Ryazan State University, Ryazan, September 25–28, 2018. Part 1, Tome 168 (2019), pp. 114-122. http://geodesic.mathdoc.fr/item/INTO_2019_168_a13/

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