Geodesic
On the Value Function to Differential Games with Simple Motions and~Piecewise Linear Data
Contributions to game theory and management, Tome 2 (2009), pp. 450-460.

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Positional differential games with simple dynamics are considered under assumption that at least one of two input functions (the Hamiltonian and the cost terminal function) is piecewise linear and positively homogeneous. The structure of the value function of the differential game is investigated in the framework of the theory of minimax (or/and viscosity) solutions for Hamilton–Jacobi equations. Inequalities are provided to estimate the value function. Cases of explicit formulas for the value function are pointed out.
Keywords: positional differential games, value function, Hamilton–Jacobi equations, minimax solutions, viscosity solutions, Hopf formulas, piecewise linear functions. 
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Lyubov G. Shagalova. On the Value Function to Differential Games with Simple Motions and~Piecewise Linear Data. Contributions to game theory and management, Tome 2 (2009), pp. 450-460. http://geodesic.mathdoc.fr/item/CGTM_2009_2_a33/

[1] Bardi M., Evans L. C., “On Hopf's formulas for solutions of Hamilton–Jacobi equations”, Nonlinear Analysis, Theory, Methods, Appl., 8:11 (1984), 1373–1381 | DOI | MR | Zbl

[2] Clarke F. H., Optimization and Nonsmooth Analysis, Wiley, New York, 1983 | MR | Zbl

[3] Crandall M. G., Lions P. L., “Viscosity solutions of Hamilton–Jacobi equations”, Trans. Amer. Math. Soc., 277:1 (1983), 1–42 | DOI | MR | Zbl

[4] Demyanov V. F., Rubinov A. M., Quasidifferential Calculus, Optimization Software, New York, 1986 | MR | Zbl

[5] Hopf E., “Generalized solutions of nonlinear equations of first order”, J. Math. Mech., 14 (1965), 951–973 | MR | Zbl

[6] Krasovskii N. N., Subbotin A. I., Positional Differential Games, Nauka, M., 1974 (in Russian) | MR | Zbl

[7] Melzer M., “On the expressibility of piecewise-linear continuous functions as the difference of two piecewise-linear convex functions”, Mathematical Programming Study, 29 (1986), 118–134 | DOI | MR | Zbl

[8] Petrosjan L. A., Differential Pursuit Games, Izdat. Leningrad. Univ., Leningrad, 1977 (in Russian) | MR

[9] Pshenichnyi B. N., Sagaidak M. I., “Differential games of prescribed duration”, Cybernetics, 6:2 (1970), 72–83 | DOI | MR

[10] Shagalova L. G., “A Piecewise linear minimax solution of the Hamilton–Jacobi equation”, Proceedings of IFAC Conference on Nonsmooth and Discontinuous Problems of Control and Optimization, eds. Kirillova F. M., Batukhtin V. D., Pergamon, Elsevier Science, Oxford, 1999, 193–197 | MR

[11] Subbotin A. I., Minimax Inequalities and Hamilton–Jacobi Equations, Nauka, M., 1991 (in Russian) | MR

[12] Subbotin A. I., Generalized Solutions of First-Order PDEs. The Dynamical Optimization Perspective, Birkhauser, Boston, 1995 | MR

[13] Subbotin A. I., Shagalova L. G., “A piecewise linear solution of the Cauchy problem for the Hamilton–Jacobi equation”, Russian Acad. Sci. Dokl. Math., 46:1 (1993), 144–148 | MR

[14] Ukhobotov V. I., “Differential game with simple motion”, Soviet Mathematics, 35:8 (1991), 67–70 | MR | Zbl