# Viscosity solutions of the Isaacs equation οn an attainable set

Applicationes Mathematicae (1994)

- Volume: 22, Issue: 2, page 181-192
- ISSN: 1233-7234

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topZaremba, Leszek. "Viscosity solutions of the Isaacs equation οn an attainable set." Applicationes Mathematicae 22.2 (1994): 181-192. <http://eudml.org/doc/219090>.

@article{Zaremba1994,

abstract = {We apply a modification of the viscosity solution concept introduced in [8] to the Isaacs equation defined on the set attainable from a given set of initial conditions. We extend the notion of a lower strategy introduced by us in [17] to a more general setting to prove that the lower and upper values of a differential game are subsolutions (resp. supersolutions) in our sense to the upper (resp. lower) Isaacs equation of the differential game. Our basic restriction is that the variable duration time of the game is bounded above by a certain number T>0. In order to obtain our results, we prove the Bellman optimality principle of dynamic programming for differential games.},

author = {Zaremba, Leszek},

journal = {Applicationes Mathematicae},

keywords = {Isaacs equation; dynamic programming; differential game; viscosity solution; Hamilton- Jacobi-Isaacs equations},

language = {eng},

number = {2},

pages = {181-192},

title = {Viscosity solutions of the Isaacs equation οn an attainable set},

url = {http://eudml.org/doc/219090},

volume = {22},

year = {1994},

}

TY - JOUR

AU - Zaremba, Leszek

TI - Viscosity solutions of the Isaacs equation οn an attainable set

JO - Applicationes Mathematicae

PY - 1994

VL - 22

IS - 2

SP - 181

EP - 192

AB - We apply a modification of the viscosity solution concept introduced in [8] to the Isaacs equation defined on the set attainable from a given set of initial conditions. We extend the notion of a lower strategy introduced by us in [17] to a more general setting to prove that the lower and upper values of a differential game are subsolutions (resp. supersolutions) in our sense to the upper (resp. lower) Isaacs equation of the differential game. Our basic restriction is that the variable duration time of the game is bounded above by a certain number T>0. In order to obtain our results, we prove the Bellman optimality principle of dynamic programming for differential games.

LA - eng

KW - Isaacs equation; dynamic programming; differential game; viscosity solution; Hamilton- Jacobi-Isaacs equations

UR - http://eudml.org/doc/219090

ER -

## References

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- [2] M. Crandall and P. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), 1-42.
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- [8] H. Ishii, J.-L. Menaldi and L. Zaremba, Viscosity solutions of the Bellman equation on an attainable set, Problems Control Inform. Theory 20 (1991), 317-328. Zbl0757.49022
- [9] N. Krasovskiĭ and A. Subbotin, Positional Differential Games, Nauka, Moscow, 1974 (in Russian).
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- [11] P. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pitman, Boston, 1982.
- [12] O. Oleĭnik and S. Kruzhkov, Quasi-linear second order parabolic equations with several independent variables, Uspekhi Mat. Nauk 16 (5) (1961), 115-155 (in Russian).
- [13] P. Souganidis, Existence of viscosity solutions of Hamilton-Jacobi equations, J. Differential Equations 56 (1985), 345-390. Zbl0506.35020
- [14] A. Subbotin, A generalization of the fundamental equation of the theory of differential games, Dokl. Akad. Nauk SSSR 254 (1980), 293-297 (in Russian).
- [15] A. Subbotin, Existence and uniqueness results for Hamilton-Jacobi equation, Nonlinear Anal., to appear.
- [16] A. Subbotin and A. Taras'ev, Stability properties of the value function of a differential game and viscosity solutions of Hamilton-Jacobi equations, Problems Control Inform. Theory 15 (1986), 451-463. Zbl0631.90106
- [17] L. S. Zaremba, Optimality principles of dynamic programming in differential games, J. Math. Anal. Appl. 138 (1989), 43-51. Zbl0681.90101

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