Hamilton–Jacobi equations and two-person zero-sum differential games with unbounded controls

Hong Qiu; Jiongmin Yong

ESAIM: Control, Optimisation and Calculus of Variations (2013)

  • Volume: 19, Issue: 2, page 404-437
  • ISSN: 1292-8119

Abstract

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A two-person zero-sum differential game with unbounded controls is considered. Under proper coercivity conditions, the upper and lower value functions are characterized as the unique viscosity solutions to the corresponding upper and lower Hamilton–Jacobi–Isaacs equations, respectively. Consequently, when the Isaacs’ condition is satisfied, the upper and lower value functions coincide, leading to the existence of the value function of the differential game. Due to the unboundedness of the controls, the corresponding upper and lower Hamiltonians grow super linearly in the gradient of the upper and lower value functions, respectively. A uniqueness theorem of viscosity solution to Hamilton–Jacobi equations involving such kind of Hamiltonian is proved, without relying on the convexity/concavity of the Hamiltonian. Also, it is shown that the assumed coercivity conditions guaranteeing the finiteness of the upper and lower value functions are sharp in some sense.

How to cite

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Qiu, Hong, and Yong, Jiongmin. "Hamilton–Jacobi equations and two-person zero-sum differential games with unbounded controls." ESAIM: Control, Optimisation and Calculus of Variations 19.2 (2013): 404-437. <http://eudml.org/doc/272807>.

@article{Qiu2013,
abstract = {A two-person zero-sum differential game with unbounded controls is considered. Under proper coercivity conditions, the upper and lower value functions are characterized as the unique viscosity solutions to the corresponding upper and lower Hamilton–Jacobi–Isaacs equations, respectively. Consequently, when the Isaacs’ condition is satisfied, the upper and lower value functions coincide, leading to the existence of the value function of the differential game. Due to the unboundedness of the controls, the corresponding upper and lower Hamiltonians grow super linearly in the gradient of the upper and lower value functions, respectively. A uniqueness theorem of viscosity solution to Hamilton–Jacobi equations involving such kind of Hamiltonian is proved, without relying on the convexity/concavity of the Hamiltonian. Also, it is shown that the assumed coercivity conditions guaranteeing the finiteness of the upper and lower value functions are sharp in some sense.},
author = {Qiu, Hong, Yong, Jiongmin},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {two-person zero-sum differential games; unbounded control; Hamilton–Jacobi equation; viscosity solution; Hamilton-Jacobi equation},
language = {eng},
number = {2},
pages = {404-437},
publisher = {EDP-Sciences},
title = {Hamilton–Jacobi equations and two-person zero-sum differential games with unbounded controls},
url = {http://eudml.org/doc/272807},
volume = {19},
year = {2013},
}

TY - JOUR
AU - Qiu, Hong
AU - Yong, Jiongmin
TI - Hamilton–Jacobi equations and two-person zero-sum differential games with unbounded controls
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2013
PB - EDP-Sciences
VL - 19
IS - 2
SP - 404
EP - 437
AB - A two-person zero-sum differential game with unbounded controls is considered. Under proper coercivity conditions, the upper and lower value functions are characterized as the unique viscosity solutions to the corresponding upper and lower Hamilton–Jacobi–Isaacs equations, respectively. Consequently, when the Isaacs’ condition is satisfied, the upper and lower value functions coincide, leading to the existence of the value function of the differential game. Due to the unboundedness of the controls, the corresponding upper and lower Hamiltonians grow super linearly in the gradient of the upper and lower value functions, respectively. A uniqueness theorem of viscosity solution to Hamilton–Jacobi equations involving such kind of Hamiltonian is proved, without relying on the convexity/concavity of the Hamiltonian. Also, it is shown that the assumed coercivity conditions guaranteeing the finiteness of the upper and lower value functions are sharp in some sense.
LA - eng
KW - two-person zero-sum differential games; unbounded control; Hamilton–Jacobi equation; viscosity solution; Hamilton-Jacobi equation
UR - http://eudml.org/doc/272807
ER -

References

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