Smooth solutions of systems of quasilinear parabolic equations

Alain Bensoussan; Jens Frehse

ESAIM: Control, Optimisation and Calculus of Variations (2002)

  • Volume: 8, page 169-193
  • ISSN: 1292-8119

Abstract

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We consider in this article diagonal parabolic systems arising in the context of stochastic differential games. We address the issue of finding smooth solutions of the system. Such a regularity result is extremely important to derive an optimal feedback proving the existence of a Nash point of a certain class of stochastic differential games. Unlike in the case of scalar equation, smoothness of solutions is not achieved in general. A special structure of the nonlinear hamiltonian seems to be the adequate one to achieve the regularity property. A key step in the theory is to prove the existence of Hölder solution.

How to cite

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Bensoussan, Alain, and Frehse, Jens. "Smooth solutions of systems of quasilinear parabolic equations." ESAIM: Control, Optimisation and Calculus of Variations 8 (2002): 169-193. <http://eudml.org/doc/245506>.

@article{Bensoussan2002,
abstract = {We consider in this article diagonal parabolic systems arising in the context of stochastic differential games. We address the issue of finding smooth solutions of the system. Such a regularity result is extremely important to derive an optimal feedback proving the existence of a Nash point of a certain class of stochastic differential games. Unlike in the case of scalar equation, smoothness of solutions is not achieved in general. A special structure of the nonlinear hamiltonian seems to be the adequate one to achieve the regularity property. A key step in the theory is to prove the existence of Hölder solution.},
author = {Bensoussan, Alain, Frehse, Jens},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {parabolic equations; quasilinear; game theory; regularity; stochastic optimal control; smallness condition; specific structure; maximum principle; Green function; hamiltonian; quadratic nonlinearity},
language = {eng},
pages = {169-193},
publisher = {EDP-Sciences},
title = {Smooth solutions of systems of quasilinear parabolic equations},
url = {http://eudml.org/doc/245506},
volume = {8},
year = {2002},
}

TY - JOUR
AU - Bensoussan, Alain
AU - Frehse, Jens
TI - Smooth solutions of systems of quasilinear parabolic equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2002
PB - EDP-Sciences
VL - 8
SP - 169
EP - 193
AB - We consider in this article diagonal parabolic systems arising in the context of stochastic differential games. We address the issue of finding smooth solutions of the system. Such a regularity result is extremely important to derive an optimal feedback proving the existence of a Nash point of a certain class of stochastic differential games. Unlike in the case of scalar equation, smoothness of solutions is not achieved in general. A special structure of the nonlinear hamiltonian seems to be the adequate one to achieve the regularity property. A key step in the theory is to prove the existence of Hölder solution.
LA - eng
KW - parabolic equations; quasilinear; game theory; regularity; stochastic optimal control; smallness condition; specific structure; maximum principle; Green function; hamiltonian; quadratic nonlinearity
UR - http://eudml.org/doc/245506
ER -

References

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  14. [14] J. Leray and J.-L. Lions, Quelques résultats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty–Browder. Bull. Soc. Math. France 93 (1965) 97-107. Zbl0132.10502
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