Smooth Solutions of systems of quasilinear parabolic equations

Alain Bensoussan; Jens Frehse

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • 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 (2010): 169-193. <http://eudml.org/doc/90644>.

@article{Bensoussan2010,
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; stochastic optimal control},
language = {eng},
month = {3},
pages = {169-193},
publisher = {EDP Sciences},
title = {Smooth Solutions of systems of quasilinear parabolic equations},
url = {http://eudml.org/doc/90644},
volume = {8},
year = {2010},
}

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
DA - 2010/3//
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; stochastic optimal control
UR - http://eudml.org/doc/90644
ER -

References

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  12. J. Frehse, Bellman Systems of Stochastic Differential Games with three Players in Optimal Control and Partial Differential Equations, edited by J.L. Menaldi, E. Rofman and A. Sulem. IOS Press (2001).  
  13. S. Hildebrandt and K.-O. Widman, Some regularity results for quasilinear elliptic systems of second order. Math. Z.142 (1975) 67-86.  
  14. J. Leray and J.-L. Lions, Quelques résultats de Viv sik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder. Bull. Soc. Math. France93 (1965) 97-107.  
  15. O.A. Ladyvzenskaja, V.A. Solonnikov and N.N. Ural'ceva, Linear and quasilinear equations of parabolic type. American Mathematical Society, Providence, R.I. (1967).  
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  17. M. Wiegner, Ein optimaler Regularitätssatz für schwache Lösungen gewisser elliptischer Systeme. Math. Z.147 (1976) 21-28. Copyright EDP Sciences, SMAI 2002  

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