The adjoint method, recently introduced by Evans, is used to study obstacle problems, weakly coupled systems, cell problems for weakly coupled systems of Hamilton - Jacobi equations, and weakly coupled systems of obstacle type. In particular, new results about the speed of convergence of some approximation procedures are derived.
Mots clés : adjoint methods, cell problems, Hamilton − Jacobi equations, obstacle problems, weakly coupled systems, weak KAM theory
@article{COCV_2013__19_3_754_0,
author = {Cagnetti, Filippo and Gomes, Diogo and Tran, Hung Vinh},
title = {Adjoint methods for obstacle problems and weakly coupled systems of {PDE}},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {754--779},
publisher = {EDP-Sciences},
volume = {19},
number = {3},
year = {2013},
doi = {10.1051/cocv/2012032},
mrnumber = {3092361},
zbl = {1273.35090},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2012032/}
}
TY - JOUR AU - Cagnetti, Filippo AU - Gomes, Diogo AU - Tran, Hung Vinh TI - Adjoint methods for obstacle problems and weakly coupled systems of PDE JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 754 EP - 779 VL - 19 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2012032/ DO - 10.1051/cocv/2012032 LA - en ID - COCV_2013__19_3_754_0 ER -
%0 Journal Article %A Cagnetti, Filippo %A Gomes, Diogo %A Tran, Hung Vinh %T Adjoint methods for obstacle problems and weakly coupled systems of PDE %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 754-779 %V 19 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2012032/ %R 10.1051/cocv/2012032 %G en %F COCV_2013__19_3_754_0
Cagnetti, Filippo; Gomes, Diogo; Tran, Hung Vinh. Adjoint methods for obstacle problems and weakly coupled systems of PDE. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 3, pp. 754-779. doi : 10.1051/cocv/2012032. http://www.numdam.org/articles/10.1051/cocv/2012032/
[1] and , Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26 (1988) 1133-1148. | MR | Zbl
[2] and , Optimal switching for ordinary differential equations. SIAM J. Control Optim. 22 (1984) 143-161. | MR | Zbl
[3] , and , Aubry-Mather measures in the nonconvex setting. SIAM J. Math. Anal. 43 (2011) 2601-2629. | MR | Zbl
[4] and , Comparison results for a class of weakly coupled systems of eikonal equations. Hokkaido Math. J. 37 (2008) 349-362. | MR
[5] , , and , Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Commun. Pure Appl. Anal. 8 (2009) 1291-1302. | MR | Zbl
[6] and , Viscosity solutions for weakly coupled systems of Hamilton-Jacobi equations. Proc. London Math. Soc. 63 (1991) 212-240. | MR | Zbl
[7] and , Adjoint methods for the infinity Laplacian partial differential equation. Arch. Ration. Mech. Anal. 201 (2011) 87-113. | MR | Zbl
[8] , Adjoint and compensated compactness methods for Hamilton-Jacobi PDE. Arch. Ration. Mech. Anal. 197 (2010) 1053-1088. | MR | Zbl
[9] , A stochastic analogue of Aubry-Mather theory. Nonlinearity 15 (2002) 581-603. | MR | Zbl
[10] and , Viscosity solutions for monotone systems of second-order elliptic PDEs. Commun. Partial Differ. Equ. 16 (1991) 1095-1128. | MR | Zbl
[11] and , On the rate of convergence of solutions for the singular perturbations of gradient obstacle problems. Funkcial. Ekvac. 33 (1990) 551-562. | MR | Zbl
[12] , Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics. Pitman (Advanced Publishing Program), Boston, Mass. 69 (1982). | MR | Zbl
[13] , and , Homogenization of Hamilton-Jacobi equations, Preliminary Version, (1988).
[14] , Adjoint methods for static Hamilton-Jacobi equations. Calc. Var. Partial Differ. Equ. 41 (2011) 301-319. | MR | Zbl
Cité par Sources :
