An alternative approach for the analysis and the numerical approximation of ODEs, using a variational framework, is presented. It is based on the natural and elementary idea of minimizing the residual of the differential equation measured in a usual norm. Typical existence results for Cauchy problems can thus be recovered, and finer sets of assumptions for existence are made explicit. We treat, in particular, the cases of an explicit ODE and a differential inclusion. This approach also allows for a whole strategy to approximate numerically the solution. It is briefly indicated here as it will be pursued systematically and in a much more broad fashion in a subsequent paper.
Mots-clés : variational methods, convexity, coercivity, value function
@article{COCV_2009__15_1_139_0,
author = {Amat, Sergio and Pedregal, Pablo},
title = {A variational approach to implicit {ODEs} and differential inclusions},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {139--148},
publisher = {EDP-Sciences},
volume = {15},
number = {1},
year = {2009},
doi = {10.1051/cocv:2008020},
mrnumber = {2488572},
zbl = {1172.34002},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2008020/}
}
TY - JOUR AU - Amat, Sergio AU - Pedregal, Pablo TI - A variational approach to implicit ODEs and differential inclusions JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2009 SP - 139 EP - 148 VL - 15 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2008020/ DO - 10.1051/cocv:2008020 LA - en ID - COCV_2009__15_1_139_0 ER -
%0 Journal Article %A Amat, Sergio %A Pedregal, Pablo %T A variational approach to implicit ODEs and differential inclusions %J ESAIM: Control, Optimisation and Calculus of Variations %D 2009 %P 139-148 %V 15 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2008020/ %R 10.1051/cocv:2008020 %G en %F COCV_2009__15_1_139_0
Amat, Sergio; Pedregal, Pablo. A variational approach to implicit ODEs and differential inclusions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 139-148. doi : 10.1051/cocv:2008020. http://www.numdam.org/articles/10.1051/cocv:2008020/
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