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 ${L}^{p}$ 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.

Keywords: 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://archive.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://archive.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://archive.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, Volume 15 (2009) no. 1, pp. 139-148. doi : 10.1051/cocv:2008020. http://archive.numdam.org/articles/10.1051/cocv:2008020/

[1] Least-squares finite element methods. Proc. ICM2006 III (2006) 1137-1162. | MR | Zbl

and ,[2] Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955). | MR | Zbl

and ,[3] Direct Methods in the Calculus of Variations. Springer (1989). | MR | Zbl

,[4] Controlled Markov Processes and Viscosity Solutions. Springer, New York (2006). | MR | Zbl

and ,[5] Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. John Wiley and Sons Ltd. (1991). | MR | Zbl

,[6] Introduction to the Theory of Differential Inclusions, Graduate Studies in Mathematics 41. American Mathematical Society (2002). | MR | Zbl

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