A variational approach to implicit ODEs and differential inclusions
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 139-148.

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.

DOI : https://doi.org/10.1051/cocv:2008020
Classification : 34A12,  49J05,  65L05
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},
zbl = {1172.34002},
mrnumber = {2488572},
language = {en},
url = {www.numdam.org/item/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/item/COCV_2009__15_1_139_0/

[1] P. Bochev and M. Gunzburger, Least-squares finite element methods. Proc. ICM2006 III (2006) 1137-1162. | MR 2275722 | Zbl 1100.65098

[2] E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955). | MR 69338 | Zbl 0064.33002

[3] B. Dacorogna, Direct Methods in the Calculus of Variations. Springer (1989). | MR 990890 | Zbl 0703.49001

[4] W.H. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions. Springer, New York (2006). | MR 2179357 | Zbl 1105.60005

[5] J.D. Lambert, Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. John Wiley and Sons Ltd. (1991). | MR 1127425 | Zbl 0745.65049

[6] G.V. Smirnov, Introduction to the Theory of Differential Inclusions, Graduate Studies in Mathematics 41. American Mathematical Society (2002). | MR 1867542 | Zbl 0992.34001