# A variational approach to implicit ODEs and differential inclusions

ESAIM: Control, Optimisation and Calculus of Variations (2009)

- Volume: 15, Issue: 1, page 139-148
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topAmat, Sergio, and Pedregal, Pablo. "A variational approach to implicit ODEs and differential inclusions." ESAIM: Control, Optimisation and Calculus of Variations 15.1 (2009): 139-148. <http://eudml.org/doc/250571>.

@article{Amat2009,

abstract = {
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 Lp 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.
},

author = {Amat, Sergio, Pedregal, Pablo},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Variational methods; convexity; coercivity; value function; variational methods},

language = {eng},

month = {1},

number = {1},

pages = {139-148},

publisher = {EDP Sciences},

title = {A variational approach to implicit ODEs and differential inclusions},

url = {http://eudml.org/doc/250571},

volume = {15},

year = {2009},

}

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

DA - 2009/1//

PB - EDP Sciences

VL - 15

IS - 1

SP - 139

EP - 148

AB -
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 Lp 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.

LA - eng

KW - Variational methods; convexity; coercivity; value function; variational methods

UR - http://eudml.org/doc/250571

ER -

## References

top- P. Bochev and M. Gunzburger, Least-squares finite element methods. Proc. ICM2006III (2006) 1137–1162. Zbl1100.65098
- E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955). Zbl0064.33002
- B. Dacorogna, Direct Methods in the Calculus of Variations. Springer (1989). Zbl0703.49001
- W.H. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions. Springer, New York (2006). Zbl1105.60005
- J.D. Lambert, Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. John Wiley and Sons Ltd. (1991). Zbl0745.65049
- G.V. Smirnov, Introduction to the Theory of Differential Inclusions, Graduate Studies in Mathematics41. American Mathematical Society (2002).

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.