A variational approach to implicit ODEs and differential inclusions

Sergio Amat; Pablo Pedregal

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

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

Abstract

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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.

How to cite

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Amat, 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

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

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