Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-16T03:27:03.827Z Has data issue: false hasContentIssue false
  • English
  • Français

A variational approach to implicit ODEs and differential inclusions

Published online by Cambridge University Press:  23 January 2009

Sergio Amat  and
Pablo Pedregal
Sergio Amat
Affiliation:
Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Spain. [email protected]
Pablo Pedregal
Affiliation:
E.T.S. Ingenieros Industriales, Universidad de Castilla La Mancha, Campus de Ciudad Real, Spain. [email protected]
Get access

Abstract

An alternative approach for the analysis and the numericalapproximation of ODEs, using a variational framework, ispresented. 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 berecovered, 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 methodsconvexitycoercivityvalue function
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 1 , January 2009 , pp. 139 - 148
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

P. Bochev and M. Gunzburger, Least-squares finite element methods. Proc. ICM2006 III (2006) 1137–1162.
E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955).
B. Dacorogna, Direct Methods in the Calculus of Variations. Springer (1989).
W.H. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions. Springer, New York (2006).
J.D. Lambert, Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. John Wiley and Sons Ltd. (1991).
G.V. Smirnov, Introduction to the Theory of Differential Inclusions, Graduate Studies in Mathematics 41. American Mathematical Society (2002).