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A variational approach to implicit ODEs and differential inclusions

Published online by Cambridge University Press:  23 January 2009

Sergio Amat
Affiliation:
Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Spain. sergio.amat@upct.es
Pablo Pedregal
Affiliation:
E.T.S. Ingenieros Industriales, Universidad de Castilla La Mancha, Campus de Ciudad Real, Spain. Pablo.Pedregal@uclm.es
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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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2008

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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).

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