ESAIM: Control, Optimisation and Calculus of Variations

Research Article

A variational approach to implicit ODEs and differential inclusions

Amat, Sergioa1 and Pedregal, Pabloa2

a1 Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Spain.

a2 E.T.S. Ingenieros Industriales, Universidad de Castilla La Mancha, Campus de Ciudad Real, Spain.


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.

(Received March 14 2007)

(Online publication January 23 2009)

Key Words:

  • Variational methods;
  • convexity;
  • coercivity;
  • value function

Mathematics Subject Classification:

  • 34A12;
  • 49J05;
  • 65L05