Some applications of BV functions in optimal controls and calculus of variations

¹ Departamento de Matemática Aplicada y Ciencias de la Computación E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria Av. Los Castros s/n, 39071 Santander, Spain
² Institut für Mathematik, Universität Graz Heinrichstrasse 36, A-8010 Graz, Austria
³ Departamento de Matemáticas, Estadística y Computación Facultad de Ciencias, Universidad de Cantabria Av. Los Castros s/n, 39071 Santander, Spain

Abstract

In this paper we discuss the use of bounded variation functions in the study of some optimal control problems as well as in the calculus of variations. The bounded variation functions are well adapted to the study of parameter identification problems, such as the coefficients of an elliptic or parabolic operator. These functions are also convenient for the image recovery problems. These problems are well formulated in the space BV(Ω) in the sense that they have a solution under reasonable assumptions. The numerical approximation of these problems is interesting because of the non separability of the space BV(Ω). A very surprising fact is the influence of the chosen norm, among all the possible equivalent norms in BV(Ω), on the convergence of the numerical approximations. Indeed the approximation by piecewise constant functions fails if the norm is not properly chosen.