Pointwise minimization of supplemented variational problems

Peter Kosmol¹ ; Dieter Müller-Wichards²

¹ Mathematisches Seminar Universität Kiel Ludewig-Meyn-Str. 4 24098 Kiel, Germany
² Fachbereich Elektrotechnik/Informatik Hochschule f. Angewandte Wissenschaften Berliner Tor 7 20099 Hamburg, Germany

Colloquium Mathematicum, Tome 101 (2004) no. 1, pp. 25-49.

Résumé

We describe an approach to variational problems, where the solutions appear as pointwise (finite-dimensional) minima for fixed $t$ of the supplemented Lagrangian. The minimization is performed simultaneously with respect to the state variable $x$ and $\dot x$, as opposed to Pontryagin's maximum principle, where optimization is done only with respect to the $\dot x$-variable. We use the idea of the equivalent problems of Carathéodory employing suitable (and simple) supplements to the original minimization problem. Whereas Carathéodory considers equivalent problems by use of solutions of the Hamilton–Jacobi partial differential equations, we shall demonstrate that quadratic supplements can be constructed such that the supplemented Lagrangian is convex in the vicinity of the solution. In this way, the fundamental theorems of the calculus of variations are obtained. In particular, we avoid any employment of field theory.

Détail

DOI : 10.4064/cm101-1-3

Keywords: describe approach variational problems where solutions appear pointwise finite dimensional minima fixed supplemented lagrangian minimization performed simultaneously respect state variable dot opposed pontryagins maximum principle where optimization done only respect dot x variable idea equivalent problems carath odory employing suitable simple supplements original minimization problem whereas carath odory considers equivalent problems solutions hamilton jacobi partial differential equations shall demonstrate quadratic supplements constructed supplemented lagrangian convex vicinity solution fundamental theorems calculus variations obtained particular avoid employment field theory