Pointwise minimization of supplemented variational problems

Tom 101 / 2004

Peter Kosmol, Dieter Müller-Wichards Colloquium Mathematicum 101 (2004), 25-49 MSC: 49K15, 49K99. DOI: 10.4064/cm101-1-3


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.


  • Peter KosmolMathematisches Seminar
    Universität Kiel
    Ludewig-Meyn-Str. 4
    24098 Kiel, Germany
  • Dieter Müller-WichardsFachbereich Elektrotechnik/Informatik
    Hochschule f. Angewandte Wissenschaften
    Berliner Tor 7
    20099 Hamburg, Germany

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