Pointwise minimization of supplemented variational problems

Peter Kosmol, and Dieter Müller-Wichards. "Pointwise minimization of supplemented variational problems." Colloquium Mathematicae 101.1 (2004): 25-49. <http://eudml.org/doc/283867>.

@article{PeterKosmol2004,
abstract = {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 ẋ, as opposed to Pontryagin's maximum principle, where optimization is done only with respect to the ẋ-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.},
author = {Peter Kosmol, Dieter Müller-Wichards},
journal = {Colloquium Mathematicae},
keywords = {variational problems; convexification; pointwise minimization; necessary and sufficient conditions; weak and strong local minima; fundamental theorems; extremals and extremaloids; equivalent problems; linear and quadratic supplements},
language = {eng},
number = {1},
pages = {25-49},
title = {Pointwise minimization of supplemented variational problems},
url = {http://eudml.org/doc/283867},
volume = {101},
year = {2004},
}

TY - JOUR
AU - Peter Kosmol
AU - Dieter Müller-Wichards
TI - Pointwise minimization of supplemented variational problems
JO - Colloquium Mathematicae
PY - 2004
VL - 101
IS - 1
SP - 25
EP - 49
AB - 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 ẋ, as opposed to Pontryagin's maximum principle, where optimization is done only with respect to the ẋ-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.
LA - eng
KW - variational problems; convexification; pointwise minimization; necessary and sufficient conditions; weak and strong local minima; fundamental theorems; extremals and extremaloids; equivalent problems; linear and quadratic supplements
UR - http://eudml.org/doc/283867
ER -