Draft:Convex compactification
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In mathematics, the convex compactification is a compactification which is simultaneously a convex subset in a locally convex space. The convex compactification can be used for Relaxation (as continuous extension) of various problems in variational calculus and optimization theory. The additional linear structure allows e.g. for developing a differential calculus and more advanced considerations e.g. in relaxation in variational calculus or optimization theory.[1] It may capture both fast oscillations and concentration effects in optimal controls or solutions of variational problems. They are known under the names of relaxed or chattering controls (or sometimes bang-bang controls) in optimal control problems.
The linear structure gives rise to various maximum principles as first-order necessary optimality conditions, known in optimal-control theory as Pontryagin's maximum principle. In variational calculus, the relaxed problems can serve for modelling of various microstructures arising in modelling Ferroics, i.e. various materials exhibiting e.g. Ferroelasticity (as Shape-memory alloys) or Ferromagnetism. The first-order optimality conditions for the relaxed problems leads Weierstrass-type maximum principle.
In Partial differential equations, relaxation leads to the concept of measure-valued solutions.
Example[edit]
- The set of Young measures[2][3] arising from bounded sets in Lebesgue spaces.
- The set of DiPerna-Majda measures [4][5] arising from bounded sets in Lebesgue spaces.
See also[edit]
References[edit]
Notes[edit]
- ^ Roubíček, T. (1991). "Convex compactifications and special extensions of optimization problems". Nonlinear Analysis, Th., Methods, Appl. Vol. 16. pp. 117–1126.
- ^ Young, L. C. (1937). "Generalized curves and the existence of an attained absolute minimum in the Calculus of Variations". Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie. Classe III. Vol. XXX, no. 7–9. pp. 211–234. JFM 63.1064.01. Zbl 0019.21901.
- ^ Ball, J.M. (1989). "A version of the fundamental theorem for young measures". PDEs and Continuum Models of Phase Transitions. Lecture Notes in Physics. Vol. 344. pp. 207–215. doi:10.1007/BFb0024945. ISBN 3-540-51617-4.
{{cite book}}:|journal=ignored (help) - ^ M.Kružík, T.Roubíček (1997). "On the measures of DiPerna and Majda. (A preprint Feb.1995 DOI: 10.13140/RG.2.2.32737.66408)". Mathematica Bohemica. Vol. 122. pp. 383–399.
- ^ J.J.Alibert, G.Bouchitté (1997). "Non-uniform integrability and generalized young measure". J. Convex Anal. Vol. 4. pp. 129–147.
Sources[edit]
- L.C. Florescu, C. Godet-Thobie (2012), Young measures and compactness in measure spaces, Berlin: W. de Gruyter, ISBN 9783110280517
- P. Pedregal (1997), Parametrized Measures and Variational Principles, Basel: Birkhäuser, ISBN 978-3-0348-9815-7
- Roubíček, T. (2020), Relaxation in Optimization Theory and Variational Calculus (2nd ed.), Berlin: W. de Gruyter, ISBN 978-3-11-014542-7
- Young, L. C. (1969), Lectures on the Calculus of Variations and Optimal Control, Philadelphia–London–Toronto: W. B. Saunders, pp. xi+331, ISBN 978-0-7216-9640-9, MR 0259704, Zbl 0177.37801
Category:convex analysis
Category:Compactification (mathematics)