In mathematics, specifically in convex analysis, the convex compactification is a compactification which is simultaneously a convex subset in a locally convex space in functional analysis. 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.
The notion was introduced by Roubíček in 1991.[1]
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
editReferences
editNotes
edit- ^ a b Roubíček, Tomáš (1991), "Convex compactifications and special extensions of optimization problems", Nonlinear Analysis, 16 (12): 1117–1126, doi:10.1016/0362-546X(91)90199-B, MR 1111622
- ^ 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, XXX (7–9): 211–234, JFM 63.1064.01, Zbl 0019.21901
- ^ Ball, J. M. (1989), "A version of the fundamental theorem for Young measures", in Rascle, M.; Serre, D.; Slemrod, M. (eds.), PDEs and continuum models of phase transitions: Proceedings of the N.S.F.–C.N.R.S. Joint Seminar held in Nice, January 18–22, 1988, Lecture Notes in Physics, vol. 344, Berlin: Springer, pp. 207–215, doi:10.1007/BFb0024945, ISBN 3-540-51617-4, MR 1036070
- ^ Kružík, Martin; Roubíček, Tomáš (1997), "On the measures of DiPerna and Majda", Mathematica Bohemica, 122 (4): 383–399, MR 1489400
- ^ Alibert, J. J.; Bouchitté, G. (1997), "Non-uniform integrability and generalized Young measures", Journal of Convex Analysis, 4 (1): 129–147, MR 1459885
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, Tomáš (2020-11-09), "Theory of Convex Compactifications", Relaxation in Optimization Theory and Variational Calculus, De Gruyter Series in Nonlinear Analysis applications (2nd ed.), De Gruyter, doi:10.1515/9783110590852-002, ISBN 978-3-11-059085-2
- 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