convex compactification
{{Short description|Concept of mathematics in convex analysis}}
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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.{{citation
| last = Roubíček | first = Tomáš
| doi = 10.1016/0362-546X(91)90199-B
| issue = 12
| journal = Nonlinear Analysis
| mr = 1111622
| pages = 1117–1126
| title = Convex compactifications and special extensions of optimization problems
| volume = 16
| year = 1991}} 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
- The set of Young measures{{citation | last = Young | first = L. C. | author-link =
| title = Generalized curves and the existence of an attained absolute minimum in the calculus of variations
| journal = Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie | volume = XXX | series =Classe III
| issue = 7–9 | pages = 211–234 | year=1937 | url=http://rcin.org.pl/dlibra/editions-content?id=69114 | jfm =63.1064.01 | zbl =0019.21901 }}{{citation
| last = Ball | first = J. M.
| editor1-last = Rascle | editor1-first = M.
| editor2-last = Serre | editor2-first = D.
| editor3-last = Slemrod | editor3-first = M.
| contribution = A version of the fundamental theorem for Young measures
| doi = 10.1007/BFb0024945
| isbn = 3-540-51617-4
| location = Berlin
| mr = 1036070
| pages = 207–215
| publisher = Springer
| series = Lecture Notes in Physics
| title = 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
| volume = 344
| year = 1989}} arising from bounded sets in Lebesgue spaces.
- The set of DiPerna-Majda measures {{citation
| last1 = Kružík | first1 = Martin
| last2 = Roubíček | first2 = Tomáš
| issue = 4
| journal = Mathematica Bohemica
| mr = 1489400
| pages = 383–399
| title = On the measures of DiPerna and Majda
| volume = 122
| year = 1997}}{{citation
| last1 = Alibert | first1 = J. J.
| last2 = Bouchitté | first2 = G.
| issue = 1
| journal = Journal of Convex Analysis
| mr = 1459885
| pages = 129–147
| title = Non-uniform integrability and generalized Young measures
| volume = 4
| year = 1997}} arising from bounded sets in Lebesgue spaces.
See also
References
=Notes=
{{Reflist}}
=Sources=
- {{Citation | author= L.C. Florescu, C. Godet-Thobie | title= Young measures and compactness in measure spaces | publisher=W. de Gruyter | place = Berlin | year=2012 | isbn=9783110280517}}
- {{Citation | author=P. Pedregal | title=Parametrized Measures and Variational Principles| publisher=Birkhäuser |location=Basel|year=1997|isbn=978-3-0348-9815-7}}
- {{cite book | first = Tomáš | last = Roubíček | title=Relaxation in Optimization Theory and Variational Calculus | chapter= Theory of Convex Compactifications | publisher= De Gruyter | date=2020-11-09 | isbn=978-3-11-059085-2 | doi=10.1515/9783110590852-002 | edition = 2nd | series = De Gruyter Series in Nonlinear Analysis applications }}
- {{Citation | last = Young | first = L. C. | author-link = | title = Lectures on the Calculus of Variations and Optimal Control | place = Philadelphia–London–Toronto | publisher = W. B. Saunders | year = 1969 | pages = xi+331 | url = https://books.google.com/books?id=YQpRAAAAMAAJ
| doi = | id = | isbn = 978-0-7216-9640-9| mr = 0259704 | zbl = 0177.37801}}
{{DEFAULTSORT:convex compactification}}