Relaxation in Optimization Theory and Variational Calculus

0536072 - ÚT 2021 RIV DE eng B - Monography
Roubíček, Tomáš
Relaxation in Optimization Theory and Variational Calculus.
2nd Edition. - Berlin: Walter de Gruyter, 2020. 474 s. De Gruyter Series in Nonlinear Analysis and Applications, 4. ISBN 978-3-11-058962-7. ISSN 0941-813X
R&D Projects: GA ČR(CZ) GA19-04956S
Institutional support: RVO:61388998
Keywords : continuous extenstions * convex compactifications * young measures * optimal control * partial differential equations * variational problems
OECD category: Applied mathematics
https://www.degruyter.com/view/title/537021

In this monograph, relaxation means an extension of optimization or variational problems in a certain natural way, typically by some continuity. Beside compactness, also convexity is the main attribute of the extended (so-called relaxed) problems,which yields existence and stability of their solutions as well as optimality conditions, or allows for using some fixed-point theorems, especially in noncooperative game theory. The book offers an exposition of an original abstract theory of convex compactifications and of application to various concrete problems in optimal control and calculus of variations. Classical Young measures and their numerous generalizations are used to capture various fast oscillation and possibly concentration phenomena in the limits. In particular, one can construct convex locally compact metrizable envelopes of Lebesgue spaces, typically occuring in original (nonrelaxed) problems and here embedded densely into such envelopes. Also numerical aspects are pursued. This new edition reflects in particular also the achievements within quarter a century that passed from the first edition. Many improvements of the presentation and expansions have been added, too.
Permanent Link: http://hdl.handle.net/11104/0316587