期刊
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
卷 59, 期 2, 页码 1293-1311出版社
SIAM PUBLICATIONS
DOI: 10.1137/20M1361572
关键词
relaxed controls; convex compactifications; Young measures; Choquet theory; optimal control of parabolic equations; existence; maximum principle; Filippov-Roxin theory
资金
- CSF (Czech Science Foundation) [19-04956S]
- Centre of Excellence for Nonlinear Dynamic Behaviour of Advanced Materials in Engineering [CZ.02.1.01/0.0/0.0/15 003/0000493]
- [RVO: 61388998]
Nonconvex optimal control problems governed by evolution problems in infinite-dimensional spaces are addressed by extending the control on a convex compactification to ensure existence of solutions and simplify analysis. A compromise convex compactification is devised using classical techniques for Young measures and Choquet theory, applied to parabolic optimal control problems for existence and optimality conditions.
Nonconvex optimal control problems governed by evolution problems in infinite-dimensional spaces (as, e.g., parabolic boundary-value problems) needs a continuous (and possibly also smooth) extension on some (preferably convex) compactification, called relaxation, to guarantee existence of their solutions and to facilitate analysis by relatively conventional tools. When the control is valued in some subsets of Lebesgue spaces, the usual extensions are either too coarse (allowing in fact only very restricted nonlinearities) or too fine (being nonmetrizable). To overcome these drawbacks, a compromising convex compactification is here devised, combining classical techniques for Young measures with Choquet theory. This is applied to parabolic optimal control problems as far as existence and optimality conditions concerns.
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