Sib. Èlektron. Mat. Izv., 2017, Volume 14, Pages 1349–1372
Differentical equations, dynamical systems and optimal control
Maximization problems for eigenvalues of linear elliptic operators
V. Yu. Goncharov
Moscow Aviation Institute (National Research University),
Volokolamskoe Shosse, 4,
125993, Moscow, Russia
Maximization problems for eigenvalues of elliptic operators are considered.
The problems under investigation are optimal control problems in coefficients,
admissible controls form a weak* compact set of essentially bounded measurable functions,
and convexity hypotheses on the coefficients of operators are made.
The purpose of this article is twofold:
(i) to derive necessary optimality conditions, which form a basis for efficient numerical solution;
(ii) to describe the structure of the set of solutions for such a problem,
to prove uniqueness criteria,
and to characterize the case of non-uniqueness.
The main idea of the article is that, even in the case of multiple eigenvalues, one can derive necessary optimality conditions, which involve only one eigenfunction.
The derived necessary optimality conditions also make it possible to replace the original non-smooth extremal problem
the problem of finding a saddle point of a certain concrete functional.
Applications of the results to optimal design problems for non-homogeneous columns and three-layered plates are given.
eigenvalue optimization, elliptic boundary-value problems, control in coefficients, uniqueness criteria, optimality conditions, saddle points, multiple eigenvalues, optimal structural design, non-homogeneous column,
PDF file (260 kB)
MSC: 35Q93, 49K15, 49K20
Received April 25, 2016, published December 7, 2017
V. Yu. Goncharov, “Maximization problems for eigenvalues of linear elliptic operators”, Sib. Èlektron. Mat. Izv., 14 (2017), 1349–1372
Citation in format AMSBIB
\paper Maximization problems for eigenvalues of linear elliptic operators
\jour Sib. \`Elektron. Mat. Izv.
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