| Issue |
ESAIM: COCV
Volume 27, 2021
|
|
|---|---|---|
| Article Number | 79 | |
| Number of page(s) | 35 | |
| DOI | https://doi.org/10.1051/cocv/2021074 | |
| Published online | 22 July 2021 | |
Constrained nonsmooth problems of the calculus of variations*
Institute for Problems in Mechanical Engineering, Russian Academy of Sciences,
Saint Petersburg,
199178 Russia.
** Corresponding author: dmv@ipme.ru
Received:
29
April
2020
Accepted:
26
June
2021
The paper is devoted to an analysis of optimality conditions for nonsmooth multidimensional problems of the calculus of variations with various types of constraints, such as additional constraints at the boundary and isoperimetric constraints. To derive optimality conditions, we study generalised concepts of differentiability of nonsmooth functions called codifferentiability and quasidifferentiability. Under some natural and easily verifiable assumptions we prove that a nonsmooth integral functional defined on the Sobolev space is continuously codifferentiable and compute its codifferential and quasidifferential. Then we apply general optimality conditions for nonsmooth optimisation problems in Banach spaces to obtain optimality conditions for nonsmooth problems of the calculus of variations. Through a series of simple examples we demonstrate that our optimality conditions are sometimes better than existing ones in terms of various subdifferentials, in the sense that our optimality conditions can detect the non-optimality of a given point, when subdifferential-based optimality conditions fail to disqualify this point as non-optimal.
Mathematics Subject Classification: 49K10 / 49J52 / 90C48
Key words: Nonsmooth analysis / calculus of variations / optimality conditions / codifferential / quasidifferential
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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