An optimal control problem is considered where the state of the system is described by a variational inequality for the operator w → εΔ²w - φ(‖∇w‖²)Δw. A set of nonnegative functions φ is used as a control region. The problem is shown to have a solution for every fixed ε > 0. Moreover, the solvability of the limit optimal control problem corresponding to ε = 0 is proved. A compactness property of the solutions of the optimal control problems for ε > 0 and their relation with the limit problem are established. This type of operator arises in the theory of nonlinear plates, and the choice of a most suitable function φ is of interest for applications [2]. The problem of control of the function w has been studied in [4] for the operator under consideration, and some statements of this work will be used. Nonstationary problems with analogous operators were analyzed in [6,7]. Some general results on control of second-order variational inequalities can be found in [1]. The first section of this paper deals with the control problem for our fourth-order operator, the second considers a second-order operator, and the third studies the relationship between the solutions of the two problems.
@article{bwmeta1.element.bwnjournal-article-bcpv27z1p225bwm,
author = {Khludnev, A.},
title = {An optimal control problem for a fourth-order variational inequality},
journal = {Banach Center Publications},
volume = {27},
year = {1992},
pages = {225-231},
zbl = {0807.49002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv27z1p225bwm}
}
Khludnev, A. An optimal control problem for a fourth-order variational inequality. Banach Center Publications, Tome 27 (1992) pp. 225-231. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv27z1p225bwm/
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[003] [4] A. M. Khludnev, On limit passages in optimal control problems for a fourth-order operator, ibid. 25 (8) (1989), 1427-1435 (in Russian). | Zbl 0688.49006
[004] [5] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Springer, 1972.
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