English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Pro

Presents the zaia usage as premium feature

ZAIA

Presents AI Discovery search usage as a premium feature

AI Discovery

Presents AI Discovery explain and summary usage as a premium feature

Content Analysis

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Presents the pdf analysis as premium feature

PDF Analysis

1500 Top-up

3000 Top-up

Zendy Basic

2500 Top-up

Zendy Unlimited

Presents the access of premium content as premium feature

Premium Content

An optimal control problem is considered where the state of the system is described by a variational inequality for the operator w → e∆w−φ(‖∇w‖)∆w. A set of nonnegative functions φ is used as a control region. The problem is shown to have a solution for every fixed e > 0. Moreover, the solvability of the limit optimal control problem corresponding to e = 0 is proved. A compactness property of the solutions of the optimal control problems for e > 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.

An optimal control problem for a fourth-order variational inequality