# An optimal control problem for a fourth-order variational inequality

• Volume: 27, Issue: 1, page 225-231
• ISSN: 0137-6934

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## Abstract

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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.

## How to cite

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Khludnev, A.. "An optimal control problem for a fourth-order variational inequality." Banach Center Publications 27.1 (1992): 225-231. <http://eudml.org/doc/262655>.

@article{Khludnev1992,
abstract = {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.},
author = {Khludnev, A.},
journal = {Banach Center Publications},
keywords = {elastic plate theory; optimal control problem; variational inequality},
language = {eng},
number = {1},
pages = {225-231},
title = {An optimal control problem for a fourth-order variational inequality},
url = {http://eudml.org/doc/262655},
volume = {27},
year = {1992},
}

TY - JOUR
AU - Khludnev, A.
TI - An optimal control problem for a fourth-order variational inequality
JO - Banach Center Publications
PY - 1992
VL - 27
IS - 1
SP - 225
EP - 231
AB - 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.
LA - eng
KW - elastic plate theory; optimal control problem; variational inequality
UR - http://eudml.org/doc/262655
ER -

## References

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1. [1] V. Barbu, Optimal Control of Variational Inequalities, Res. Notes in Math. 100, Pitman, 1984.
2. [2] E. I. Grigolyuk and G. M. Kulikow, On a simplified method of solution of nonlinear problems in elastic plate and shell theory, in: Some Applied Problems of Plate and Shell Theory, Moscow University, 1981, 94-121 (in Russian).
3. [3] A. M. Khludnev, A boundary-value problem for a system of equations with a monotone operator, Differentsial'nye Uravneniya 16 (10) (1980), 1843-1849 (in Russian). Zbl0452.35017
4. [4] A. M. Khludnev, On limit passages in optimal control problems for a fourth-order operator, ibid. 25 (8) (1989), 1427-1435 (in Russian). Zbl0688.49006
5. [5] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Springer, 1972.
6. [6] S. I. Pokhozhaev, On a class of quasilinear hyperbolic equations, Mat. Sb. 96 (1) (1975), 152-166 (in Russian).
7. [7] S. I. Pokhozhaev, On a quasilinear hyperbolic Kirchhoff equation, Differentsial'nye Uravneniya 21 (1) (1985), 101-108 (in Russian). Zbl0584.35073

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