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OPTIMAL CONTROL PROBLEMS GOVERNED BY MARGUERRE–VON KÁRMÁN EQUATIONS

Part of: Thin bodies, structures Elastic materials Equations of mathematical physics and other areas of application Optimality conditions Existence theories

Published online by Cambridge University Press:  01 January 2025

MOHAMMED EL AMINE RIAHI  and
ABDERREZAK GHEZAL Open the ORCID record for ABDERREZAK GHEZAL [Opens in a new window]
MOHAMMED EL AMINE RIAHI
Affiliation:
Laboratory of Applied Mathematics, Faculty of Mathematics and Material Sciences, Kasdi Merbah Ouargla University , Algeria e-mail: riahi.math@gmail.com
ABDERREZAK GHEZAL*
Affiliation:
Laboratory of Applied Mathematics, Faculty of Mathematics and Material Sciences, Kasdi Merbah Ouargla University , Algeria e-mail: riahi.math@gmail.com
*
e-mail: ghezal.ark@gmail.com
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Abstract

We consider the Marguerre–von Kármán equations that model the deformation of a thin, nonlinearly elastic, shallow shell, subjected to a specific class of boundary conditions of von Kármán’s type. Next, we reduce these equations to a single equation with a cubic operator following Berger’s classical method, whose second member depends on the function defining the middle surface of the shallow shell and the resultant of the vertical forces acting on the shallow shell. We also prove the existence and uniqueness of a weak solution to the reduced equation. Then, we prove the existence theorem for the optimal control problem governed by Marguerre–von Kármán equations, with a control variable on the resultant of the vertical forces. Using the Fréchet differentiability of the state function with respect to the control variable, we prove the uniqueness of the optimal control and derive the necessary optimality condition. As a result, this work addresses the more general geometry of Marguerre–von Kármán shallow shells to study the quadratic cost optimal control problems governed by these equations.

Keywords

Marguerre–von Kármán equationsoptimal controlnecessary optimality conditionnonlinear elasticityFréchet differentiability

MSC classification

Primary: 35Q93: PDEs in connection with control and optimization
Secondary: 74B20: Nonlinear elasticity 49J20: Optimal control problems involving partial differential equations 49K20: Problems involving partial differential equations 74K25: Shells

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 67 , 2025 , e38
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of the Australian Mathematical Publishing Association Inc

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