Volume 29, 2023
|Number of page(s)||16|
|Published online||30 March 2023|
Optimal control of a parabolic equation with memory
Departmento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria,
2 Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA
* Corresponding author: email@example.com
Accepted: 1 March 2023
An optimal control problem for a semilinear parabolic partial differential equation with memory is considered. The well-posedness as well as the first and the second order differentiability of the state equation is established by means of Schauder fixed point theorem and the implicity function theorem. For the corresponding optimal control problem with the quadratic cost functional, the existence of optimal control is proved. The first and the second order necessary conditions are presented, including the investigation of the adjoint equations which are linear parabolic equations with a measure as a coefficient of the operator. Finally, the sufficiency of the second order optimality condition for the local optimal control is proved.
Mathematics Subject Classification: 49J20 / 49K20 / 35K58 / 45K05
Key words: Parabolic partial differential equation with memory / optimal control / optimality conditions
This author was supported by MCIN/ AEI/10.13039/501100011033/ under research project PID2020-114837GB-I00.
© The authors. Published by EDP Sciences, SMAI 2023
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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