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We consider a heat conduction problem <italic>S</italic> with mixed boundary conditions in an <italic>n</italic>-dimensional domain Ω with regular boundary and a family of problems <italic>Sα</italic> with also mixed boundary conditions in Ω, where <italic>α ></italic> 0 is the heat transfer coefficient on the portion of the boundary Γ1. In relation to these state systems, we formulate <italic>Neumann boundary</italic> optimal control problems on the heat flux <italic>q</italic> which is definite on the complementary portion Γ2 of the boundary of Ω. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system state and the adjoint state when the heat transfer coefficient <italic>α</italic> goes to infinity. Furthermore, we formulate particular <italic>boundary</italic> optimal control problems on a real parameter <italic>λ</italic>, in relation to the parabolic problems <italic>S</italic> and <italic>Sα</italic> and to mixed elliptic problems <italic>P</italic> and <italic>Pα</italic>. We find an explicit form for the optimal controls, we prove monotony properties and we obtain convergence results when the parameter time goes to infinity.
</abstract>

We consider a heat conduction problem S with mixed boundary conditions in an n-dimensional domain Ω with regular boundary and a family of problems Sα with also mixed boundary conditions in Ω, where α > 0 is the heat transfer coefficient on the portion of the boundary Γ1. In relation to these state systems, we formulate Neumann boundary optimal control problems on the heat flux q which is definite on the complementary portion Γ2 of the boundary of Ω. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system state and the adjoint state when the heat transfer coefficient α goes to infinity. Furthermore, we formulate particular boundary optimal control problems on a real parameter λ, in relation to the parabolic problems S and Sα and to mixed elliptic problems P and Pα. We find an explicit form for the optimal controls, we prove monotony properties and we obtain convergence results when the parameter time goes to infinity.

Neumann boundary optimal control problems governed by parabolic variational equalities

Control and Cybernetics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

We consider a heat conduction problem S with mixed boundary conditions in an n-dimensional domain Ω with regular boundary and a family of problems Sα with also...

Neumann boundary optimal control problems governed by parabolic variational equalities

Published Online: Jun 28, 2022

Page range: 227 - 252

Received: Sep 01, 2020

Accepted: Mar 01, 2021

DOI: https://doi.org/10.2478/candc-2021-0012

Keywords
parabolic variational equalities, optimal control, mixed boundary conditions, optimality conditions, convergence

© 2021 Carolina M. Bollo et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Neumann boundary optimal control problems governed by parabolic variational equalities

Published Online: Jun 28, 2022

Page range: 227 - 252

Received: Sep 01, 2020

Accepted: Mar 01, 2021

DOI: https://doi.org/10.2478/candc-2021-0012

Keywordsparabolic variational equalities, optimal control, mixed boundary conditions, optimality conditions, convergence

© 2021 Carolina M. Bollo et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
parabolic variational equalities, optimal control, mixed boundary conditions, optimality conditions, convergence