<p>We prove the approximate controllability, for the interior final observation, of the solution of a linear heat equation on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega \times (0,T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, when the control is placed on a small part of the boundary <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$l_{\varepsilon }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>l</mi> <mi>ε</mi> </msub> </math></EquationSource> </InlineEquation> which is heterogeneous (with a critical scale) and where we assume a Robin-type boundary condition. One of the motivations is related to some atmosphere – deep ocean models in climatology. We consider the special case of two-dimensional domains <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> which requires suitable coefficients in the Robin-term and in the control. Firstly, we apply the homogenization process proving that the solution of the microscopic problem converges, as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varepsilon \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, to a function <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$u_{0}(x,t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> that is the unique solution to a suitable global state parabolic problem with a Robin-type boundary condition on a part of the boundary. We consider a microscopic optimal control problem and prove the weak convergence of the state and the optimal control. Finally, we prove the approximate controllability by passing to the limit in a penalty parameter of the cost functional. The conclusion gives a certain mathematical justification to some arguments used by ecologists but it brings to light also some limitations that must be assumed on the local controls to conclude that the result is globally satisfactory.</p>

From a small reactive part of the boundary to the whole interior non-reactive domain: critical scale homogenization, optimal control and controllability

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2025年7月3日

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J. I. Díaz,
A .V. Podolskiy,
T. A. Shaposhnikova

摘要

We prove the approximate controllability, for the interior final observation, of the solution of a linear heat equation on $\Omega \times (0,T)$ $Ω \times (0, T)$ , when the control is placed on a small part of the boundary $l_{\varepsilon }$ $l_{ε}$ which is heterogeneous (with a critical scale) and where we assume a Robin-type boundary condition. One of the motivations is related to some atmosphere – deep ocean models in climatology. We consider the special case of two-dimensional domains $\Omega $ $Ω$ which requires suitable coefficients in the Robin-term and in the control. Firstly, we apply the homogenization process proving that the solution of the microscopic problem converges, as $\varepsilon \rightarrow 0$ $ε \to 0$ , to a function $u_{0}(x,t)$ $u_{0} (x, t)$ that is the unique solution to a suitable global state parabolic problem with a Robin-type boundary condition on a part of the boundary. We consider a microscopic optimal control problem and prove the weak convergence of the state and the optimal control. Finally, we prove the approximate controllability by passing to the limit in a penalty parameter of the cost functional. The conclusion gives a certain mathematical justification to some arguments used by ecologists but it brings to light also some limitations that must be assumed on the local controls to conclude that the result is globally satisfactory.