Think globally, act locally: approximate controllability through homogenization of an optimal control problem with control on the boundary of certain particles

Abstract The phrase "Think globally, act locally" became prominent in the context of sustainable development and environmental examples des fonctions activism, encouraging individuals and communities to address global challenges by taking local actions. In this paper, we offer a mathematical framework in which such metaphor can be understood in terms of the global homogenized formulation of some suitable control problem stated in a domain $$\Omega _{\varepsilon }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>ε</mml:mi> </mml:msub> </mml:math> , a part of a given domain $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> , which is exterior to a periodic distribution of many particles. We assume a linear heat equation in $$\Omega _{\varepsilon }\times (0,T)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>ε</mml:mi> </mml:msub> <mml:mo>×</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and a Robin-type boundary condition on the boundary of the particles. We prove the “approximate controllability” of the problem, with a final observation, when the control is implemented only on the boundary of certain particles. Firstly, we apply the homogenization process, proving that the solution of the microscopic problem converges, as $$\varepsilon \rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , to a function $$u_{0}(x,t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> that is the unique solution to a suitable global state parabolic problem. 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. This conclusion gives a certain mathematical justification for the popular phrase used by ecologists. Moreover, it brings to light some limitations that must be assumed on the local controls to conclude that the result is globally satisfactory.