RT Journal Article
T1 Think globally, act locally: approximate controllability through homogenization of an optimal control problem with control on the boundary of certain particles
A1 Díaz Díaz, Jesús Ildefonso
A1 Podolskiy, A. V.
A1 Shaposhnikova, T. A.
AB 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 ε, a part of a given domain , which is exterior to a periodic distribution of many particles. We assume a linear heat equation in ε × (0, T ) 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 ε → 0, to a function u0(x, t) 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.
PB Springer
SN 1578-7303
SN 1579-1505
YR 2025
FD 2025
LK https://hdl.handle.net/20.500.14352/124679
UL https://hdl.handle.net/20.500.14352/124679
LA eng
NO 2025 Acuerdos transformativos CRUE
NO Ministerio de Ciencia e Innovación
DS Docta Complutense
RD 18 dic 2025