On a constrained approximate controllability problem for the heat equation

Citation
Jh. Ortega et E. Zuazua, On a constrained approximate controllability problem for the heat equation, J OPTIM TH, 108(1), 2001, pp. 29-64
Citations number
13
Categorie Soggetti
Engineering Mathematics
Journal title
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
ISSN journal
00223239 → ACNP
Volume
108
Issue
1
Year of publication
2001
Pages
29 - 64
Database
ISI
SICI code
0022-3239(200101)108:1<29:OACACP>2.0.ZU;2-A
Abstract
In this work, we study an approximate control problem for the heat equation , with a nonstandard but rather natural restriction on the solution. It is well known that approximate controllability holds. On the other hand, the t otal mass of the solutions of the uncontrolled system is constant in time. Therefore, it is natural to analyze whether approximate controllability hol ds supposing the total mass of the solution to be a given constant along th e trajectory. Under this additional restriction, approximate controllabilit y is not always true. For instance, this property fails when Omega is a bal l. We prove that the system is generically controllable; that is, given an open regular bounded domain Omega, there exists an arbitrarily small smooth deformation u, such that the system is approximately controllable in the n ew domain Omega + u under this constraint. We reduce our control problem to a nonstandard uniqueness problem. This uniqueness property is shown to hol d generically with respect to the domain.