Feedback controllability for blowup points of the heat equation. (English. French summary) Zbl 1503.35089
Summary: This paper is concerned with a controllability problem of blowup points for the heat equation. It can be described as follows: In the absence of control, the solution to the linear heat equation globally exists in a bounded domain \(\Omega \). We wonder whether for a given time \(T>0\) and a point \(a\) in this domain, we can find a feedback control, acting on a given internal subset \(\omega\) of this domain, such that the corresponding solution to the heat equation blows up at time \(T\) and holds \(a\) as a unique blowup point. In this paper, we positively answer the question, when \(a\in\omega\). On the contrary, when \(a\in\Omega\smallsetminus\overline{\omega}\), we show this is not possible, for any open-loop or feedback control which guarantees that the solution is \(L^\infty(\Omega)\) whenever it exists.
MSC:
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35B44 | Blow-up in context of PDEs |
| 93B05 | Controllability |
| 93B52 | Feedback control |
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