Feedback controllability for blowup points of the heat equation

Abstract

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 Ω. 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 ω 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∈ω. On the contrary, when a∈Ω∖ω‾, we show this is not possible, for any open-loop or feedback control which guarantees that the solution is L∞(Ω) whenever it exists.