On the null-controllability of the heat equation in unbounded domains. (English) Zbl 1079.35018

Miller, Luc

Bull. Sci. Math. 129, No. 2, 175-185 (2005).

The paper is devoted to the null-controllability of the heat equation with Dirichlet condition in unbounded domains. This is an open problem which has been recently underscored. Two aspects of the problem are discussed. First, the author gives a geometric necessary condition that characterizes “controlling capacity” of a subset for interior null-controllability in the Euclidean setting. The proof of the statement is based on heat kernel estimates. Second, the author specifies a class of null-controllable heat equations on unbounded product domains. Rather simple examples, namely, an infinite strip in the plane controlled from one boundary and an infinite rod controlled from an inner infinite rod, are considered. Proving the results, the author uses the idea that the null-controllability of an abstract control system and its null-controllability cost are invariant with respect to taking its tensor product with a system generated by a non-positive self-adjoint operator. The paper continues the author’s investigations in control theory.

Reviewer: Vyacheslav I. Maksimov (Ekaterinburg)

MSC:

35B37	PDE in connection with control problems (MSC2000)
93B05	Controllability
35K20	Initial-boundary value problems for second-order parabolic equations

Keywords:

heat equation; controllability; Dirichlet condition; kernel estimates

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