Null controllability of the heat equation in unbounded domains by a finite measure control region

Abstract

Abstract. Motivated by two recent works of Micu and Zuazua and Cabanillas, De Menezes andZuazua, we study the null controllability of the heat equation in unbounded domains, typically R+ or R N . Considering an unbounded and disconnected control region of the form ω := ∪nωn, we prove two null controllability results: under some technical assumption on the control parts ωn, we prove that every initial datum in some weighted L 2 space can be controlled to zero by usual control functions, and every initial datum in L 2 (Ω) can be controlled to zero using control functions in a weighted L 2 space. At last we give several examples in which the control region has a finite measure and our null controllability results apply.Mathematics Subject Classification. 35K05, 93B05, 93B07.

Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition

“…In the present paper, we extend the results of [7] to the heat equation controlled by the Neumann boundary condition (see (1), (2)). Analogs of all results of paper [7] are obtained here.…”

“…This problem is considered in spaces of Sobolev type (see details in Section 2). Control problems for the heat equation were studied in unbounded domains in [1,2,7,13,14,4]. In particular, in [14], the null-controllability problem for equation (1) controlled by the Dirichlet boundary condition w(0, •) = u, t ∈ (0, T ), (5) under initial condition (3) with L 2 -control (u ∈ L 2 (0, T )) was studied in a weighted Sobolev space of negative order.…”

Reachability and Controllability Problems for the Heat Equation on a Half-Axis

“…However controlability problems for the heat equation on domains unbounded with respect to spatial variables were not fully investigated. These problems for this equation were studied in [1,2,8,9,11]. In particular, in [9], null-controllability problem for control system (1.1)-(1.3) with L 2 -control (u ∈ L 2 (0, T )) was investigated in a weighted Sobolev space of negative order.…”

Observability inequalities for the heat equation with bounded potentials on the whole space

“…It showed that, for some parabolic equations in an unbounded domain Ω ⊂ R N , the observability inequality holds when observations are made over a subset ω ⊂ Ω, with Ω\ω bounded. For other similar results, we refer the readers to [3,7,18,20,31].…”