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Interior controllability of the nD semilinear heat equation. (English) Zbl 1243.93020

Summary: In this paper, we prove the interior approximate controllability of the following semilinear heat equation

z t (t,x)=Δz(t,x)+1 ω u(t,x)+f(t,z,u(t,x))in(0,τ]×Ω,z=0,on(0,τ)×Ω,z(0,x)=z 0 (x),xΩ,

where Ω is a bounded domain in N (N1),z 0 L 2 (Ω), ω is an open nonempty subset of Ω, and 1 ω denotes the characteristic function of the set ω. The distributed control u belong to L 2 ([0,τ];L 2 (Ω;)) and the nonlinear function f:[0,τ]×× is smooth enough and there are a,b,c, with c-1, such that

sup (t,z,u)Q τ |f(t,z,u)-az-cu-b|<,

where Q τ =[0,τ]××. Under this condition, we prove the following statement: For all open nonempty subset ω of Ω the system is approximately controllable on [0,τ]. Moreover, we could exhibit a sequence of controls steering the nonlinear system (1) from an initial state z 0 to an ϵ neighborhood of the final state z 1 at time τ>0, which is very important from a practical and numerical point of view.

MSC:
93B05Controllability
93C25Control systems in abstract spaces
35K05Heat equation