Abstract
In this article, we study the null-controllability of a heat equation in a domain composed of two media of different constant conductivities. In particular, we are interested in the behavior of the system when the conductivity of the medium on which the control does not act goes to infinity, corresponding at the limit to a perfectly conductive medium. In that case, and under suitable geometric conditions, we obtain a uniform null-controllability result. Our strategy is based on the analysis of the controllability of the corresponding wave operators and the transmutation technique, which explains the geometric conditions.
Highlights
To prove Theorem 1.1, we first remark that it is equivalent to prove a uniform observability result for the adjoint heat equation, that is the existence of a time T > 0 and a constant C = C(Ω, ω, T, σ2) > 0 such that for all σ1 σ2, the solution zσ of
One could try to follow the approach in [26, 27], which would consist in first deducing from Theorem 1.4 a uniform controllability result for the wave equation, and use a transmutation technique to deduce a uniform controllability result for the corresponding heat equation
This is in principle possible, but in our case this would be delicate since the observability estimate (1.14) implies the controllability of the corresponding wave equation with a control bounded by the norm of the initial datum in L2(Ω) × H−1(Ω)
Summary
To prove Theorem 1.1, we first remark that it is equivalent to prove a uniform observability result for the adjoint heat equation (on which we perform the change of time t → T − t as usual), that is the existence of a time T > 0 and a constant C = C(Ω, ω, T, σ2) > 0 such that for all σ1 σ2, the solution zσ of. One could try to follow the approach in [26, 27], which would consist in first deducing from Theorem 1.4 a uniform controllability result for the wave equation, and use a transmutation technique to deduce a uniform controllability result for the corresponding heat equation This is in principle possible, but in our case this would be delicate since the observability estimate (1.14) implies the controllability of the corresponding wave equation with a control bounded by the norm of the initial datum in L2(Ω) × H−1(Ω). To avoid these difficulties and follow the dependence in σ more clearly, we have chosen to use the transmutation technique of [11, 12], and to avoid the use of negative Sobolev spaces
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