Abstract
Let the matrix operator L = D∂xx + q (x) A0, with D = diag (1, ν), ν ≠ 1, q ∈ L∞ (0, π), and A0 is a Jordan block of order 1. We analyze the boundary null controllability for the system yt − Ly = 0. When [see formula in PDF] and q is constant, q = 1 for instance, there exists a family of root vectors of [see formula in PDF] forming a Riesz basis of L2(0,π;ℝ2). Moreover F. Ammar Khodja et al. [J. Funct. Anal. 267 (2014) 2077–2151] shows the existence of a minimal time of control depending on condensation of eigenvalues of [see formula in PDF], that is to say the existence of T0 (ν) such that the system is null controllable at time T > T0 (ν) and not null controllable at time T < T0 (ν). In the same paper, the authors prove that for all τ ∈ [0, + ∞], there exists ν ∈] 0, + ∞ [ such that T0 (ν) = τ. When q depends on x, the property of Riesz basis is no more guaranteed. This leads to a new phenomena: simultaneous condensation of eigenvalues and eigenfunctions. This condensation affects the time of null controllability.
Highlights
Introduction and main resultsThis paper deals with the controllability of two coupled one-dimensional parabolic equations, with different diffusion coefficients, where the control is exerted at one boundary point
The authors used the method of moments of Fattorini-Russell to give a necessary and sufficient condition of null controllability at any time T > 0 for system (1.5)
We study the null controllability properties of system (1.5) to the case where D = Id and
Summary
This paper deals with the controllability of two coupled one-dimensional parabolic equations, with different diffusion coefficients, where the control is exerted at one boundary point. This result was generalized by [1] to the case n ≥ 2, m ≥ 1 In these two papers, the authors used the method of moments of Fattorini-Russell to give a necessary and sufficient condition of null controllability at any time T > 0 for system (1.5). The authors used the method of moments of Fattorini-Russell to give a necessary and sufficient condition of null controllability at any time T > 0 for system (1.5) In both cases, the sequence of eigenvalues Λ = {Λk}k≥1 ⊂ R+. Λ = {Λk}k≥1 which does eigenfunctions is complete not but satisfy the it is not a Rgaiepszcobnadsiisti(osneeaPpproepa.ri2n.g4)info(r1s.o4m) heo√wνev∈/erQt∗+heansedquqe∈ncLe∞o(f0a,sπs)oc.iAatseda consequence, we will see that a minimal time of control T0 ∈ [0, +∞], depends simultaneously of condensation of eigenvalues and associated eigenfunctions of (L∗, D(L∗)) To this end, we will use the block methods moment developed by A.
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