Carleman Estimates for a Class of Degenerate Parabolic Operators
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The proof is based on the choice of suitable weighted functions and Hardy-type inequalities and deduce null controllability results for the degenerate one-dimensional heat equation with the same boundary conditions as above.Abstract:
Given $\alpha \in [0,2)$ and $f \in L^2 ((0,T)\times(0,1))$, we derive new Carleman estimates for the degenerate parabolic problem $w_t + (x^\alpha w_x) _x =f$, where $(t,x) \in (0,T) \times (0,1)$, associated to the boundary conditions $w(t,1)=0$ and $w(t,0)=0$ if $0 \leq \alpha <1$ or $(x^\alpha w_x)(t,0)=0$ if $1\leq \alpha <2$ The proof is based on the choice of suitable weighted functions and Hardy-type inequalities As a consequence, for all $0 \leq \alpha <2$ and $\omega\subset\subset(0,1)$, we deduce null controllability results for the degenerate one-dimensional heat equation $u_t - (x^\alpha u_x)_x = h \chi _\omega$ with the same boundary conditions as aboveread more
Citations
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Carleman estimates for parabolic equations and applications
TL;DR: In this paper, a review of the applicability of the Carleman estimate to the estimation of solutions and inverse problems is presented. But this review is limited to parabolic equations.
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Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form
TL;DR: In this article, the authors give null controllability results for some degenerate parabolic equations in non-divergence form on a bounded interval, where the coefficient of the second order term degenerates at the extreme points of the domain.
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Null controllability for the heat equation with singular inverse-square potentials
TL;DR: In this article, the null controllability of the heat equation perturbed by a singular inverse-square potential arising in quantum mechanics and combustion theory was proved within the range of subcritical coefficients of the singular potential, provided the control acts on an annular set around the singularity.
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Control and Stabilization Properties for a Singular Heat Equation with an Inverse-Square Potential
TL;DR: In this article, the authors analyzed the control properties of parabolic equations with a singular potential, where μ is a real number, and showed that it is possible to stabilize the corresponding systems uniformly with respect to ϵ ≥ 0, due to the presence of explosive modes which concentrate around the singularity.
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Exact boundary controllability of 1-d parabolic and hyperbolic degenerate equations ∗
TL;DR: The boundary controllability of a class of one-dimensional degenerate equations is studied and sharp observability estimates for these equations are proved using nonharmonic Fourier series.
References
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Hardy-type inequalities
Bohumír Opic,Alois Kufner +1 more
TL;DR: In this article, the one-dimensional Hardy inequality is defined as an imbedding of a weighted Sobolev space into a weighted Lebesgue space and the n-dimensional case special weights.
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Contróle Exact De Léquation De La Chaleur
G. Lebeau,Luc Robbiano +1 more
TL;DR: In this paper, De La Chaleur et al. present the Controle Exact De Lequation De La CHaleur (CEDE) for Communications in Partial Differential Equations.