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# Observability for heat equations

Description

Observability for heat equations Kim Dang PHUNG Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China. Université d?Orléans, Laboratoire de Mathématiques - Analyse, Probabilités, Modélisation - Orléans, CNRS FR CNRS 2964, 45067 Orléans cedex 2, France. E-mail: Abstract This talk describes di?erent approaches to get the observability for heat equations without the use of Carleman inequalities. Contents 1 The heat equation and observability 2 2 Our motivation 3 3 Our strategy 4 3.1 Proof of Hölder continuous dependence from one point in time ) Sum of Laplacian eigenfunctions . . . . . . . . . . . . . . . 4 3.2 Proof of Hölder continuous dependence from one point in time ) Observability . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.3 Proof of Hölder continuous dependence from one point in time ) Re?ned Observability . . . . . . . . . . . . . . . . . . . . . 7 4 What I hope 9 4.1 Logarithmic convexity method . . . . . . . . . . . . . . . . . . . 9 4.2 Weighted logarithmic convexity method .

• weighted logarithmic

• logarithmic convexity

• convexity method

• empty open

• quantitative unique

• lebeau-robbiano strategy

• heat equations

Subjects

##### Logarithmically convex function

Informations

heat equations
Kim Dang PHUNG Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China. Université dOrléans, Laboratoire de Mathématiques - Analyse, Probabilités, Modélisation - Orléans, CNRS FR CNRS 2964, 45067 Orléans cedex 2, France. E-mail: kim_dang_phung@yahoo.fr
Observability
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Abstract
This talk describes di¤erent approaches to get the observability for heat equations without the use of Carleman inequalities.
The heat equation and observability
1
Our motivation
2
Our strategy 3.1 Proof of "Hölder continuous dependence from one point in time" ) . . . . . . . . . . . . . . ."Sum of Laplacian eigenfunctions" 3.2 Proof of "Hölder continuous dependence from one point in time" )"Observability" . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Proof of "Hölder continuous dependence from one point in time" )"Rened Observability" . . . . . . . . . . . . . . . . . . . . .
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What I hope 4.1 Logarithmic convexity method . . . . . . . . . . . . . . . . . . . 4.2 Weighted logarithmic convexity method . . . . . . . . . . . . . .
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5 What I can do 13 5.1 The frequency function . . . . . . . . . . . . . . . . . . . . . . . . 13 5.2 The frequency function with weight . . . . . . . . . . . . . . . . . 15 5.3 The heat equation with space-time potential . . . . . . . . . . . . 21
This talk was done when the author visited School of Mathematics & Statistics, Northeast Normal University, Changchun, China, (July 4-21, 2011).
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Contents
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What already exists 6.1 Monotonicity formula . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Proof of Lemma B . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Proof of Lemma A . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Proof of Lemma C . . . . . . . . . . . . . . . . . . . . . . 6.2 Quantitative unique continuation property for the Laplacian . . . 6.3 Quantitative unique continuation property for the elliptic opera-tor@t2+ . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 6.4 The heat equation and the Hölder continuous dependence from one point in time . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 The heat equation and observability
We consider the heat equation in the solutionu=u(x; t) @tuu= 0in(0;+1), <:8u(;0)2L2(), u= 0on@(0;+1),
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(1.1)
living in a bounded open setinRn,n1, either convex orC2and connected, with boundary@that the above problem is well-posed and. It is well-known have a unique solutionu2C[0; T] ;L2()\L20; T;H01()for allT >0.
The observability problem consists in proving the following estimate Zju(x; T)j2dxCZ0TZ!x; t)j2dxdt ju(
for some constantC >0independent on the initial data. Here,T >0and!is a non-empty open subset in.
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In the literature, two ways allow to prove such observability estimate. One is due to the work of Fursikov and Imanuvilov based on global Carleman in-equalities (see [FI]). The other proof is established by Lebeau and Robbiano (see [LR]. See [Le] for an english version). We resume the Lebeau-Robbiano strategy as follows. ku(; T)k2L2()CZ0TZju(x; t)j2dxdt ! * controllability in nite and innite dimension * j=1X;::;Njajj2CeCpNZ!j=1X;::;Najej(x)2dx for anyfajgwhere(ej; j)solves the eigenvalue problem with Dirichlet bound-ary conditions (0< 12   being the corresponding eigenvalues).
* For any >0and any non-trivial'2C01((0; T)), there areC >0and 2(0;1), such that for anyw2H2((0; T))with@t2+ w=fand wj@= 0, it holds kwkH1((;T))CkwkH1((0;T))kfkL2((0;T))+k'wkL2((0;T)1) The above interpolation inequality is proved using Carleman inequalities.
Recently, a shortcut of the Lebeau-Robbiano strategy is given in [M].
2 Our motivation
In application to bang-bang control (see [W]), we need the following rened observability estimate from measure set in time. ku(; T)kL2()CZEZ!ju(x; t)jdxdt for some constantC >0independent on the initial data. Here,E(0; T)is a measurable set of positive measure and!is a non-empty open subset in.
Further, we want to be able to extend the proof to heat equations with space-time potentials.
The approach describes in this talk is linked to parabolic quantitative unique continuation (see [BT], [Li], [L], [P], [EFV], [K], [KT] and references therein).
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