//img.uscri.be/pth/febe35bf8c20d5b0a1e97a6b57b4266126f0fde6
Cet ouvrage fait partie de la bibliothèque YouScribe
Obtenez un accès à la bibliothèque pour le lire en ligne
En savoir plus

Observability for heat equations

37 pages
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


Voir plus Voir moins
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
for
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" . . . . . . . . . . . . . . . . . . . . .
4 4 4 7
3
9 9 10
What I hope 4.1 Logarithmic convexity method . . . . . . . . . . . . . . . . . . . 4.2 Weighted logarithmic convexity method . . . . . . . . . . . . . .
4
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).
3
Contents
1
2
6
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),
22 23 24
24 26 30
31
32
(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.
2
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).
3