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

Sujets

• Northeast Normal University
• Logarithmically convex function

Informations

• Publié par: pefav
• Nombre de lectures: 11
• Langue: English

Extrait

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: [e-mail]

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

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9 9 10

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

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@ t 2 + . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 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) @ t uu= 0in(0;+1), < : 8 u(;0)2L 2 (), u= 0on@(0;+1),

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24 26 30

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(1.1)

living in a bounded open setinR n ,n1, either convex orC 2 and connected, with boundary@that the above problem is well-posed and. It is well-known have a unique solutionu2C  [0; T] ;L 2 ()  \L 2  0; T;H 01 ()  for allT >0.

The observability problem consists in proving the following estimate Z  ju(x; T)j 2 dxC Z 0 T Z ! x; t)j 2 dxdt 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)k 2 L2() C Z 0 T Z ju(x; t)j 2 dxdt ! * controllability in nite and innite dimension *  j=1 X ;::;N ja j j 2 Ce C p  N Z !      j=1 X ;::;N a j e j (x)       2 dx for anyfa j gwhere(e j ;  j )solves the eigenvalue problem with Dirichlet bound- ary conditions (0<  1  2    being the corresponding eigenvalues).

* For any >0and any non-trivial'2C 0 1 ((0; T)), there areC >0and 2(0;1), such that for anyw2H 2 ((0; T))with  @ t 2 +   w=fand w j@ = 0, it holds kwk H 1 ((;T)) Ckwk  H 1 ((0;T))  kfk L 2 ((0;T)) +k'wk L 2 ((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)k L2() C Z E Z ! 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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