Some uniqueness results for the determination of forces acting over - PowerPoint PPT Presentation

Some uniqueness results for the determination of forces acting over Germain-Lagrange plates Three different techniques A. Kawano Escola Politecnica da Universidade de Sao Paulo A. Kawano (Poli-USP) Germain-Lagrange plates May 2014 1 / 49

Introduction Presentation of the Germain-Lagrange Operator Plan Introduction 1 Presentation of the Germain-Lagrange Operator Presentation of the inverse problems Inverse Problems 2 Unbounded plate Rectangular Plate Plates with arbitrary shapes A. Kawano (Poli-USP) Germain-Lagrange plates May 2014 2 / 49

Introduction Presentation of the Germain-Lagrange Operator Basic Equation The Germain-Lagrange equation models the vibration of a plate. Is is given by ∂ 2 u ∂ t 2 + △ 2 u = h , where u is the displacement and h is the loading. A. Kawano (Poli-USP) Germain-Lagrange plates May 2014 3 / 49

Introduction Presentation of the Germain-Lagrange Operator Fundamental solution We can work out an explicit expression for a fundamental solution φ of the Germain-Lagrange operator. It is given by � + ∞ sen [ r 2 t ] φ ( t , x ) = H ( t ) J 0 ( r | x | ) d r . (1) 2 π r 0 A. Kawano (Poli-USP) Germain-Lagrange plates May 2014 4 / 49

Introduction Presentation of the Germain-Lagrange Operator The Germain-Lagrange operator is not hypo-elliptic We have two properties: The operator P = △ 2 + ∂ 2 ∂ t 2 is not hypo-elliptic. Proof The speed of propagation of signals in a Germain-Lagrange plate is infinite. Proof A. Kawano (Poli-USP) Germain-Lagrange plates May 2014 5 / 49

Introduction Presentation of the inverse problems Plan Introduction 1 Presentation of the Germain-Lagrange Operator Presentation of the inverse problems Inverse Problems 2 Unbounded plate Rectangular Plate Plates with arbitrary shapes A. Kawano (Poli-USP) Germain-Lagrange plates May 2014 6 / 49

Introduction Presentation of the inverse problems Inverse problems Statement of the problems presented here We consider the problem of proving that loads h of the form h ( t , x ) = g ( t ) f ( x ) in the equation ∂ 2 u ∂ t 2 + △ 2 u = h = g ( t ) f ( x ) , can be determined uniquely from data about the displacement of a set of points in the plate. In general, f is a distribution and g a function of time that is class C 1 . A. Kawano (Poli-USP) Germain-Lagrange plates May 2014 7 / 49

Introduction Presentation of the inverse problems Inverse problems Statement of the problems presented here Given the fact that the speed of propagation is infinite, the data for the inverse problem is taken in arbitrary small intervals of time and space. The problems we are going to present are: Unbounded plate that extends over all R 2 . 1 Rectangular plate. 2 Bounded plate with arbitrary shape. 3 A. Kawano (Poli-USP) Germain-Lagrange plates May 2014 8 / 49

Inverse Problems Unbounded plate Plan Introduction 1 Presentation of the Germain-Lagrange Operator Presentation of the inverse problems Inverse Problems 2 Unbounded plate Rectangular Plate Plates with arbitrary shapes A. Kawano (Poli-USP) Germain-Lagrange plates May 2014 9 / 49

Inverse Problems Unbounded plate Unbounded Plate  ∂ 2 u ∂ t 2 + △ 2 u = g ⊗ f , in ] 0 , + ∞ ) × R 2 ,   u ( 0 ) = 0 , (2)   ∂ u ∂ t ( 0 ) = 0 , where f ∈ E ′ ( R 2 ) (To be determined); ∃ T 0 > 0 such that g ∈ C ([ 0 , T 0 ]) and g ( 0 ) � = 0. A. Kawano (Poli-USP) Germain-Lagrange plates May 2014 10 / 49

Inverse Problems Unbounded plate Data for the identification of f ∈ E ′ ( R 2 ) Theorem 1 (For the unbounded case) If ∃ T 0 > 0 such that g ∈ C ([ 0 , T 0 ]) and g ( 0 ) � = 0 , and f ∈ E ′ ( R 2 ) , then for arbitrary 0 < T < T 0 , the knowledge of the set Γ ] 0 , T [ × Ω = {� u , ψ ⊗ ϕ � : ψ ∈ C ∞ c (] 0 , T [) , ϕ ∈ C ∞ c (Ω) } , where Ω ⊂ R 2 is arbitrary, is enough to determine uniquely f ∈ E ′ ( R 2 ) . A. Kawano (Poli-USP) Germain-Lagrange plates May 2014 11 / 49

Inverse Problems Unbounded plate Idea of the proof Technique: spherical means The steps for the proof are: 1 Factor into two Schrodinger operators Write explicitly the solution in terms of a convolution with a 2 fundamental solution of the equation. It is well known that the only solution w ∈ C ∞ ([ 0 , + ∞ ) , S ′ ( R 2 )) , supported in [ 0 , + ∞ ) , that (9) is given by satisfies w ( t , x ) = E t ∗ f , where the convolution is performed only in the spatial variable, and for t > 0, 4 πı t e ı � x � 2 1 4 t . E t ( x ) = 3 Aplication of a spherical means result A. Kawano (Poli-USP) Germain-Lagrange plates May 2014 12 / 49

Inverse Problems Rectangular Plate Plan Introduction 1 Presentation of the Germain-Lagrange Operator Presentation of the inverse problems Inverse Problems 2 Unbounded plate Rectangular Plate Plates with arbitrary shapes A. Kawano (Poli-USP) Germain-Lagrange plates May 2014 13 / 49

Inverse Problems Rectangular Plate Rectangular Plate  ρ h ∂ 2 ˜ u = g ⊗ ˜ t 2 + D △ 2 ˜ u  Q , in ] 0 , + ∞ ) × (] 0 , L 1 [ × ] 0 , L 2 [) ,   ∂ ˜ u = ∂ ˜ u ˜ t = 0 , at t = 0 , (3) ∂ ˜ � �  � ˜ �  ∂ 2 ˜  u (˜ ν 2 (˜ ∀ ˜ u γ ∂ ˜ t ) = γ ∂ ˜ t ) = 0 , t ≥ 0 . R R ∂ ˜ where g : [ 0 , + ∞ ) → R is a C 1 function with g ( 0 ) � = 0, ˜ u stands for the vertical displacement, ∂ ˜ R is the boundary of the rectangle R =] 0 , L 1 [ × ] 0 , L 2 [ , ν is the normal to ∂ ˜ ˜ R , where it is defined, and 1 R : H 1 (˜ 2 ( ∂ ˜ γ ∂ ˜ R ) → H R ) is the trace operator u �→ u | ∂ ˜ R . The constants (3) stand respectively for mass density, plate ρ, h , D > 0 that appear in thickness and flexural rigidity. A. Kawano (Poli-USP) Germain-Lagrange plates May 2014 14 / 49

Inverse Problems Rectangular Plate Data for the rectangular plate problem The set �� ˜ � � t ∈ [ 0 , ˜ c (˜ Γ a u (˜ : ˜ T a ] , ψ ∈ C ∞ Ω = t ) , ψ Ω) , (4) T a , ˜ ˜ where ˜ Ω ⊂ ˜ R is any arbitrary non-empty line segment parallel to one of the rectangle sides, which we take as being the Ox axis, containing any point ( x , y 0 ) ∈ ˜ R such that sen ( y 0 n ) � = 0, ∀ n ∈ N , is enough for the identification of ˜ Q ∈ L 2 (˜ R ) , provided that � ˜ T a > L 1 L 2 ρ h D . 4 π The set �� ∂ ˜ � � u Γ c x (˜ : ˜ t ∈ [ 0 , ˜ T c ] , ψ ∈ C ∞ T c , [ 0 , L 2 ] = t , 0 , · ) , ψ c ([ 0 , L 2 ]) , (5) ˜ ∂ ˜ where ˜ T c > 0 can be arbitrarily small, is enough for the determination of ˜ Q ∈ L 2 (˜ R ) . A. Kawano (Poli-USP) Germain-Lagrange plates May 2014 15 / 49

Inverse Problems Rectangular Plate Steps in the proof Technique: almost periodic distributions 1. Representation of the loading 2. Representation of the solution 3. Application of the data 4. The main ingredient: Almost Periodic Distributions Application of almost periodic distributions property 5. A. Kawano (Poli-USP) Germain-Lagrange plates May 2014 16 / 49

Inverse Problems Plates with arbitrary shapes Plan Introduction 1 Presentation of the Germain-Lagrange Operator Presentation of the inverse problems Inverse Problems 2 Unbounded plate Rectangular Plate Plates with arbitrary shapes A. Kawano (Poli-USP) Germain-Lagrange plates May 2014 17 / 49

Inverse Problems Plates with arbitrary shapes Plates with arbitrary shapes Let Ω ⊂ R 2 be any bounded subset with smooth and regular boundary. Let T 0 > 0 and N ∈ N . Consider the plate equation  ∂ 2 u ∂ t 2 + △ 2 u = � N   n = 1 g n ⊗ f n , in ] 0 , T 0 [ × Ω ,  (6) u = ∂ u ∂ t = 0 , at t = 0 ,    γ ∂ Ω ( u ( t )) = γ ∂ Ω ( △ u ( t )) = 0 , ∀ t ∈ [ 0 , T 0 [ , where the set { g n : n ∈ { 1 , . . . , N }} ⊂ C 1 ([ 0 , T 0 ]) is linearly independent, and f n ∈ H − 1 (Ω) , ∀ n ∈ { 1 , . . . , N } . The vertical displacement is represented by u , and γ ∂ Ω : H 1 (˜ 1 2 ( ∂ ˜ R ) → H R ) is the trace operator u �→ u | ∂ Ω . A. Kawano (Poli-USP) Germain-Lagrange plates May 2014 18 / 49

Inverse Problems Plates with arbitrary shapes Data for plates with arbitrary shapes Let O ⊂ R 2 be any open neighborhood of G + ( x ) ⊂ R 2 . Now define VG + ( x ) = O ∩ Ω . Also, define for any x 0 ∈ R 2 \ Ω and T < T 0 the set ˜ Γ T , G + ( x 0 ) ( u ) = { ( t , φ, � u ( t , · ) , φ � ) : t ∈ [ 0 , T ] , φ ∈ C ∞ c ( VG + ( x 0 )) } . (7) Observe that from ˜ Γ T , G + ( x 0 ) ( u ) it is not possible to extract information about derivatives of u with respect to the space variable at the boundary, since all φ ∈ C ∞ c ( VG + ( x 0 )) are already null near ∂ Ω . ˜ Γ T , G + ( x 0 ) ( u ) is in fact interior data. Define for any x 0 ∈ R 2 \ Ω and T < T 0 the set �� � ∂ u ( t , · ) �� � ˆ : t ∈ [ 0 , T ] , φ ∈ C ∞ Γ T , G + ( x 0 ) ( u ) = t , φ, , φ c ( G + ( x 0 )) , ∂ν (8) A. Kawano (Poli-USP) Germain-Lagrange plates May 2014 19 / 49

Inverse Problems Plates with arbitrary shapes Theorem Theorem 1 For Ω ⊂ R 2 open and bounded, given x 0 ∈ R 2 \ Ω , and any (6) , any one of the sets 0 < T < T 0 , then if u is a solution of Γ T , G + ( x 0 ) ( u ) or ˆ ˜ Γ T , G + ( x 0 ) ( u ) is enough to uniquely determine the set (6) . { f 1 , . . . , f N } ⊂ H − 1 (Ω) in A. Kawano (Poli-USP) Germain-Lagrange plates May 2014 20 / 49