Simultaneous Identification of Source, Initial Conditions and Asynchronous Sources in the Vibration Problem of Euler-Bernoulli Beams Alexandre Kawano University of S˜ ao Paulo (Brazil) November – 2017 Alexandre Kawano (University of S˜ ao Paulo)Simultaneous Identification of Source, Initial Conditions and Asynchronous Sources in the Vib November – 2017 1 / 41
Abstract In this article we show under what conditions it is possible to uniquely identify simultaneously the source and initial conditions in a vibrating Euler-Bernoulli beam, when the available data is the observation of the displacement of a point during an arbitrary small interval of time. A counterexample is also shown to indicate that if some conditions are not satisfied then the unique identification is impossible. Alexandre Kawano (University of S˜ ao Paulo)Simultaneous Identification of Source, Initial Conditions and Asynchronous Sources in the Vib November – 2017 2 / 41
Introduction The equation The equation that appears in this work is the Euler-Bernoulli equation that describes the motion of an elastic beam under dynamic loading. Alexandre Kawano (University of S˜ ao Paulo)Simultaneous Identification of Source, Initial Conditions and Asynchronous Sources in the Vib November – 2017 3 / 41
Introduction The problem � � = � J ρ ∂ 2 w ∂ t 2 + ∂ 2 EI ∂ 2 w j =1 g j ⊗ f j , in ]0 , T 0 [ × ]0 , L [ , ∂ x 2 ∂ x 2 w (0) = w 0 , in ]0 , L [ , ∂ w ∂ t (0 , x ) = v 0 , in ]0 , L [ , w ( t , ξ ) = ∂ w ∂ x ( t , ξ ) = 0 , ∀ t ∈ [0 , T 0 [ , ∀ ξ ∈ { 0 , L } , (1) where ρ ∈ C ∞ ([0 , L ]), ρ > 0, is the mass density, EI ∈ C ∞ ([0 , L ]), EI > 0, is the rigidity, { g 1 , g 2 , . . . , g J } ⊂ C J [0 , T 0 [ is such that · · · 1 g 1 (0) g J (0) 0 g ′ 1 (0) · · · g ′ J (0) [ G (0)] = . . . . (2) . . . . . . . . g ( J ) g ( J ) 0 1 (0) · · · (0) J is invertible. The set of functions { f 1 , . . . , f J } ⊂ H − 2 (]0 , L [) describe the spatial loading imposed to the beam and w is the displacement. Alexandre Kawano (University of S˜ ao Paulo)Simultaneous Identification of Source, Initial Conditions and Asynchronous Sources in the Vib November – 2017 4 / 41
Introduction The problem We will prove that if the initial velocity is known, the force spatial distribution { f 1 , . . . , f J } ⊂ H − 2 (]0 , L [) and the initial position can be simultaneously identified uniquely given the knowledge of the set Γ = { ( w ( t , x ) : ( t , x ) ∈ [0 , T ] × Ω 0 } , (3) where 0 < T < T 0 and Ω 0 ⊂ [0 , L ], non empty open set, can be arbitrarily small. Furthermore, the initial velocity v 0 ∈ L 2 (]0 , L [) can also be uniquely identified along with the initial position w 0 ∈ L 2 (]0 , L [) and the forcing terms { f 1 , . . . , f J } if it is also available the final velocity (knowledge of the displacement profile is not necessary) of the beam at T 0 and the data (3). Alexandre Kawano (University of S˜ ao Paulo)Simultaneous Identification of Source, Initial Conditions and Asynchronous Sources in the Vib November – 2017 5 / 41
Introduction Counter example Let u ( t , x ) = h ( t ) ϕ ( x ). Then it automatically satisfies � � � � ρ∂ 2 u ∂ t 2 + ∂ 2 EI ∂ 2 u = ∂ 2 h + h ∂ 2 EI ∂ 2 ϕ ∂ t 2 ρϕ . ∂ x 2 ∂ x 2 ∂ x 2 ∂ x 2 � �� � � �� � g 1 f 1 g 2 f 2 The initial conditions are � u (0 , x ) = h (0) ϕ ( x ) , ∀ x ∈ ]0 , L ] , u t (0 , x ) = h ′ (0) ϕ ( x ) , ∀ x ∈ ]0 , L ] . Consider a situation in which h (0) � = 0, h ′ (0) � = 0, ϕ �≡ 0, but ϕ | Ω 0 = 0. In this case, the forcing terms f 1 , f 2 and the initial conditions are not null, but w | [0 , T ] × Ω 0 = 0. That is, the data (3) is insufficient to fix uniquely the loading { f 1 , f 2 } . Alexandre Kawano (University of S˜ ao Paulo)Simultaneous Identification of Source, Initial Conditions and Asynchronous Sources in the Vib November – 2017 6 / 41
Introduction However, we are going to see that if the initial position is null, the set of functions g 1 , g 2 satisfy a certain condition and w | [0 , T ] × Ω 0 = 0, then necessarily f 1 = 0 and f 2 = 0. Alexandre Kawano (University of S˜ ao Paulo)Simultaneous Identification of Source, Initial Conditions and Asynchronous Sources in the Vib November – 2017 7 / 41
The direct problem Solution of the direct problem Associated eigenproblem Consider the eigenvalue problem for S n ∈ H = H 2 0 (]0 , L [): � � � 1 ∂ 2 EI ∂ 2 S = λ n S , in ]0 , L [ , ρ ∂ x 2 ∂ x 2 (4) S (0) = S ( L ) = S ′ (0) = S ′ ( L ) = 0 . Alexandre Kawano (University of S˜ ao Paulo)Simultaneous Identification of Source, Initial Conditions and Asynchronous Sources in the Vib November – 2017 8 / 41
The direct problem Solution of the direct problem Associated eigenproblem With respect to the internal product, � EI � ∂ 2 φ 1 ∂ x 2 , ∂ 2 φ 2 � φ 1 , φ 2 � H = , (5) ρ ∂ x 2 L 2 ρ (0 , L ) where � L ( φ 1 , φ 2 ) L 2 ρ (0 , L ) = ρ ( x ) φ 1 ( x ) φ 2 ( x ) d x , 0 � � the operator φ T ∂ 2 EI ∂ 2 φ �→ 1 is self adjoint. Then the set of ∂ x 2 ∂ x 2 ρ eigenvectors of this problem forms an enumerable orthonormal basis ( S n ) n ∈ N of H that is also orthogonal in L 2 ρ (0 , L ). Furthermore, O ( λ n ) = n 4 , S n ∈ C ∞ ([0 , L ]), λ n > 0 and ( S n , S n ) L 2 ρ (0 , L ) = 1 /λ n , ∀ n ∈ N . (6) Alexandre Kawano (University of S˜ ao Paulo)Simultaneous Identification of Source, Initial Conditions and Asynchronous Sources in the Vib November – 2017 9 / 41
The direct problem Solution of the direct problem Elements of the dual of H Any Q ∈ H ∗ , can be expressed as � Q = β n λ n S n , (7) n ∈ N where ( β n ) n ∈ N ∈ ℓ 2 . In fact, by Riez Theorem there is a � n ∈ N β n S n ∈ H , with ( β n ) n ∈ N ∈ ℓ 2 , such that � � Q ( φ ) = � φ , β n S n � H = ( φ, λ n β n S n ) L 2 ρ . n ∈ N n ∈ N Then for any Q ∈ H ∗ there is a sequence ( β n ) n ∈ N ∈ ℓ 2 such that Q = � n ∈ N λ n β n S n . Alexandre Kawano (University of S˜ ao Paulo)Simultaneous Identification of Source, Initial Conditions and Asynchronous Sources in the Vib November – 2017 10 / 41
The direct problem Solution of the direct problem Elements of the dual of H Any component of the spatial force distribution f j ∈ H ∗ , j ∈ { 1 , . . . J } can be expressed as � f j ρ = A j , n λ n S n , n ∈ N for ( A j , n ) ∈ ℓ 2 . Alexandre Kawano (University of S˜ ao Paulo)Simultaneous Identification of Source, Initial Conditions and Asynchronous Sources in the Vib November – 2017 11 / 41
The direct problem Solution of the direct problem Elements of the dual of H The initial position w 0 ∈ L 2 (]0 , L [) and the initial velocity v 0 ∈ L 2 (]0 , L [) are represented respectively by � � � � w 0 = W n λ n S n , v 0 = V n λ n S n , n ∈ N n ∈ N where W n = ( w 0 , √ λ n S n ) L 2 ρ . The expression for ( V n ) n ∈ N ∈ ℓ 2 ( N ) is analogous. Alexandre Kawano (University of S˜ ao Paulo)Simultaneous Identification of Source, Initial Conditions and Asynchronous Sources in the Vib November – 2017 12 / 41
The direct problem Solution of the direct problem The solution Using the Galerkin Method, we get a formal solution of (1) given by � � � � � w ( t , x ) = V n sin( λ n t ) S n ( x ) + W n λ n cos( λ n t ) S n ( x ) n ∈ N n ∈ N � t (8) J � � � � + g j ( t − τ ) λ n sin( λ n τ ) S n ( x ) d τ. A j , n 0 j =1 n ∈ N By substitution we can see that (8) is a solution of the first equation of problem (1) in the sense of distributions. Note that al initial and boundary conditions are satisfied. Alexandre Kawano (University of S˜ ao Paulo)Simultaneous Identification of Source, Initial Conditions and Asynchronous Sources in the Vib November – 2017 13 / 41
The direct problem Solution of the direct problem Uniqueness for the direct problem From the fact that ( S n ) n ∈ N forms an orthonormal basis in H and � S n � 2 ρ = 1 /λ n , ∀ n ∈ N , we obtain the following proposition. L 2 Proposition w ∈ C ([0 , T 0 ] , H 2 (]0 , L [)) ∩ C 1 ([0 , T 0 ] , H ∗ ) . Besides, w (0) , w ′ (0) ∈ L 2 ρ (0 , L ) . Using a method analogous to the energy method applied to the wave equation found, for example, in [Evans(1991)], we can see that (1) admits at most one solution in C ([0 , T 0 ] , H 2 (]0 , L [)) ∩ C 1 ([0 , T 0 ] , H ∗ ). Alexandre Kawano (University of S˜ ao Paulo)Simultaneous Identification of Source, Initial Conditions and Asynchronous Sources in the Vib November – 2017 14 / 41
Inverse problem Preparation for the solution of the inverse problem Rewritting the solution of the direct problem The solution (8) can be rewritten in another form as � � w ( t , x ) = λ n S n ( x ) W n n ∈ N � t � � � � 1 × ( λ n τ ) S n ( x ) − W n λ n sin( + V n λ n cos( λ n τ ) S n ( x )) d τ 0 n ∈ N � t � J � � � g j ( t − τ ) + A j , n λ n sin( λ n τ ) S n ( x ) d τ. 0 j =1 n ∈ N Alexandre Kawano (University of S˜ ao Paulo)Simultaneous Identification of Source, Initial Conditions and Asynchronous Sources in the Vib November – 2017 15 / 41
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