Some remarks on homogenization and exact controllability for the one-dimensional wave equation
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Some remarks on homogenization and exact controllability for the one-dimensional wave equation. Pablo Pedregal Depto. Matemáticas, ETSI Industriales Universidad de Castilla- La Mancha. Francisco Periago Depto. Matemática Aplicada y Estadística, ETSI Industriales

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Some remarks on homogenization and exact controllability for the one-dimensional wave equation

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Some remarks on homogenization and exact controllability for the one dimensional wave equation

Some remarks on homogenization and exact controllability for the one-dimensional wave equation

Pablo Pedregal

Depto. Matemáticas, ETSI Industriales

Universidad de Castilla- La Mancha

Francisco Periago

Depto. Matemática Aplicada y Estadística, ETSI Industriales

Universidad Politécnica de Cartagena


Some remarks on homogenization and exact controllability for the one dimensional wave equation

THE ONE-DIMENSIONAL WAVE EQUATION


Some remarks on homogenization and exact controllability for the one dimensional wave equation

CONVERGENCE OF THE ENERGY

The convergence of the energy holds whenever


Some remarks on homogenization and exact controllability for the one dimensional wave equation

FIRST REMARK ON HOMOGENIZATION

Remark 1 (Convergence of the conormal derivatives)


Some remarks on homogenization and exact controllability for the one dimensional wave equation

IDEA OF THE PROOF

S. Brahim-Otsmane, G. Francfort and F. Murat (1992)


Some remarks on homogenization and exact controllability for the one dimensional wave equation

UNIFORM EXACT CONTROLLABILITY

Yes

Enrique Fernández-Cara, Enrique Zuazua (2001)

No


Some remarks on homogenization and exact controllability for the one dimensional wave equation

HOMOGENIZATION

M. Avellaneda, C. Bardos and J. Rauch (1992)


Some remarks on homogenization and exact controllability for the one dimensional wave equation

HOMOGENIZATION


Some remarks on homogenization and exact controllability for the one dimensional wave equation

CURES FOR THIS BAD BEHAVIOUR!

*C. Castro, 1999. Uniform exact controllability and convergence of controls for the projection of the solutions over the subspaces generated by the eigenfunctions corresponding to low (and high) frequencies.

Other interesting questions to analyze are

1.To identify, if there exists, the class of non-resonant initial data

2.If we wish to control all the initial data, then we must add more control elements on the system (for instance, in the form of an internal control)


Some remarks on homogenization and exact controllability for the one dimensional wave equation

INITIAL DATA

We have found a class of initial data of the adjoint system for which there is convergence of the cononormal derivatives. This gives us a class of non-resonant initial data for the control system.


Some remarks on homogenization and exact controllability for the one dimensional wave equation

A CONTROLLABILITY RESULT

As a result of the convergence of the conormal derivatives we have:


Some remarks on homogenization and exact controllability for the one dimensional wave equation

OPEN PROBLEM

To identify the class of non-resonant initial data


Some remarks on homogenization and exact controllability for the one dimensional wave equation

INTERNAL FEEDBACK CONTROL

Result


Some remarks on homogenization and exact controllability for the one dimensional wave equation

IDEA OF THE PROOF


Some remarks on homogenization and exact controllability for the one dimensional wave equation

IDEA OF THE PROOF

The main advantage of this approach is that we have explicit formulae for both state and controls


Some remarks on homogenization and exact controllability for the one dimensional wave equation

AN EXAMPLE


Some remarks on homogenization and exact controllability for the one dimensional wave equation

SECOND REMARK ON HOMOGENIZATION

The above limit may be represented through the Young Measure associated with the gradient of the solution of the wave equation


Some remarks on homogenization and exact controllability for the one dimensional wave equation

A SHORT COURSE ON YOUNG MEASURES

Existence Theorem (L. C. Young ’40 – J. M. Ball ’89)

Definition


Some remarks on homogenization and exact controllability for the one dimensional wave equation

SECOND REMARK ON HOMOGENIZATION

Goal: to compute the Young Measure associated with


Some remarks on homogenization and exact controllability for the one dimensional wave equation

SECOND REMARK ON HOMOGENIZATION

Remark 2

Proof = corrector + properties of Young measures


Some remarks on homogenization and exact controllability for the one dimensional wave equation

INTERNAL EXACT CONTROLLABILITY

J. L. Lions proved that

As a consequence of the computation of the Young measure,

which shows that the limit of the strain of the oscillating system is greater than the strain of the limit system


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