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

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

Universidad Politécnica de Cartagena

THE ONE-DIMENSIONAL WAVE EQUATION the one-dimensional wave equation

CONVERGENCE OF THE ENERGY the one-dimensional wave equation

The convergence of the energy holds whenever

FIRST REMARK ON HOMOGENIZATION the one-dimensional wave equation

Remark 1 (Convergence of the conormal derivatives)

IDEA OF THE PROOF the one-dimensional wave equation

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

UNIFORM EXACT CONTROLLABILITY the one-dimensional wave equation

Yes

Enrique Fernández-Cara, Enrique Zuazua (2001)

No

HOMOGENIZATION the one-dimensional wave equation

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

HOMOGENIZATION the one-dimensional wave equation

CURES FOR THIS BAD BEHAVIOUR! the one-dimensional wave equation

*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)

INITIAL DATA the one-dimensional wave equation

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.

A CONTROLLABILITY RESULT the one-dimensional wave equation

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

OPEN PROBLEM the one-dimensional wave equation

To identify the class of non-resonant initial data

INTERNAL FEEDBACK CONTROL the one-dimensional wave equation

Result

IDEA OF THE PROOF the one-dimensional wave equation

IDEA OF THE PROOF the one-dimensional wave equation

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

AN EXAMPLE the one-dimensional wave equation

SECOND REMARK ON HOMOGENIZATION the one-dimensional wave equation

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

A SHORT COURSE ON YOUNG MEASURES the one-dimensional wave equation

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

Definition

SECOND REMARK ON HOMOGENIZATION the one-dimensional wave equation

Goal: to compute the Young Measure associated with

SECOND REMARK ON HOMOGENIZATION the one-dimensional wave equation

Remark 2

Proof = corrector + properties of Young measures

INTERNAL EXACT CONTROLLABILITY the one-dimensional wave equation

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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