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

THE ONE-DIMENSIONAL WAVE EQUATION

CONVERGENCE OF THE ENERGY

The convergence of the energy holds whenever

FIRST REMARK ON HOMOGENIZATION

Remark 1 (Convergence of the conormal derivatives)

IDEA OF THE PROOF

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

UNIFORM EXACT CONTROLLABILITY

Yes

No

Enrique Fernández-Cara, Enrique Zuazua (2001)

HOMOGENIZATION

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

HOMOGENIZATION

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.

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)

Other interesting questions to analyze are

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.

A CONTROLLABILITY RESULT

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

OPEN PROBLEM

To identify the class of non-resonant initial data

INTERNAL FEEDBACK CONTROL

Result

IDEA OF THE PROOF

IDEA OF THE PROOF

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

AN EXAMPLE

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

A SHORT COURSE ON YOUNG MEASURES

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

Definition

SECOND REMARK ON HOMOGENIZATION

Goal: to compute the Young Measure associated with

SECOND REMARK ON HOMOGENIZATION

Remark 2

Proof = corrector + properties of Young measures

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