1. 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

2. THE ONE-DIMENSIONAL WAVE EQUATION

3. CONVERGENCE OF THE ENERGY
The convergence of the energy holds whenever

4. FIRST REMARK ON HOMOGENIZATION
Remark 1 (Convergence of the conormal derivatives)

5. IDEA OF THE PROOF
S. Brahim-Otsmane, G. Francfort and F. Murat (1992)

6. UNIFORM EXACT CONTROLLABILITY
Yes
Enrique Fernández-Cara, Enrique Zuazua (2001) No

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

8. HOMOGENIZATION

9. CURES FOR THIS BAD BEHAVIOUR!
* C. Castro, 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)

10. 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.

11. A CONTROLLABILITY RESULT
As a result of the convergence of the conormal derivatives we have:

12. OPEN PROBLEM
To identify the class of non-resonant initial data

13. INTERNAL FEEDBACK CONTROL
Result

14. IDEA OF THE PROOF

15. IDEA OF THE PROOF
The main advantage of this approach is that we have explicit formulae for both state and controls

16. AN EXAMPLE

17. 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

18. A SHORT COURSE ON YOUNG MEASURES
Existence Theorem (L. C. Young ’40 – J. M. Ball ’89)
Definition

19. SECOND REMARK ON HOMOGENIZATION
Goal: to compute the Young Measure associated with

20. SECOND REMARK ON HOMOGENIZATION
Proof = corrector + properties of Young measures

21. 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