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 Enrique Fernández-Cara, Enrique Zuazua (2001) No

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. 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 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 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