Quarterly of Applied Mathematics

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

Author(s): Pablo Pedregal; Francisco Periago
Journal: Quart. Appl. Math. 64 (2006), 529-546.
MSC (2000): Primary 35B27, 35L05, 93B05
Posted: June 15, 2006
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Abstract: This paper contains three results concerning the homogenization and exact controllability for the one-dimensional wave equation. First, we give sufficient conditions on the initial data to ensure the convergence of the conormal derivatives associated with the wave equation with a rapidly oscillating coefficient and zero Dirichlet boundary conditions. Secondly, we apply this result to prove the existence of a class of initial data whose associated boundary controls are uniformly bounded and obtain some information (in particular, its limit behavior) on this class of data. Finally, we prove that all initial data in $L^2\times H^{-1}$ may be uniformly controlled but at the price of adding an internal feedback control in our system. The main advantage of this last procedure is that we have explicit formulae for both states and controls.

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Pablo Pedregal
Affiliation: Departamento de Matemáticas, ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain
Email: pablo.pedregal@uclm.es

Francisco Periago
Affiliation: Departamento de Matemática Aplicada y Estadística, ETSI Industriales, Universidad Politécnica de Cartagena, 30203 Cartagena, Spain
Email: f.periago@upct.es

PII: S0033-569X-06-01022-4
Received by editor(s): November 30, 2005
Posted: June 15, 2006
Additional Notes: The first author was supported by project MTM2004-07114 from Ministerio de Educación y Ciencia (Spain) and PAI05-029 from JCCM (Castilla-La Mancha, Spain)
The second author was supported by projects MTM2004-07114 from Ministerio de Educación y Ciencia (Spain) and 00675/PI/04 from Fundación Séneca (Murcia, Spain)
Copyright of article: Copyright 2006, Brown University

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