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Sensors and boundary state reconstruction of hyperbolic systems

Zerrik E. H., Bourray H., Ben Hadid S.

International Journal of Applied Mathematics and Computer Science

2010

Vol. 20, no 2

227-238

This paper deals with the problem of regional observability of hyperbolic systems in the case where the subregion of interest is a boundary part of the system evolution domain. We give a definition and establish characterizations in connection with the sensor structure. Then we show that it is possible to reconstruct the system state on a subregion of the boundary. The developed approach, based on the Hilbert uniqueness method (Lions, 1988), leads to a reconstruction algorithm. The obtained results are illustrated with numerical examples and simulations.

An output controllability problem for semilinear distributed hyperbolic systems

Zerrik E., Larhrissi R., Bourray H.

International Journal of Applied Mathematics and Computer Science

2007

Vol. 17, no 4

437-446

The paper aims at extending the notion of regional controllability developed for linear systems to the semilinear hyperbolic case. We begin with an asymptotically linear system and the approach is based on an extension of the Hilbert uniqueness method and Schauder’s fixed point theorem. The analytical case is then tackled using generalized inverse techniques and converted to a fixed point problem leading to an algorithm which is successfully implemented numerically and illustrated with examples.