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On aspects of stabilization and feedback control of partial differential equations

In this talk feedback control and stabilization problems for the wave equation and the viscous Burgers equation are considered. In the first part we consider an optimal feedback control problem for the wave equation. For a semi-discrete formulation of the problem a feedback law is derived from the dynamic programming principle which requires to solve a Hamilton-Jacobi-Bellman (HJB) equation. Because of the curse of dimensionality, classical discretization methods based on finite elements make the numerical resolution by the HJB approach infeasible. Therefore, an approximation based on spectral elements is used to discretize the wave equation. The effect of noise is considered and numerical simulations are presented to show the relevance of the approach. Furthermore, an extension on sparse grids is discussed. In the second part of the talk a feedback stabilization problem of the viscous Burgers equation to a nonstationary trajectory is considered. The feedback law is derived from a differential Riccati-like equation. Estimates for the dimension of the finite-dimensional internal controller are derived and numerical examples are presented.

CMAP UMR 7641 École Polytechnique CNRS, Route de Saclay, 91128 Palaiseau Cedex France, Tél: +33 1 69 33 46 23 Fax: +33 1 69 33 46 46