Adaptive control of an anti-stable wave PDE.

*(English)*Zbl 1219.93055Summary: Adaptive control of PDEs is a problem of nonlinear dynamic feedback design for an infinite-dimensional system. The problem is nonlinear even when the PDE is linear. Past papers on adaptive control of unstable PDEs with unmatched parametric uncertainties have considered only parabolic PDEs and first-order hyperbolic PDEs. In this paper we introduce several tools for approaching adaptive control problems of second-order-in-time PDEs. We present these tools through a benchmark example of an unstable wave equation with an unmatched (non-collocated) anti-damping term, which serves both as a source of instability and of parametric uncertainty. This plant has infinitely many eigenvalues arbitrarily far to the right of the imaginary axis and they reside on a vertical line whose position is completely unknown. The key effort in the design is to avoid the appearance of the second time derivative of the parameter estimate in the error system.