Adaptive control of an anti-stable wave PDE

Past papers on adaptive control of unstable PDEs with unmatched parametric uncertainties have considered only parabolic PDEs and first-order hyperbolic PDEs. In this note 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. The key effort in the design is to avoid the appearance of the second time derivative of the parameter estimate in the error system.