Indirect controller design for non-parabolic partial differential equations

There is by a well-known theory of approximation of PDEs where the semigroup is analytic.
However, this assumption is not always satisfied. Of particular interest are a number of
applications involving wave propagation. There are many different approximation methods
for these systems that are fine for simulation. But the qualitative behaviour of the
approximated eigenvalues can be quite different. Controller and estimator design with
different approximations can yield different results. Of particular interest is optimal linear
quadratic control of systems that may be only asymptotically stabilizable. For linear systems,
this issue only arises with infinite-dimensional systems. Sufficient conditions which
guarantee when approximations to the optimal feedback result in the cost converging to the
optimal cost have recently been obtained.