Riccati equations for second order spatially invariant partial differential systems

Recently, the class of spatially invariant systems was introduced with motivating examples of partial differential equations on an infinite domain. For these it was shown that by taking Fourier transforms, one obtains infinitely many finite-dimensional systems with a scalar parameter. The idea is that, for the LQR controller design for these systems, one can solve the parameterized LQR-Riccati equation pointwise. While for simple first order systems like the heat equations this approach works, for second order systems like wave or beam equations it is easy to construct examples for which this approach fails. Here we give a correct formulation for second order partial differential systems including wave and beam type equations.