Localization of the LQR Feedback Kernel in Spatially-Invariant Problems over Sobolev Spaces
We consider the LQR controller design problem for spatially-invariant systems on the real line where the state space is a Sobolev space. Such problems arise when dealing with systems describing wave or beam-bending motion. We demonstrate that the optimal state feedback is a spatial convolution operator with an exponentially decaying kernel, enabling implementation with a localized architecture. We generalize analogous results for the L2 setting and provide a rigorous explanation of numerical results previously observed in the Sobolev space setting. The main tool utilized is a transformation from a Sobolev to an L2 space, which is constructed from a spectral factorization of the spatial frequency weighting matrix of the Sobolev norm. We show the equivalence of the two problems in terms of the solvability conditions of the LQR problem. As a case study, we analyze the wave equation; we provide analytical expressions for the dependence of the decay rate of the optimal LQR feedback convolution kernel on wave speed and the LQR cost weights.