Spatially invariant embeddings of systems with boundaries

We consider certain spatially distributed optimal control problems where the spatial domains are finite intervals with boundaries. The optimal control design procedures for spatially invariant systems are normally not applicable to such bounded spatial domains. For problems that possess certain symmetries, we show how to apply spatially invariant techniques using embeddings. In this note, we report on such “embeddable” problems. As an application, we consider LQR problems for systems posed as PDEs on finite intervals, where it will turn out that the solution for the finite-extent system equals that of its spatially invariant counterpart plus a term that corrects for the boundary conditions. We also show that this decomposition can be understood as a Toeplitz plus Hankel decomposition of the state feedback gain operator, with the Toeplitz part governing the feedback in the interior domain, while the Hankel part provides the needed corrections near the boundaries.