Riccati equations and LQ-optimal control for a class of hyperbolic PDEs
Anthony Hastir, Birgit Jacob, Hans Zwart
TL;DR
This work advances LQ-optimal control for a class of boundary-controlled hyperbolic PDEs with unbounded input/output by deriving an explicit solution to the operator-node Riccati equation via a system-node representation. It connects finite-dimensional CAREs to infinite-dimensional operator Riccati equations and develops a Weiss--Weiss spectral-factorization framework, highlighting that the standard $(I+D^*D)^{-1}$ term must be replaced by $(\Omega^*\Omega)^{-1}$ with $\Omega=P^{1/2}$ to handle unbounded operators. An explicit boundary-transport example demonstrates that the extended Riccati formulation is necessary to correctly recover the LQ-optimal feedback, linking the operator-node solution to a scalar CARE in the example. The results provide a concrete pathway to closed-form LQ control for a broad class of hyperbolic PDEs and clarify the role of spectral factorization and Popov coercivity in infinite-dimensional settings, with avenues for generalization and dual Kalman-filter-type problems.
Abstract
We derive an explicit solution to the operator Riccati equation solving the Linear-Quadratic (LQ) optimal control problem for a class of boundary controlled hyperbolic partial differential equations (PDEs). Different descriptions of the system are used to obtain different representations of the operator Riccati equation. By means of an example, we illustrate the importance of considering an extended operator Riccati equation to solve the LQ-optimal control problem for our class of systems.