A geometric characterization of the slow space of the Hamiltonian system arising from the singular LQR problem

In this paper we first characterize the slow space of a given state-space system. We provide this characterization in terms of an eigenspace of the corresponding Rosenbrock matrix pair. We also characterize the “good” slow space in terms of a stable eigenspace of the Rosenbrock matrix pair. Moreover, we show how the dimensions of these subspaces can be calculated from the determinant of the Rosenbrock matrix pencil. Then, we apply these results to the Hamiltonian system arising from the singular linear quadratic regulator (LQR) problem and explore a few interesting properties of the good slow space of this Hamiltonian system. Finally, we provide a feedback law to achieve the smooth optimal solutions.