The optimal linear-quadratic time-invariant regulator with cheap control

The infinite-time linear-quadratic regulator is considered as the weighting on the control energy tends to zero (cheap control). First, a study is made of the qualitative behavior of the limiting optimal state and control trajectories. In particular, the orders of initial singularity are found and related to the excess of poles over zeros in the plant. Secondly, it is found for which initial conditions the limiting minimum cost is zero (perfect regulation). This generalizes an earlier result of Kwakernaak and Sivan. Finally, a simple extension is made to the steady-state LQG problem with cheap control and accurate observations.