The stable regulator problem and its inverse

This paper provides a detailed survey and extension of certain properties of the stable regulator problem: determine
\min\inf{u} \int\min{0}\max{\infty} x'Qx + u'u dt
subject to
\dot{x} = Fx + Gu; x(0) = x_{0}; \liminf{t \rightarrow \infty} x(t) = 0
where
Q
is not necessarily sign definite. First, equivalence conditions recently given by Willems for the existence of the minimum are extended to include statements in terms of the Hamiltonian matrix and spectral factorization. This provides a precise relation between the time-domain and frequency-domain solutions to the problems. Second, the inverse problem of whether a given feedback
u = -Kx
is optimal for some
Q
is easily resolved, as is the redundancy problem of distinct Q
1
and Q
2
, resulting in the same optimal control.