On the existence of optimal solutions for the linear system quadratic cost problem

The linear system
\dot{x} = Ax +bu
with the quadratic cost function
\int\min{0}\max{\infty}(x'Hx+u^{2})dt
is considered. The following equivalent conditions are shown to be necessary and sufficient for the existence of a solution to the optimization problem: 1) the existence of a positive definite solution for the algebraic matrix Riccati equation; 2) the existence of a vector
k
such that the matrix
A-bk'
is asymptotically stables and 3) the cancellations in the vector
(sI-A)^{-1}b
are only of stable factors.