On the existence of optimal solutions for the linear system quadratic cost problem
Raphael Sivan, S. Shalvi · IEEE Transactions on Automatic Control · 1968
The linear system\dot{x} = Ax +buwith the quadratic cost function\int\min{0}\max{\infty}(x'Hx+u^{2})dtis 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 vectorksuch that the matrixA-bk'is asymptotically stables and 3) the cancellations in the vector(sI-A)^{-1}bare only of stable factors.