On the existence of optimal stabilizing controls

A rather general sufficient condition is given for the stability of linear systems that may be used to extend the limited state LQR problem, or Lyapunov's stability criterion. Note: Let
\dot{x}= Ax + Bu + D\upsilon
y=Cx
and
J= \int (\frac{1}{2}x^{T}C^{T}_{o}x + \frac{1}{2} u^{v}tRu)dt
then
u
is called the control
\upsilon
is called the influence
y
is called the output and y
o
is called the synthetic output where
y_{o}=C_{o}x
. If we choose
u=Fy
as a control we get the autonomous linear system
\dot{x}=A(F)x + D\upsilon
where
A(F)=A-BFC
. We may not choose the control to be a function of the synthetic output, nor may we in any way choose the influence.