Consider a finite-dimensional stable time-invariant system described by \[ i(t)=[A+\sum^ \infty_{k=1}N_ kv_ k(t)]x(t)+Bu(t)+Dw(t) \]
\[ y(t)=Cx(t), E[w(t)w(r)^ T]=W\delta (t-r), W>0, \] \(w(t)\), \(v_ n(t)\) independent, zero-mean Gaussian white noise. Assume the control law to be of the form \(u(t)=Gx(t)\). The authors give necessary and sufficient conditions for the existence of a feedback gain matrix \(G\) assigning a specified value \(\overline X\) to the state covariance
\(X=\lim_{t\to \infty,r\to 0}E[x(t)x(t)^ T]\). A set of feedback gain matrices is found such that the closed-loop system achieves the assigned \(\overline X\), and a methodology of constrained variance designing is developed.