Weighted least squares stationary approximations to linear systems

The problem of replacing the time-varying linear system
\dot{X} = A(t)X
by a stationary one
\dot{Y} = BY
is investigated. The matrix
B
is selected so that
X(t) = Y(t)
in the interval [
0, T
]. Several quadratic criteria are proposed to aid in determining suitable candidate systems. One criterion for choosing
B
is initial condition dependent, and another bounds the "worst case" homogeneous system performance. Both of these criteria produce weighted least squares fits to
A(t)
.