New equations for the time-dependent regulator problem

This correspondence presents a derivation of an alternate set of equations for determination of the optimal feedback gain matrix for the time-dependent linear regulator problem. In contrast to the usual Riccati equation approach, the new system is composed of only
nm
equations, where
m
is the number of systems input terminals. As payment for this reduction in number of equations, however, the new system is not entirely general, being restricted to constant control matrices
G
, with arbitrary dynamics and cost matrices
F
and
Q
, respectively.