Redundancy in linear optimum regulator problem

The optimum linear regulator problem is the determination of the input to \dot{x} = Fx + Gu that minimizes \int \liminf{0} \limsup{\infty} x^{T}Qx + u^{T}u dt , and the well-known solution is a feedback law u = K^{T}x . It is known that the problem statement is redundant, in that distinct matrices Q 1 and Q 2 can yield the same feedback law K . Such matrices are called equivalent, and a simple test for equivalence is available for the single-input case. This note generalizes the test to the multivariable case.