Trajectory insensitivity of optimal feedback systems

It is shown that for a single-input linear time-invariant plant in phase-variable canonic form, such that the control minimizes a quadratic performance index, the integral of a quadratic form of the state sensitivity using feedback control is less than the corresponding one using open-loop control, for any arbitrary positive semidefinite nonzero weighting matrix. Classes of state representations, which are not in phase-variable canonic form for which the relative insensitivity holds, are given.