Efficiency and optimality in constrained variance control

For the class of linear discrete systems with square nonsingular disturbance matrix we consider the constrained variance control problem in which there are constraints on the magnitudes of each of the state variances. This is a special case of a generalized constrained cost problem. Assuming that a solution exists to the constrained cost problem, multiobjective optimization theory is used to develop sufficient conditions for this probieal to have an efficient (nondominated, Pareto-optimal) solution and for any efficient solution to solve an associated scalar optimiaization problem, it is then proved that these sufficient conditions are satisfied for the constrained variance problem. Therefore. if a solution exists to the constrained variance problem, then there exists a solution which also solves an optional linear quadratic problem which weights only the state variances.