Families of linear-quadratic problems: Continuity properties

In this paper we investigate for given one-parameter families of linear time-invariant finite-dimensional systems the parameter dependence of the linear-quadratic optimal cost, optimal control inputs, optimal state-trajectories, and optimal outputs. It is shown that results that have been obtained in the past in the context of the problem of cheap control can in fact be generalized to a much broader class of parameter-dependent cost-functionals, including cost functionals in which for every parameter value the weighting matrix of the control inputs is singular. Essentially, only two assumptions on the parameter dependence of the cost-functionals are required in order to have continuity of the optimal cost and optimal control inputs with respect to the underlying parameter. One assumption is concerned with the continuity of the weighting matrices with respect to this parameter, the other with the monotonicy of the weighting matrices with respect to the parameter. Instrumental in our development is a characterization of the linear-quadratic optimal cost in terms of the so-called dissipation inequality. The results obtained are applied to the problem of cheap control and to a problem of priority control. The latter provides an example of a family of quadratic cosl-functionals with a polynomial parameter dependence.