Singular linear-quadratic control for semistabilization

Singular control has been extensively studied for the linear-quadratic regulator problem for achieving asymptotic stability of controlled linear systems with cheap control. In this paper, the singular control problem is extended to guarantee a weaker form of closed-loop stability, namely, semistability, which is of paramount importance for consensus control of network dynamical systems. Specifically, we show that the optimal state-feedback controller for the semistable linear-quadratic regulator problem can be solved using an algebraic Riccati equation. This result allows us to address the semistable singular control problem and obtain several expressions for the minimum cost of a semistabilizing singular controller. In particular, after establishing connections between Qui and Davison´s formula for the minimum cost of a cheap controller and our results, we prove that the cost of a semistabilizing singular controller is zero if and only if the controlled system is minimum phase and right invertible.