The quadratic matrix inequality in singular $H_\infty$ control with state feedback

In this paper the standard $H_\infty $ control problem using state feedback is considered. Given a linear, time-invariant, finite-dimensional system, this problem consists of finding a static state feedback such that the resulting closed-loop transfer matrix has $H_\infty $ norm smaller than some a priori given upper bound. In addition it is required that the closed-loop system is internally stable. Conditions for the existence of a suitable state feedback are formulated in terms of a quadratic matrix inequality, reminiscent of the dissipation inequality of singular linear quadratic optimal control. Where the direct feedthrough matrix of the control input is injective, the results presented here specialize to known results in terms of solvability of a certain indefinite algebraic Riccati equation.