The general nonstrict algebraic Riccati inequality

Abstract

If (A, B) is stabilizable, it is well known how to verify the solvability or the existence of positive definite solutions of the nonstrict algebraic Riccati inequality (ARI) $\text{A∗X + XA − XBB∗X + Q ≥0}$. This paper serves to demonstrate how far it is presently possible to generalize these results if the uncontrollable modes of (A, B) are not restricted. We prove a new reduction principle which may be formulated as follows: The ARI has a solution iff a certain reduced-order Riccati equation (for a stabilizable system), a linear equation, and a reduced-order nonstrict Lyapunov inequality (corresponding to the uncontrollable modes on the imaginary axis) have solutions. Basically, the quadratic Riccati inequality is hence reduced to a linear Lyapunov inequality. We discuss how this principle may be applied to actually check the solvability of the ARI. As the main motivation for our work we briefly explain the consequences of these results for the general state-feedback H_∞-optimization problem.