Residual bounds of approximate solutions of the algebraic Riccati equation

Summary.

Let $\tilde{X} \geq 0$ approximate the unique Hermitian positive semi-definite solution $X$ to the algebraic Riccati equation (ARE) \[G+A^{\rm H}X+XA-XFX=0,\] where $F, G \geq 0$, $(A,F)$ is stabilizable, and $(A,G)$ is detectable. Let \[\hat{R}=G+A^{\rm H}\tilde{X}+\tilde{X}A-\tilde{X}F\tilde{X} \] be the residual of the ARE with respect to $\tilde{X}$, and define the linear operator $\vec L$ by \[ \vec LH=(A-F\tilde{X})^{\rm H}H+H(A-F\tilde{X}),\;\;\;\;\;\; H=H^{\rm H} \in {\C}^{n \times n}. \] By applying a new forward perturbation bound to the optimal backward perturbation corresponding to the approximate solution $\tilde{X}$, we obtained the following result: If $A-F\tilde{X}$ is stable, and if $4\|\vec L^{-1}\|$ $\|\vec L^{-1}\hat{R}\|$ $\|F\| <1$ for any unitarily invariant norm $\|\;\|$, then \[ \frac{\|\tilde{X}-X\|}{\|\tilde{X}\|} \leq \frac{2}{1+\sqrt{1-4\|\vec L^{-1}\| \|\vec L^{-1}\hat{R}\|{\ts}\|F\|}}\cdot \frac{\|\vec L^{-1}\hat{R}\|}{\|\tilde{X}\|}. \]