Further results on the measurement of solution bounds of the generalized Lyapunov equations

New lower matrix bounds for the solution of the continuous algebraic lyapunov equation

“…Consider the CALE (1) with: Then, the unique positive definite solution of (1) is: With \documentclass{article} \footskip=0pc\pagestyle{empty} \begin{document} $\alpha\,{=}\,1$ \end{document}, the lower matrix bounds for the solution P are found by Theorems 2.1 and 2.3, respectively, to be In fact, it can be seen that \documentclass{article} \footskip=0pc\pagestyle{empty} \begin{document} $P_{L2}\,{=}\,P_{exact}$ \end{document}. Since Q is singular, the matrix bounds proposed in 3, 6–8, 10–12, 14, 16, 17 cannot work here. The lower bound P L 11 gives the trivial bound P ≥0.…”

“…Often in control applications, such as in 1–4, the exact solution of the CALE is not required, but rather a bound on its solution. As such, a number of works have been presented over the last three decades for deriving solution bounds of this equation 5–20, many of which are summarised in 9. Solution bounds include extremal eigenvalue, trace, determinant, summation, product and matrix bounds.…”