Talk:Lyapunov equation

Weak Lyapunov First theorem[edit]

Does it exist a weak formulation for that theorem?

I need the proof that

Theorem. If there exist

P >= 0 and Q > 0 satisfying A^T P + PA + Q = 0 then the linear system is globally Lyapunov stable

. The quadratic function V(z) = z^T Pz is a Lyapunov function that can be used to verify stability.

Or also (if possible, I have no idea about the proof of the theorem)

Theorem. If there exist

P >= 0 and Q >= 0

satisfying A^T P + PA + Q = 0 then the linear system is globally

Lyapunov stable

. The quadratic function V(z) = z^T Pz is a Lyapunov function that can be used to verify stability.

194.206.211.87 08:59, 16 May 2007 (UTC)[reply]

Easily Computable Analytic Solution[edit]

I'd like to see a better explanation of the "easily computable" solution. It also requires a citation; there is nothing to back up the math here.

PrintStar (talk) 15:15, 30 November 2010 (UTC)[reply]

This is the Stein equation. Did Lyapunov ever even think about this equation?[edit]

This equation is known in mathematics as the Stein equation, in particular it is the symmetric Stein equation. For example

A functional approach to the Stein equation - ScienceDirect doi:10.1016/j.laa.2006.07.025 (core.ac.uk) A Note on the -Stein Matrix Equation (hindawi.com) ... and a lot more literature can be given.

I myself have published it in the engineering literature as "Discrete Lyapunov Equation" because the referees, who apparently did not know better insisted I not call it the Stein equation. When we asked why, they could not give any explanation.

What have people got against Stein? 173.68.125.17 (talk) 16:21, 7 February 2023 (UTC)[reply]

Unfortunately, this happens a lot: List of misnamed theorems

I edited the page and added that the discrete Lyapunov equation is also known as Stein equation. Saung Tadashi (talk) 16:32, 7 February 2023 (UTC)[reply]