Talk:Lyapunov equation

Weak Lyapunov First theorem

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)

Easily Computable Analytic Solution

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)

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

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)

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)