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On the question of the existence of extremal Lyapunov functions. (Russian) Zbl 0572.34042
The paper deals with the asymptotically stable n-dimensional system \(\dot x=Ax\) with a matrix A of diagonal-block type. It is shown that for \(n\geq 3\) the problem of finding a matrix \(H_ 0\) as a solution of Lyapunov’s equation in the class G(H) of positive definite matrices such that \[ \lambda_{\max}(H_ 0)/\lambda_{\min}(H_ 0)=\inf_{H\in G(H)}(\lambda_{\max}(H)/\lambda_{\min}(H)) \] (\(\lambda\) \({}_{\max}(H)\) and \(\lambda_{\min}(H)\) are the greatest and the least eigenvalue of the matrix H respectively) does not have only one solution.
Reviewer: G.A.Leonov
34D20 Stability of solutions to ordinary differential equations
34A30 Linear ordinary differential equations and systems, general