Some remarks on the extremal Lyapunov function for linear systems (Q800552)

Let us consider an asymptotically stable system of linear equations with coefficients (1) \(\dot x=Ax\), \(x\in R^ n\). For the solution of system (1), which satisfies the initial condition \(x(x_ 0,t_ 0)=x_ 0\), the following estimate holds \[ \| x(x_ 0,t)\|\leq \sqrt{\lambda_{\max}(H)/\lambda_{\min}(H)}\| x_ 0\|, \] where \(\lambda_{\max}(H)\), \(\lambda_{\min}(H)\) are respectively the greatest and the smallest eigenvalues of the symmetric positive-definite matrix H, which is a solution to the Lyapunov equation \(A^ TH+HA=-C.\) Matrix C is also symmetric and positive definite. Let \(H_ 0\) satisfy \(\lambda_{\max}(H_ 0)/\lambda_{\min}(H_ 0)=\inf_{H}\{\lambda_{\max}(H)/\lambda_{\min}(H)\}\). We consider the existence of \(V_ 0(x)=(x,H_ 0x)\) for a system (1) of arbitrary order.