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On an algorithm for the determination of the Lyapunov function which gives the extremal integral estimate. (Russian) Zbl 0572.34047
The paper deals with the problem of constructing the Lyapunov function \(v(x)=x^*H_ 0x\) for obtaining an extremal estimate for the solution to the linear system \(\dot x=Ax\). The matrix \(H_ 0\) fulfills the following condition: \(\phi (H_ 0,C_ 0)=\sup_{H\in G}\{\phi (H,C)\},\) where \(\phi (H,C)=(\lambda_{\min}(C)/\lambda_{\max}(H))\sqrt{\lambda_{\min}(H)/\;lambda_{\max}(H)}\), G is the space of positive definite matrices, which satisfy Lyapunov’s equation \(A^ TH+HA=-C\), \(\lambda_{\max}(\cdot)\) and \(\lambda_{\min}(\cdot)\) are the greatest and smallest eigenvalue of the corresponding matrices.
Reviewer: G.A.Leonov
MSC:
34D20 Stability of solutions to ordinary differential equations
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