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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
##### Keywords:
Lyapunov function; Lyapunov’s equation