On an algorithm for the determination of the Lyapunov function which gives the extremal integral estimate. (Russian) Zbl 0572.34047

Khusainov, D. Ya.; Ivokhin, E. V.

Vestn. Kiev. Univ., Model. Optimizatsiya Slozhnykh Sist. 4, 62-67 (1985).

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