ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

15. Numerical Analysis and Scientific Computing

1991 MS Classification: 15, 34, 65 Shtykel, Tatyana, Humboldt-Universität zu Berlin, Germany On the criterion of asymptotical stability for index-1 tractable DAE. The index-1 tractable differential-algebraic equation Ax′(t) + Bx(t) = 0 is considered. The Lyapunov stability of the trivial solution is discussed. The asymptotical stability of the trivial solution can be fully characterized by the quantitative parameter æ(A, B) = 2∥A∥∥B∥∥H∥, where

H = \frac{1}{2π} ∫_{−∞}^{∞}(A+BQ)^{∗}(iξA+B)^{−∗}P^{∗}F P (iξA+B)^{−1}(A+BQ)dξ +Q^{∗}F Q.

Here the matrices P and Q are the projectors onto the invariante subspaces of the matrix pencil {A, B} associated with the finite and infinite eigenvalues, respectively, and F = (A + BQ)^{−∗}(A + BQ)^{−1}. The parameter æ(A, B) < ∞ if and only if the index of the pencil {A, B} is not greater than 1 and all its finite eigenvalues have negative real parts. A methods of reducing the pencil {A, B} with index 1 to the Kronecker canonical form and of computing the projectors P, Q and the parameter æ(A, B) are described.