×

zbMATH — the first resource for mathematics

The singular case of the problem of absolute stability of systems of ordinary differential equations. (English. Russian original) Zbl 0687.34040
Sib. Math. J. 30, No. 1, 150-153 (1989); translation from Sib. Mat. Zh. 30, No. 1(173), 194-198 (1989).
Let \[ J(\xi (.))=\int^{\infty}_{-\infty}\xi (i\omega)^*\Pi (i\omega)\xi (i\omega)d\omega +2 Re\int^{\infty}_{- \infty}r(i\omega)^*\xi (i\omega)d\omega, \] be a functional on \(H^ 2({\mathbb{C}}^ m)\). Let \(\Pi\) (p) be an \(m\times m\) matrix with rational entries, \(\Pi (i\omega)^*=\Pi (i\omega)\), r(p) a column rational m- vector and the elements of \(\Pi\) (p) and r(p) be bounded on the imaginary axis and \(r(\infty)=0\). The following theorem is proved: under previous assumptions J(\(\xi\) (.)) is semibounded if and only if the following conditions hold: 1) for every \(\omega\in {\mathbb{R}}\), \(\Pi\) (i\(\omega)\) is hermitian and nonnegative; 2) for any \(\xi (.)\in H^ 2({\mathbb{C}}^ n)\) with \(\Pi (i\omega)\xi (i\omega)=0\) the function \(r(.)^*\xi (.)\) belongs to \(H^ 2({\mathbb{C}})\).

MSC:
34D20 Stability of solutions to ordinary differential equations
93D10 Popov-type stability of feedback systems
34G10 Linear differential equations in abstract spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. L. Likhtarnikov and V. A. Yakubovich, Appendix to: V. Rezvana, Absolute Stability of Automatic Systems with Delay [in Russian], Nauka, Moscow (1983).
[2] D. Z. Arov and V. A. Yakubovich, ?Semiboundedness condition for quadratic functionals on Hardy spaces,? Vestn. Leningr. Gos. Univ. (LGU), Ser. Mat., Mekh., Astron., No. 1, 11-18 (1982). · Zbl 0493.30029
[3] D. V. Yakubovich, ?Algorithm for completing a rectangular polynomial matrix to a square one with given determinant,? Kibern. Vychisl. Tekh., Kiev., No. 62, 85-89 (1984). · Zbl 0602.15017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.