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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
##### Keywords:
Popov-type stability
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##### 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
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