The singular case of the problem of absolute stability of systems of ordinary differential equations (Q1262993)

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 \(H^ 2({\mathbb{C}})\).