Eigenvalue Problem for Lagrangian Systems. VI

The gyroscopic Lagrangian system $\text{ξ¨+Aξ̇+Hξ(t)=0}$ on the finite‐dimensional complex Hilbert space E is shown to be stable if and only if there exist Hermitian operators iα and P > 0 such that A = Pα + αP and H = P(¼ + α²)P. Structural stability is shown to be equivalent to the uniqueness of P and α, and to the existence of a Liapunov operator on E × E of the form Lp(T), where $\mathrm{L=}\left(\begin{array}{cc}\mathrm{-iA}& \mathrm{-i}\\ i& 0\end{array}\right)\mathrm{,\hspace{0.17em}T=}\left(\begin{array}{cc}0& \mathrm{-i}\\ \mathrm{iH}& \mathrm{iA}\end{array}\right)$ , and p(x) is a real polynomial of degree not exceeding the least of the quantities 2 dim E − 1 and 4N + 1, where N denotes the number of negative eigenvalues of H.