Eigenvalue Dynamics of an Operator Pencil in the Gyroscopic Stabilization Problem

The dynamics of eigenvalues of the quadratic pencil L (λ) := λ² R + λ\,ω\,iB + C, where ω→ ∞ is a numerical parameter and R , B , and C are complex symmetric n× n matrices, is studied. The skew-symmetric matrix Γ = ω iB expresses the gyroscopic forces in the problem described by the equation R u(t) - Γ u(t) + C u(t) =0. The case $$C<0$$ is considered. Let \κ₊, κ₋, κ₀\ be the signature of the matrix B (i.e., the numbers of positive and negative eigenvalues and the dimension of the kernel). Explicit expressions for the eigenvalues λₖ(ω) of the pencil L (λ) as ω → ∞ are obtained. It follows from these expressions that, for any sufficiently large ω > ω₀ , all the $$2n$$ eigenvalues of the pencil are semisimple, 2(n -κ₀) of them are purely imaginary, and 2κ₀ are located symmetrically with respect to the imaginary axis in the open left and right half-planes. In particular, the instability index of the problem is exactly κ₀ for large ω . Remarkably, the coefficients in the asymptotic formulas are determined explicitly and coincide with the eigenvalues of the linear pencils λ C +B , λ R +B , and P(λ R +C)P , where P is the orthogonal projection onto the kernel of B .