Stability of Gyroscopic Systems with Respect to Perturbations

Computing the closest singular matrix polynomial

“…Proof. The proof is a direct consequence of equating to zero the right hand side of the system (14).…”

“…Then using Theorem 2.6, we have that ∆ * is not a stationary point for ( 14) and consequently it is not a real multiple of M. Then (ii) does not hold true. Viceversa, assume that ∆ * is not proportional to M. Then ∆ * is not a stationary point of (14). Using Theorem 2.6 we have that d dt G ε (∆(t)) t=0 < 0, along the solution path of (14).…”

“…Theorem 2.5. Consider ∆ (t) ∈ C (d+1)n×n , with ∆ (t) F = 1 and solution of the system (14). Let σ j (t) be the simple smallest singular value of the matrix…”

“…Proof. The theorem follows directly from the gradient system structure of ( 14), substituting the expression of ∆ in (14) in the derivative of the functional G ε (t) in (10).…”

“…The stationary points of the gradient system ( 14) are possible minimizers for the functional G ε (∆) and can be characterized in the following way: Theorem 2.6. Consider ∆(t) of unit Frobenius norm, satisfying the gradient system (14). Assume that for all t, it holds G ε (∆) > 0.…”