Source
Farrell & Ioannou, “Generalized Stability Theory. Part II: Nonautonomous Operators” (1996), LIT-929, read in full in NOTE-842: §2 (eqs. 1–7), eq. 12, §4 (eqs. 13–26), Figs. 6–7.
What was actually shown
For du/dt = A(t)u, stability is the sign of the first Lyapunov exponent λ = lim sup ln‖Φ(t)‖/t (eq. 6), with Φ the propagator.
- Non-normality is necessary (eq. 12). Since d‖u‖²/dt = u†(A + A†)u, the exponent satisfies lim sup (1/t)∫λ_min dt ≤ λ ≤ lim sup (1/t)∫λ_max dt, where λ_min and λ_max are the extreme eigenvalues of (A(t) + A†(t))/2. For a normal operator λ_max is the spectral abscissa α(t) = max Re spec A(t). So if A(t) is normal and α(t) ≤ −c < 0 for every t, then λ ≤ −c. The condition must be uniform: with α(t) < 0 tending to 0, the bound gives only λ ≤ 0.
- It happens. The oscillator d/dt(x, v) = [[0, 1], [−ω²(t), 0]](x, v), with ω switching between 0.5 and 3 every T units, is neutral at every instant (eigenvalues ±iω) and non-normal. Its exponent is positive except near integer multiples of half a period (Fig. 6). A 2 × 2 operator with eigenvalues ±i, randomly rotated at each step, has a positive exponent for every angle between its eigenvectors except π/2 (Fig. 7).
- The mechanism. “Transient growth instigated at the starting time is continued with further growth when the switch to the second operator takes place, forestalling the decay that would occur in the autonomous case.”
Necessity is a bound, so what could show it wrong is an error in that line. The examples could have come out stable, and did not.
What this does not say
- That the paper shows the stable-at-every-instant case. It shows neutral cases and states that damping preserves the effect without showing it. The record closes that case in CLAIM-376.
- That any time variation destabilizes. The exponent vanishes as the switching interval goes to zero. Variation much faster than the transient growth averages out.
- Anything about random variation in general. Genericity under multiplicative noise, and how the threshold depends on non-normality and damping, are THEORY-325’s, still Proposed.