Global stability of linear systems with nonlinear perturbations: quadratic thresholds and the real stability radius gap

Let 𝐴 be Hurwitz and 𝑓 locally Lipschitz with 𝑓⁡(0) =0. Which explicit conditions on 𝑓 make the origin of ˙ 𝑥 =𝐴⁢𝑥 +𝑓⁡(𝑥) globally asymptotically stable? Using quadratic Lyapunov functions 𝑉 =⟨𝑃⁢𝑥,𝑥⟩, we show that 𝑉 is a strict certificate exactly when 𝑓 lies in a pointwise half-space, and that the largest certified sector ‖𝑓⁡(𝑥)‖ ≤𝑔⁡‖𝑥‖ is 𝑔 <𝑟𝐶⁡(𝐴), the complex stability radius. Our new results concern perturbations not confined to a ball: a direction ℎ⁡(𝑥)⁢𝑀⁢𝑥 is certified for every scalar ℎ if and only if 𝑀 is skew for some 𝑃 with 𝐴⊤⁢𝑃 +𝑃⁢𝐴 <0, and a gain–dissipation trade-off allows unbounded gain where local dissipation compensates it. We then compare 𝑟𝐶 with the real stability radius 𝑟𝑅: they coincide for normal matrices, or when the supremum defining 𝑟𝐶 is attained at 𝑤 =0, and otherwise, for an explicit 3 ×3 example, nonlinear perturbations destabilize strictly below 𝑟𝑅, bracketing the exact threshold in [0.1671,0.2001]. We also prove that 𝑟𝑁⁢𝐿 equals the radius 𝑟𝑇⁢𝑉 for time-varying perturbations when 𝑛 ≥3, which yields an exact criterion for when the quadratic threshold is the global one, and that for 𝐴3 the upper bound can be taken as 0.2000007. We also give averaged-gain and switching theorems with exact rate functions, which recover the sharp duty cycle in the burst examples, and discuss applications.