Fan, ShuyuanShuyuanFanWerner, HerbertHerbertWerner2026-10-202026-10-202026-10-06IEEE Control Systems Letters 10: 2497-2502 (2026)https://hdl.handle.net/11420/65398This letter studies level-dependent L2-attenuation certification for input-constrained nonlinear systems using a given nominal control Lyapunov function as a candidate storage function. A certificate feasibility map is derived that relates the prescribed attenuation bound, the Lyapunov sublevel set, and the minimum input bound needed for feasibility. The map establishes that tightening the attenuation requirement or enlarging the certified sublevel set yields a nondecreasing minimum input requirement. Conversely, under a fixed input limit, the attenuation bound certified by the inverse feasibility map is nondecreasing with the Lyapunov level. This feasibility structure motivates a Lyapunov-level dissipation allocation principle that strengthens dissipation on smaller levels while relaxing it on larger ones. Numerical examples illustrate the level-wise feasibility structure and the performance of the proposed Lyapunov-level dissipation allocation under input saturation.en2475-1456202624972502IEEEControl Lyapunov functionsdisturbance attenuationinput saturationnonlinear L2-gainNatural Sciences and Mathematics::537: Electricity and ElectronicsJournal Article10.1109/LCSYS.2026.3740977