Plain-English statement
An exact energy-dissipation identity plus a coercive functional inequality forces exponential decay of the energy.
Mathematical statement
If E'(t)=-aD(t) and rE(t)≤aD(t), then E(t)≤E(0)exp(-rt) for t≥0.
Intuition
The coercive inequality converts dissipation into a negative multiple of the current energy, and one-sided Gronwall propagates that local differential inequality globally in time.
Conditions
- the scalar energy curve is continuous
- the right derivative exists at every time
- the exact derivative is minus the scaled dissipation
- coercivity holds along the entire curve
Why these conditions cannot be dropped
- right derivatives match semigroups defined for nonnegative time increments
- the exact dissipation identity is what links the analytic functional inequality to time evolution
- coercivity supplies the decay rate
Proof route
- negate the coercive inequality to bound the energy derivative
- apply Mathlib's one-sided Gronwall theorem on [0,t]
- rewrite the zero-forcing Gronwall bound as an exponential
Lean interface notes
- HasDerivWithinAt uses the set Ici t to encode a right derivative
- the theorem is scalar and does not guess what E or D mean in a concrete process
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Lean statement
theorem exponential_decay_of_scaled_dissipation
{scale rate : ℝ} (curve : DissipationCurve scale)
(hcoercive : ∀ s : ℝ,
rate * curve.energy s ≤ scale * curve.dissipation s)
{t : ℝ} (ht : 0 ≤ t) :
curve.energy t ≤ curve.energy 0 * Real.exp (-rate * t) := by
simpa [sub_zero] using
exponential_decay_of_scaled_dissipation_from
curve hcoercive ht
/-- Exponential decay from every starting time forces the instantaneous
coercivity inequality.
The proof compares the energy with its exponential envelope on `[s, ∞)`. Their
difference has a local maximum at `s`; the one-sided Fermat inequality then
compares the two right derivatives. -/Open AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:114published source at 7bcd37294df1