New entry 2026-04-21-polytopic-sos-tikhonov.tex covering:
- Polytopic barrier attempt (naive Nagumo check), why it fails
(safety polytope too large for LQR contraction from anywhere),
and the Blanchini pre-image algorithm as the right fix.
- SOS polynomial barrier success on the 2-state reduced projection:
CSDP returns OPTIMAL on a degree-4 polynomial B(x1, x2). First
non-quadratic barrier artifact for this plant. Full polynomial
coefficients embedded.
- Tikhonov singular-perturbation theorem derivation for the PJ
reduction. Writes the 10-state PKE in slow-fast form with
eps=Lambda, identifies the quasi-steady manifold h(x) = PJ
formula, shows fast subsystem exponentially stable under the
prompt_critical_margin_heatup invariant. Error bound:
|x(t) - x_PJ(t)| <= C*Lambda = O(1e-4) in state units, uniform
after boundary layer. Empirical validation data (0.1% max) is
consistent with K_1 ~ 40, K_3 ~ 70 problem constants.
- apass markers for remaining open items: scram entry expansion,
heatup steam-dump Q_sg, heatup controller-ref mismatch.
The Tikhonov derivation upgrades "we ran it and 0.1% error" to
"bounded by C*Lambda where C depends on problem properties bounded
by the safety halfspaces." Rigorous rate.
Journal: 38 pages, clean build.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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