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| Format: | Recurso digital |
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Zenodo
2026
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| Accès en ligne: | https://doi.org/10.5281/zenodo.18443177 |
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- <div><strong>Preprint:</strong></div> <div>This paper formalizes observational erasure—permanent information loss from sensor coarsening—as a fundamental limit on guaranteed control. When observation maps impose symmetry on indistinguishable states, controllers face a measurability constraint: identical controls must be applied to states the sensor cannot distinguish. <strong>The paper shows this creates sharp feasibility thresholds in guaranteed reachability, below which no policy can succeed regardless of control authority.</strong></div> <div> </div> <div><strong>Main Result:</strong></div> <div>The maximum displacement a controller can achieve along a separating functional—the <strong>recovery capacity</strong>—is bounded by <strong> _rec ≤ B√(TE)</strong>, where B is the energy-displacement coupling constant, T is the time horizon, and E is the energy budget. <strong>When the erasure gap a exceeds this capacity, guaranteed reachability fails as a sharp threshold: not gradual degradation, but provable impossibility.</strong> The impossibility proof requires no dynamical symmetry assumption—only energy-displacement coupling and the measurability constraint imposed by the observation map.</div> <div> </div> <div><strong>Key Contributions:</strong></div> <div>∙ <strong>Recovery capacity bound _rec ≤ B√(TE)</strong> from Cauchy–Schwarz applied to energy-displacement coupling, yielding a sharp feasibility threshold analogous to the data rate theorem’s R > log₂|λ| for stabilization</div> <div>∙ <strong>Exact closed-form thresholds for n-fold integrators</strong> with time scaling T^((2n−1)/2) and constructive sharpness proofs via explicit optimal control</div> <div>∙ <strong>Gramian characterization for linear systems</strong>: recovery capacity equals √(v⊤W_r v)·√E, connecting standard controllability theory to erasure threshold analysis</div> <div>∙ <strong>Sensor design criterion</strong>: distinctions with erasure gap Δφ > 2B√(TE) must be preserved in the sensor pipeline for guaranteed safety</div> <div>∙ <strong>Applications to autonomous vehicle certification</strong> under SOTIF (ISO 21448:2022), explaining when sensor abstractions create provably unsafe scenarios regardless of vehicle capability</div> <div> </div> <div><strong>Scope:</strong></div> <div>The results apply to ℤ₂ <strong>observational symmetry</strong> (involution-based sensor abstractions) under <strong>L² energy constraints with deterministic dynamics</strong>. The framework complements indistinguishability characterizations (Liberzon & Mitra, HSCC 2025) with quantitative resource bounds, and expected-performance limits (Majumdar et al., RSS 2022) with guaranteed-reachability analysis. <strong>The impossibility direction is valid whenever the energy-displacement coupling bound holds; the achievability direction is proven sharp for integrator-type and linear systems.</strong> No claims of universality beyond the stated assumptions are made. The recovery capacity for linear systems is computationally equivalent to standard Gramian theory; the novelty is diagnostic and interpretive, not computational.</div>