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2026
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| Accesso online: | https://doi.org/10.5281/zenodo.18443177 |
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| _version_ | 1866901274920747008 |
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| author | Samuel Richards |
| author_facet | Samuel Richards |
| contents | <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> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18443177 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | No Reachability Without Recovery: Observational Erasure as a Fundamental Limit on Guaranteed Control Samuel Richards Autonomous vehicles Control Theory sensor design barrier certificates Hamilton-Jacobi reachability controllability Gramians control barrier functions sharp thresholds measurability constraints capture basins energy bounds observational coarsening guaranteed reachability recovery capacity observational erasure Electrical engineering, electronic engineering, information engineering Automation Applied mathematics <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> |
| title | No Reachability Without Recovery: Observational Erasure as a Fundamental Limit on Guaranteed Control |
| topic | Autonomous vehicles Control Theory sensor design barrier certificates Hamilton-Jacobi reachability controllability Gramians control barrier functions sharp thresholds measurability constraints capture basins energy bounds observational coarsening guaranteed reachability recovery capacity observational erasure Electrical engineering, electronic engineering, information engineering Automation Applied mathematics |
| url | https://doi.org/10.5281/zenodo.18443177 |