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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.17945 |
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| _version_ | 1866908604802531328 |
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| author | Andric, Sandro |
| author_facet | Andric, Sandro |
| contents | We prove an exact equality between the minimal quadratic control energy and the squared normal-quantile gap for terminal halfspaces in linear-Gaussian systems with additive control and quadratic effort $E(u) = \tfrac12\!\int u^\top M u\,dt$ where $M = B^\topΣ^{-1}B$. For terminal halfspace events, the minimal energy equals the squared normal-quantile gap divided by twice a controllability-to-noise ratio $R_T^2(w)=(w^\top W_c^M w)/(w^\top V_T w)$ and is attained by a matched-filter control. We provide an exact zero-order-hold discrete-time companion via block exponentials, relate the result to minimum-energy control, Gaussian isoperimetry, risk-sensitive/KL control, and Schrodinger bridges, and validate to high precision with Monte Carlo. We state assumptions, singular-$M$ handling, and edge cases. The statement is a compact synthesis and design-ready translator, not a universal principle. Novelty: while the ingredients (Gramians, Cauchy-Schwarz, Gaussian isoperimetry) are classical, to our knowledge the explicit quantile-energy equality with a constructive matched-filter achiever for terminal halfspaces, and its discrete-time companion, are not recorded together in the cited literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17945 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Exact Quantile-Energy Equality for Terminal Halfspaces in Linear-Gaussian Control with a Discrete-Time Companion, KL/Schrodinger Links, and High-Precision Validation Andric, Sandro Systems and Control Optimization and Control We prove an exact equality between the minimal quadratic control energy and the squared normal-quantile gap for terminal halfspaces in linear-Gaussian systems with additive control and quadratic effort $E(u) = \tfrac12\!\int u^\top M u\,dt$ where $M = B^\topΣ^{-1}B$. For terminal halfspace events, the minimal energy equals the squared normal-quantile gap divided by twice a controllability-to-noise ratio $R_T^2(w)=(w^\top W_c^M w)/(w^\top V_T w)$ and is attained by a matched-filter control. We provide an exact zero-order-hold discrete-time companion via block exponentials, relate the result to minimum-energy control, Gaussian isoperimetry, risk-sensitive/KL control, and Schrodinger bridges, and validate to high precision with Monte Carlo. We state assumptions, singular-$M$ handling, and edge cases. The statement is a compact synthesis and design-ready translator, not a universal principle. Novelty: while the ingredients (Gramians, Cauchy-Schwarz, Gaussian isoperimetry) are classical, to our knowledge the explicit quantile-energy equality with a constructive matched-filter achiever for terminal halfspaces, and its discrete-time companion, are not recorded together in the cited literature. |
| title | An Exact Quantile-Energy Equality for Terminal Halfspaces in Linear-Gaussian Control with a Discrete-Time Companion, KL/Schrodinger Links, and High-Precision Validation |
| topic | Systems and Control Optimization and Control |
| url | https://arxiv.org/abs/2510.17945 |