{"ID":2852909,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.17945","arxiv_id":"2510.17945","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","abstract":"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.","short_abstract":"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 eq...","url_abs":"https://arxiv.org/abs/2510.17945","url_pdf":"https://arxiv.org/pdf/2510.17945v1","authors":"[\"Sandro Andric\"]","published":"2025-10-20T17:12:29Z","proceeding":"eess.SY","tasks":"[\"eess.SY\",\"math.OC\"]","methods":"[]","has_code":false}