An Exact Quantile-Energy Equality for Terminal Halfspaces in Linear-Gaussian Control with a Discrete-Time Companion, KL/Schrodinger Links, and High-Precision Validation
Sandro Andric
TL;DR
This work derives an exact, closed-form link between the minimal control energy and the squared normal-quantile gap for terminal halfspaces in finite-horizon linear-Gaussian systems, closed under an $M$-weighted controllability Gramian framework. The key result $\min_u E(u)=\frac{(\Phi^{-1}(p_1)-\Phi^{-1}(p_0))^2}{2\,R_T^2(w)}$, with $R_T^2(w)=\frac{w^\top W_c^M w}{w^\top V_T w}$, is achieved by a constructive matched-filter control and extends naturally to a discrete-time companion via $R_N^2(w)=\frac{w^\top W_N^M w}{w^\top V_N w}$. The paper connects this exact quantile–energy relation to minimum-energy control, KL/risk-sensitive framing, Schrödinger bridges, and Gaussian chance constraints, and validates the continuous and discrete-time formulas with high-precision Monte Carlo experiments and a drone-altitude example. The results yield a compact, design-ready translator for energy–probability tradeoffs in terminal-halfspace constraints, enabling rapid exploration of robust control strategies under stochastic disturbances. Practical impact lies in fast, analytically tractable quantile control design for aerospace, robotics, and networked systems under Gaussian uncertainty.
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.