Control-affine Schrödinger Bridge and Generalized Bohm Potential

TL;DR

The paper establishes a precise link between the control-affine Schrödinger bridge (caSB) and Schrödinger-type wave dynamics by applying a Madelung transform to the caSB optimality system, yielding a nonlinear, complex-potential Schrödinger PDE for a wave function $oldsymbol{\psi}$ with density constraint $ ho^{m{u}}_{ ext{opt}}=oldsymbol{\psi}oldsymbol{\psi}^ atural$. The main theoretical contribution is the derivation of the complex quantum potential $V_{ exttt{caSB}}$, whose real and imaginary parts depend nonlocally on the density and dual variables, and the recovery of the optimal control from the wave-field via $m{u}_{ ext{opt}} = m{g}^{ op} abla_{m{x}}S$. In the SB special case, the formalism reduces to a Schrödinger PDE with potential $V_{ exttt{SB}}$, representing a generalization of the Bohm potential and naturally incorporating an absorption mechanism through ${ m Im}(V_{ exttt{SB}})$. The work draws a conceptual parallel to optical potentials in nuclear physics and provides a framework to view stochastic density steering as wave steering, with implications for non-equilibrium statistical mechanics and quantum-inspired control methods.

Abstract

The control-affine Schrödinger bridge concerns with a stochastic optimal control problem. Its solution is a controlled evolution of joint state probability density subject to a control-affine Itô diffusion with a given deadline connecting a given pair of initial and terminal densities. In this work, we recast the necessary conditions of optimality for the control-affine Schrödinger bridge problem as a two point boundary value problem for a quantum mechanical Schrödinger PDE with complex potential. This complex-valued potential is a generalization of the real-valued Bohm potential in quantum mechanics. Our derived potential is akin to the optical potential in nuclear physics where the real part of the potential encodes elastic scattering (transmission of wave function), and the imaginary part encodes inelastic scattering (absorption of wave function). The key takeaway is that the process noise that drives the evolution of probability densities induces an absorbing medium in the evolution of wave function. These results make new connections between control theory and non-equilibrium statistical mechanics through the lens of quantum mechanics.

Theorems & Definitions (9)

• Lemma 1: Weighted Laplacian of $R$
• proof
• Theorem 1
• proof
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5