Control-affine Schr\"odinger Bridge and Generalized Bohm Potential

TL;DR

This paper links control theory and quantum mechanics by reformulating the control-affine Schr"odinger bridge problem as a quantum PDE with a complex potential, revealing new insights into stochastic optimal control.

Contribution

It introduces a novel complex potential generalizing the Bohm potential, connecting stochastic control with quantum mechanics and non-equilibrium statistical mechanics.

Findings

Reformulation of optimality conditions as a quantum PDE

Introduction of a complex potential analogous to optical potential

Connection between process noise and absorbing medium in wave function evolution

Abstract

The control-affine Schr\"odinger bridge concerns with a stochastic optimal control problem. Its solution is a controlled evolution of joint state probability density subject to a control-affine It\^o 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 Schr\"odinger bridge problem as a two point boundary value problem for a quantum mechanical Schr\"odinger 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.