2025/04/22 by Alexis M. H. Teter, Wenqing Wang, Teter, Alexis M. H. +5
Engineering · #Advanced Control Systems Optimization #Control Systems and Identification #FOS: Computer and information sciences #FOS: Electrical engineering #FOS: Mathematics #Machine Learning (cs.LG) #Optimization and Control (math.OC) #Probability (math.PR) #Statistics Theory (math.ST) #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.2504.15753
openalex publication_date 2025/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a controllable linear time-varying (LTV) pair (\boldsymbolAt,\boldsymbolBt) and \boldsymbolQt positive semidefinite, we derive the Markov kernel for the Itô diffusion d\boldsymbolxt=\boldsymbolAt\boldsymbolxt d t + √(2)\boldsymbolBtd\boldsymbolwt with an accompanying killing of probability mass at rate (1)/(2)\boldsymbolx\top\boldsymbolQt\boldsymbolx. This Markov kernel is the Green's function for an associated linear reaction-advection-diffusion partial differential equation. Our result generalizes the recently derived kernel for the special case (\boldsymbolAt,\boldsymbolBt)=(\boldsymbol0,\boldsymbolI), and depends on the solution of an associated Riccati matrix ODE. A consequence of this result is that the linear quadratic non-Gaussian Schrödinger bridge is exactly solvable. This means that the problem of steering a controlled LTV diffusion from a given non-Gaussian distribution to another over a fixed deadline while minimizing an expected quadratic cost can be solved using dynamic Sinkhorn recursions performed with the derived kernel. Our derivation for the (\boldsymbolAt,\boldsymbolBt,\boldsymbolQt)-parametrized kernel pursues a new idea that relies on finding a state-time dependent distance-like functional given by the solution of a deterministic optimal control problem. This technique breaks away from existing methods, such as generalizing Hermite polynomials or Weyl calculus, which have seen limited success in the reaction-diffusion context. Our technique uncovers a new connection between Markov kernels, distances, and optimal control. This connection is of interest beyond its immediate application in solving the linear quadratic Schrödinger bridge problem.