LQR-Schrödinger Bridge

Updated 30 April 2026

LQR-Schrödinger Bridge is a framework that integrates stochastic steering of probability distributions with LQR's quadratic cost structure.
It employs coupled Riccati equations and reaction–diffusion PDEs to derive explicit Markov kernels and closed-loop feedback controls.
The methodology enables efficient numerical implementations and generalizations to non-Gaussian distributions, linking optimal transport with quantum mechanics.

The LQR-Schrödinger Bridge problem synthesizes the classical Schrödinger bridge—where the task is to stochastically steer probability flows between prescribed marginals—with the full structure of the linear quadratic regulator (LQR), where the cost functional is quadratic in both control and state. This unification yields a versatile paradigm for exact distributional steering in continuous and discrete time, accommodating linear system dynamics, general quadratic costs, and non-Gaussian initial and terminal distributions. It establishes deep connections between stochastic control, optimal transport, reaction-diffusion partial differential equations (PDEs), and integrable Markov kernels, with rigorous analytic and algorithmic tractability.

1. Formulation and Problem Statement

Consider a controlled linear diffusion in $\mathbb{R}^n$ over a finite time horizon $[0,T]$ . The state evolves according to

$dx_t = A(t)x_t\,dt + B(t)u_t\,dt + \sqrt{\epsilon}\,B(t)\,dw_t,$

where $A(t)$ is an $n\times n$ matrix, $B(t)$ is $n\times m$ and $(A,B)$ is uniformly controllable, and $w_t$ is an $m$ -dimensional standard Wiener process. For $[0,T]$ 0, the dynamics include noise. The initial and final state distributions are prescribed, typically as zero-mean Gaussians $[0,T]$ 1 and $[0,T]$ 2 for suitable covariance matrices.

The cost to be minimized is

$[0,T]$ 3

where $[0,T]$ 4 is positive definite and $[0,T]$ 5 symmetric (not necessarily positive definite). The control $[0,T]$ 6 belongs to the set of adapted processes of finite energy.

The task: Find $[0,T]$ 7 steering the distribution of $[0,T]$ 8 from $[0,T]$ 9 to $dx_t = A(t)x_t\,dt + B(t)u_t\,dt + \sqrt{\epsilon}\,B(t)\,dw_t,$ 0 (or, in general, between $dx_t = A(t)x_t\,dt + B(t)u_t\,dt + \sqrt{\epsilon}\,B(t)\,dw_t,$ 1 and $dx_t = A(t)x_t\,dt + B(t)u_t\,dt + \sqrt{\epsilon}\,B(t)\,dw_t,$ 2 with finite second moments), minimizing $dx_t = A(t)x_t\,dt + B(t)u_t\,dt + \sqrt{\epsilon}\,B(t)\,dw_t,$ 3 subject to the dynamics above (Chen et al., 2016, Teter et al., 22 Apr 2025).

2. Coupled Riccati Equations and LQR Structure

Under Gaussian endpoint distributions, optimal controls are linear state feedbacks: $dx_t = A(t)x_t\,dt + B(t)u_t\,dt + \sqrt{\epsilon}\,B(t)\,dw_t,$ 4 where $dx_t = A(t)x_t\,dt + B(t)u_t\,dt + \sqrt{\epsilon}\,B(t)\,dw_t,$ 5 solves the matrix Riccati ODE

$dx_t = A(t)x_t\,dt + B(t)u_t\,dt + \sqrt{\epsilon}\,B(t)\,dw_t,$ 6

subject to two-point boundary conditions involving auxiliary matrix trajectories. The dual equation, in terms of the optimal state covariance $dx_t = A(t)x_t\,dt + B(t)u_t\,dt + \sqrt{\epsilon}\,B(t)\,dw_t,$ 7, gives a coupled forward Riccati: $dx_t = A(t)x_t\,dt + B(t)u_t\,dt + \sqrt{\epsilon}\,B(t)\,dw_t,$ 8 with $dx_t = A(t)x_t\,dt + B(t)u_t\,dt + \sqrt{\epsilon}\,B(t)\,dw_t,$ 9 and $A(t)$ 0 involving $A(t)$ 1 and $A(t)$ 2. The unique solution is determined by projection onto a $A(t)$ 3-dimensional Hamiltonian flow and selecting branches to keep Gramian matrices nonsingular (Chen et al., 2016).

For general, non-Gaussian endpoint laws, the structure holds with dynamics and Sinkhorn recursions performed in functional space (Teter et al., 22 Apr 2025).

3. Markov Kernels, Reaction–Diffusion, and Exact Solvability

The core technical advancement is the construction of exact Markov kernels (Green’s functions) for the underlying reaction-diffusion PDEs encoding both advection/diffusion and quadratic killing (state cost). For time-invariant $A(t)$ 4, the kernel takes the form

$A(t)$ 5

with $A(t)$ 6 the solution of the backward Riccati ODE, $A(t)$ 7 the state transition of the closed-loop system, and $A(t)$ 8 the associated controllability Gramian under optimal feedback (Teter et al., 22 Apr 2025). For the scalar case with $A(t)$ 9, $n\times n$ 0, $n\times n$ 1, the kernel reduces to a non-radial Gaussian, revealing the connection to the Mehler kernel and quantum harmonic oscillator propagators (Teter et al., 2024, Teter et al., 2024).

Exactly solvable cases enable contractive Sinkhorn-type iterations for coupling arbitrary distributions at endpoints, with the kernel parameterizing the effective cost-to-go/distance functional arising from deterministic optimal control.

4. Schrödinger System, Sinkhorn Recursion, and Duality

The time-marginal dynamics and scaling functions underlying the optimal process satisfy a pair of coupled reaction–diffusion PDEs,

$n\times n$ 2

where $n\times n$ 3 and $n\times n$ 4 is the generator of the uncontrolled process. The product $n\times n$ 5 gives the optimally steered density (Teter et al., 22 Apr 2025, Teter et al., 2024).

Iterating the backward and forward projections with product normalization realizes a continuous-time Sinkhorn recursion in function space. For Gaussian or Gaussian-mixture approximations, all intermediate function evaluations, including the projections, may be computed in closed form. These recursions converge exponentially fast in Hilbert’s projective metric (Teter et al., 2024, Lambert, 12 Jun 2025).

5. Discrete-Time Formulation and Generalizations

The problem admits an exact discrete-time analogue. Given trajectories $n\times n$ 6, with stepwise cost

$n\times n$ 7

the optimal path probability factorizes via (dynamic programming) Gibbs–Kantorovich potentials $n\times n$ 8 and $n\times n$ 9 as

$B(t)$ 0

with $B(t)$ 1 the (non-normalized) reference measure (Lambert, 12 Jun 2025). Dual Riccati recursions for the cost-to-go potentials enable efficient forward and backward passes, resulting in Markovian transition kernels and explicit Gaussian (or generalized Gaussian) marginals.

The discrete-time LQR–Schrödinger bridge is extensible to non-Gaussian endpoint marginals via iterative proportional fitting, and generalizes Bures–Wasserstein transport by introducing time-indexed quadratic potentials that effect “curvature” in the path geometry (Lambert, 12 Jun 2025).

6. Connections: Quantum Mechanics, Optimal Transport, and Beyond

The forward–backward PDE system underlying the LQR-Schrödinger bridge, under Wick rotation, is equivalent to the Schrödinger equation for the quantum harmonic oscillator with quadratic potential. The fundamental solution is the Mehler kernel, matching the propagator for the quantum oscillator. The closed-form transition kernel structure provides a unified framework that encompasses classical heat kernel diffusion, stochastic control, quantum mechanics, and entropy-regularized optimal transport (Teter et al., 2024, Teter et al., 2024).

In the zero-noise limit ( $B(t)$ 2), the problem converges to a deterministic boundary-value optimal transport problem with general quadratic cost: $B(t)$ 3 recovering the quadratic-cost mass transport with specified marginals. The solution path is then entirely determined by the boundary conditions and the LQR structure (Chen et al., 2016).

7. Numerical Methods and Implementation Remarks

The solution pipeline for the LQR–Schrödinger bridge in both continuous and discrete time is characterized by:

Backward integration of the matrix Riccati ODE (or recursion) for the value/cost-to-go matrices.
Forward computation of the closed-loop state transition and Gramian.
Construction of the Markov kernel as a parameterized non-homogeneous Gaussian (or quasi-Gaussian) in $B(t)$ 4.
Iterative Sinkhorn-type proportional fitting in function space to enforce general (including non-Gaussian) endpoint distributions.

For practical purposes, each recursion (per time-step) is $B(t)$ 5, and convergence is empirically rapid; typically, a small number of iterations is sufficient. For mixture-of-Gaussians approximations, all required integrals decompose into explicit formulas. In non-quadratic or non-Gaussian contexts, numerical quadrature or Monte Carlo methods can be used to approximate projections, leveraging the explicit form of the kernels (Teter et al., 22 Apr 2025, Teter et al., 2024, Lambert, 12 Jun 2025).

Key References:

Chen–Georgiou–Pavon, "Optimal steering of a linear stochastic system to a final probability distribution, Part III" (Chen et al., 2016)
Blom, Chen, Georgiou, Pavon, "Markov Kernels, Distances and Optimal Control" (Teter et al., 22 Apr 2025)
Chen, Georgiou, Pavon, "Schrödinger Bridge with Quadratic State Cost is Exactly Solvable" (Teter et al., 2024)
Chen, Pavon, "Weyl Calculus and Exactly Solvable Schrödinger Bridges with Quadratic State Cost" (Teter et al., 2024)
Pavon et al., "The LQR-Schrödinger Bridge" (Lambert, 12 Jun 2025)