---
title: 'Sinkhorn MPC: Real-Time Transport Control'
url: https://www.emergentmind.com/topics/sinkhorn-mpc
type: topic
---

# Sinkhorn MPC: Real-Time Transport Control

Sinkhorn Model Predictive Control (Sinkhorn MPC) is a real-time dynamical transport algorithm for steering a finite population of agents, each governed by linear dynamics, from an initial empirical distribution to a desired target distribution. It synthesizes model predictive control (MPC) for receding-horizon trajectory optimization with entropic-regularized optimal transport, specifically using the Sinkhorn algorithm for tractable real-time assignment. The central innovation is the simultaneous integration of transport planning (target assignment) and control, enabling cost-efficient, scalable population steering under system and input constraints [2301.06492] [2203.08415].

## 1. Mathematical Formulation and Problem Structure

Consider $N$ agents, each with continuous or discrete linear time-invariant dynamics:
\[
\dot x_i(t) = A_i x_i(t) + B_i u_i(t),\quad x_i(0) = x_i^0,
\]
with state $x_i(t)\in\mathcal{X}_i$ and control $u_i(t)\in\mathcal{U}_i$. The control objective is to assign each agent to a distinct target $\{x_j^*\}_{j=1}^N$ (permutation $\sigma$), minimizing the infinite-horizon sum of per-agent costs:
\[
\min_\sigma\, \sum_{i=1}^N c_\infty^i(x_i^0, x_{\sigma(i)}^*),\qquad
c_\infty^i(x_i^0, x_j^*) = \min_{u_i} \int_0^\infty \ell_i(x_i(t), u_i(t); x_j^*)\,dt.
\]
Here $\ell_i(\cdot)$ is a nonnegative, equilibrium-detecting running cost. This direct problem is computationally intractable due to the factorial-size assignment space and coupled infinite-horizon optimal control subproblems [2301.06492, Eqns. (3)-(8)].

Sinkhorn MPC replaces the infinite-horizon problem with a finite-horizon receding surrogate. At each MPC time step, agents solve for cost $c_{T_p}^i(\check x_i, x_j^*)$ (over horizon $T_p$ with terminal constraint $x_i(T_p) = x_j^*$) and only the first input $u_i^{\mathrm{MPC}}$ is applied. Assignment is relaxed to a soft coupling $P \in \mathbb{R}_{\ge0}^{N\times N}$, minimizing a linear combination of assignment costs subject to marginal constraints:
\[
P^*(x) = \arg\min_{P\in U} \sum_{i,j=1}^N C_{ij}(x)P_{ij} - \varepsilon H(P),\qquad H(P) = -\sum_{i,j} P_{ij} (\log P_{ij} - 1),
\]
with $C_{ij}(x) = c_{T_p}^i(x_i, x_j^*)$ and $U$ the uniform doubly-stochastic polytope. The entropic regularization parameter $\varepsilon>0$ controls the smoothness of $P^*$ and interpolates between soft and hard permutations [2301.06492, Eq. (22)].

## 2. Sinkhorn Iterations and Numerical Realization

The entropic-regularized optimal transport problem is efficiently solved via Sinkhorn's matrix-scaling algorithm. The key steps involve computing the Gibbs kernel:
\[
K_{ij}(x) = \exp\left(-C_{ij}(x)/\varepsilon\right),
\]
and then iteratively updating nonnegative scaling vectors $\alpha, \beta$ to satisfy row/column marginals:
\[
\begin{align*}
\beta(l) &= (1/N)\oslash (K^\top \alpha(l)),\\
\alpha(l+1) &= (1/N)\oslash (K \beta(l)).
\end{align*}
\]
Here “$\oslash$” denotes element-wise division. The soft coupling is $P = \mathrm{diag}(\alpha) K \mathrm{diag}(\beta)$. In practical MPC integration, only a fixed (often small) number of Sinkhorn updates $S[k]$ are executed per time-step, making the approach compatible with real-time constraints and large agent populations [2203.08415, (9); 2301.06492, Eqns. (SMPC_sink) and Algorithm 1].

## 3. Receding-Horizon Sinkhorn MPC Closed-Loop Dynamics

At each discrete time $k$:
1. States $x_i[k]$ observed; finite-horizon costs $C_{ij}$ formed by solving $N^2$ QPs (parallelizable).
2. Gibbs kernel $K_{ij} = \exp\big(-C_{ij}/\varepsilon\big)$ computed.
3. $S[k]$ Sinkhorn iterations update scaling vectors; coupling $P$ approximated as above.
4. Each agent forms a barycentric target via 
   \[
   x_i^{\mathrm{tmp}}(P) = N\sum_{j=1}^N P_{ij}x_j^*.
   \]
5. Individual MPCs steer each agent towards $x_i^{\mathrm{tmp}}(P)$; plant updated to $x_i[k+1]$.

A distinguishing implementation feature is warm-starting the $\alpha$ scaling vector, significantly accelerating Sinkhorn convergence after the first iteration [2301.06492, Algorithm 1; Fig. 5].

## 4. Theoretical Analysis: Convergence, Stability, and Boundedness

### 4.1. Existence and Nature of Equilibria

Under continuity and differentiability assumptions on the cost and navigator functions, a fixed point $x^*$ exists such that $x_i^* = x_i^{\mathrm{tmp}}(P^*(x^*))$, where $P^*(x^*)$ is the equilibrium coupling [2301.06492, Proposition 1].

### 4.2. Global Convergence under Exact Coupling

If the exact optimal coupling is computed at each step (i.e., enough Sinkhorn iterations to convergence), continuous-time Sinkhorn MPC converges to the equilibrium set under strict positivity of the running cost and technical regularity conditions. The proof uses the entropic-OT cost as a Lyapunov function and applies LaSalle’s principle [2301.06492, Theorem 1].

### 4.3. Ultimate Boundedness and Discrete Realizability

In the discrete-time setting, even with a single Sinkhorn update per step, the closed-loop states are ultimately bounded. Specifically, for any $\delta>0$ and for all $k\ge\tau$,
\[
\|x_i[k]\| < \delta + \frac{\kappa_i(\nu_i)\bar r \|I - \bar A_i\|}{1-(\rho_i+\nu_i)},
\]
where $\bar A_i$ is the closed-loop MPC matrix, with spectral radius $\rho_i<1$, and $\bar r$ bounds the navigator range [2301.06492, Prop. 3; 2203.08415, Prop. 4.1].

### 4.4. Local Asymptotic Stability

For quadratic running cost and unconstrained/weakly-constrained systems, all isolated equilibria of the discrete Sinkhorn MPC scheme are locally asymptotically stable, both in the "over-smoothed" ($\varepsilon$ large) and "hard-assignment" ($\varepsilon \to 0$) regimes [2301.06492, Theorem 2; 2203.08415, 4.2]. The construction of a composite Lyapunov function combining state error and Hilbert projective metric on the Sinkhorn scaling vectors supports this claim.

## 5. Computational Complexity, Algorithmic Steps, and Scalability

The primary computational costs per step are:
- Formulation of the $N\times N$ cost matrix $C_{ij}$ (parallelizable over agents): $O(N^2 \,\mathrm{cost}_c)$.
- Sinkhorn iterations: two matrix-vector products per iteration, $O(N^2)$.
- $N$ independent MPC optimizations (closed-form in linear-quadratic settings, $O(N\times \mathrm{poly}(T_p))$ otherwise).

A single Sinkhorn iteration for $N=120$ costs $\sim$0.006 ms, compared to $\sim$0.1 s for the Hungarian method. For $N=150, 500, 800$: Sinkhorn at 0.006 ms, 0.03 ms, 0.08 ms; full LP at 0.12 s, 6.4 s, 66 s, respectively, illustrating the scalability of Sinkhorn MPC. Warm-starting further decreases per-step Sinkhorn iteration count (from $\sim$520 to $\sim$100 after first step for $N=120$) [2301.06492, Table 1, Figs. 3–5; 2203.08415, section 6].

## 6. Influence of Parameters and Practical Guidelines

- **Horizon $T_p$**: Should exceed the mixing time required for individual agents' MPC trajectories to nearly reach their (temporary) targets.
- **Entropic regularization $\varepsilon$**: Large values favor smooth, distributed assignments and improve transient response, but increase steady-state bias. Small $\varepsilon$ sharpens the assignment towards combinatorial (permutation) solutions, at the cost of increased numerical iterations and possible transient artifacts.
- **Sinkhorn step count $S$**: One iteration yields higher energy and lower quality transients. $S=5$ or more closely approximates unregularized assignment solutions.
- **Cost shaping**: Quadratic $\ell_i$ enables closed-form MPC updates; non-quadratic and input-constrained cases remain tractable but require additional QP or LP solution per agent [2301.06492, Section 5; 2203.08415, Section 6].

## 7. Numerical Examples and Benchmark Results

- **Continuous linear agents (quadratic cost)**: For $N=120$, $\varepsilon=2$, $\tau_p=100$, the method achieves rapid convergence (smooth trajectories), with per-step Sinkhorn costs as noted above.
- **Effect of $S$ and $\varepsilon$**: $S=1$ increases transient energy; $S=5$ nearly matches ideal assignment performance; lower $\varepsilon$ sharpens the steady-state but slows convergence.
- **Non-quadratic costs & input constraints**: For $\ell_i(x_i, u_i; x_j^*) = \|x_i - x_j^*\|^2 + \|u_i - \Upsilon_i\|_1$ subject to $|u_i| \leq 5$, Sinkhorn MPC executes successfully with $S=20$, $\tau_p=50$ for $N=15$, exhibiting sparse control and accurate terminal distribution [2301.06492, Figs. 6–8].

---

**References:**  
- "Entropic Model Predictive Optimal Transport over Dynamical Systems" [2301.06492].
- "Sinkhorn MPC: Model predictive optimal transport over dynamical systems" [2203.08415].

Source: https://www.emergentmind.com/topics/sinkhorn-mpc