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Schrödinger Bridges over Kinetic Swarming Models

This paper proposes a minimum-energy collective steering framework for inertial swarms subject to stochastic disturbances and governed by mean-field interaction models, utilizing Schrödinger bridge theory to derive optimal state-feedback controls that steer the system between prescribed endpoint distributions by dynamically exploiting or counteracting natural interaction forces.

Original authors: Asmaa Eldesoukey, Md Zulfiqur Haider, Italo Napolitano, Yongxin Chen, Abhishek Halder

Published 2026-08-27

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CC BY 4.0

Original authors: Asmaa Eldesoukey, Md Zulfiqur Haider, Italo Napolitano, Yongxin Chen, Abhishek Halder

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the natural world, from the swirling schools of fish to the murmuring flocks of starlings, complex order often arises from simple rules. Individual creatures do not need a central commander to coordinate their movements; instead, they react to their neighbors, aligning their speed or avoiding collisions. This phenomenon, known as emergent behavior, suggests that the collective is greater than the sum of its parts. Scientists have long studied how these groups form naturally, but a different question has recently captured the attention of researchers: how can we guide such a group to a specific shape or location within a set amount of time? This is not just about observing nature but about steering it, perhaps to guide a swarm of microscopic robots for medical treatment or to manage a crowd during an evacuation. The challenge lies in the fact that these groups are subject to random disturbances, like wind or noise, and their members are constantly influencing one another. To solve this, researchers have turned to a concept originally developed in the 1930s by the physicist Erwin Schrödinger. He imagined a scenario where particles move according to known laws, but their final positions are observed to be different from what those laws predicted. He asked a profound question: what is the most likely path these particles took to get from their starting point to that unexpected ending? This line of thinking, now called the Schrödinger bridge, provides a mathematical way to find the most efficient path between two states, treating the natural, uncontrolled movement as a baseline and calculating the smallest possible nudge needed to reach the desired destination.

A team of researchers has now applied this framework to a new and difficult class of problems involving "inertial swarms." Unlike simple models where particles move at a constant speed, these swarms consist of agents with mass that have velocity and momentum, meaning they cannot stop or turn instantly. Furthermore, these agents interact with each other through specific forces, such as the tendency to align their direction of travel or a mix of attraction and repulsion that keeps them together without crashing. The researchers wanted to know how to steer such a complex, noisy, and interacting group from a starting distribution to a target distribution in a finite amount of time while using the least amount of energy possible. They focused on two well-known models of interaction: one where agents align their velocities with their neighbors, similar to birds flocking, and another where agents are pulled together from a distance but pushed apart when they get too close, mimicking the behavior of many biological swarms.

The team developed a mathematical framework that treats the uncontrolled, natural movement of the swarm as a prior expectation. They then sought the optimal control input—a corrective force applied to every agent—that would guide the swarm to the target configuration. This control acts as a drift, a gentle push that adjusts the swarm's path. Crucially, the researchers considered two different scenarios regarding what information was available at the end of the journey. In the first scenario, they knew exactly where every agent should be and how fast it should be moving. In the second, they only knew where the agents should be located, leaving their final speeds to be determined by the most efficient path. This distinction is vital because, in many real-world applications, measuring the precise velocity of every individual in a large group is impossible; we might only see the overall shape of the crowd.

To solve these problems, the researchers derived a set of complex, coupled equations that describe how the swarm's density evolves over time. These equations are nonlinear and depend on the entire history of the swarm's interactions, making them difficult to solve directly. The team proposed a novel numerical method, a nested iterative scheme, to find solutions. This approach involves guessing a solution, refining it based on the interactions, and repeating the process until the answer stabilizes. Their simulations, conducted in a simplified one-dimensional world, revealed how the optimal controller adapts to the specific nature of the interactions. In cases where the natural tendency of the swarm helped achieve the goal, the controller exploited these forces, saving significant energy. Conversely, when the natural interactions worked against the goal, the controller actively counteracted them.

The results demonstrated that this approach is highly efficient. In one simulation involving a flocking model where agents naturally aligned their speeds, the researchers needed to steer the group to split into two distinct spatial clusters while synchronizing their velocities. The optimal controller found a way to use the agents' natural desire to align to help separate them spatially, requiring only about half the energy of a controller that ignored these interactions. In another scenario, where the goal was to infer the most likely movement of a swarm given only its starting and ending positions, the controller successfully reconstructed a plausible velocity distribution that matched the spatial constraints. The energy cost for this interacting controller was significantly lower than that of a baseline strategy that simply canceled out all interactions and treated the agents as if they were moving alone.

The researchers also tested a model driven by attractive and repulsive forces, similar to how molecules or animals maintain a cohesive group. Here, the natural tendency was for the group to spread out due to short-range repulsion. To bring the group into a tighter, more cohesive formation, the controller had to work against this natural spreading force. Even in this challenging case, where the controller had to fight the system's natural dynamics, the proposed method still outperformed a strategy that ignored the interactions entirely. The study confirms that by understanding and mathematically modeling the specific ways agents influence one another, it is possible to guide large, noisy swarms to precise configurations with remarkable efficiency. The work suggests that the most effective way to control a complex system is not to override its natural laws, but to understand them and apply the minimum necessary correction to steer the collective toward a desired future.

Technical Summary: Schrödinger Bridges over Kinetic Swarming Models

Problem Formulation
This work addresses the problem of finite-horizon minimum-energy collective steering for inertial swarms subject to stochastic disturbances. The authors consider multi-agent systems governed by kinetic models where agents interact via either Cucker–Smale alignment or Morse attraction–repulsion potentials. The objective is to steer the swarm from a prescribed initial probability distribution to a target terminal distribution within a fixed time horizon $T$ .

Two distinct scenarios are investigated regarding the specification of endpoint distributions:

Full Phase-Space Constraints: Both the initial and target distributions are specified over the full phase space (positions and velocities).
Spatial Marginal Constraints: Only the spatial marginals (positions) are prescribed at the endpoints, leaving the velocity distributions to be inferred. This models scenarios where only aggregate snapshots of the swarm are available.

The control input is a state-feedback command broadcast to all agents, aiming to regulate the collective dynamics rather than individual trajectories. The uncontrolled interacting stochastic dynamics serve as a "prior" model, and the goal is to find the minimum-energy corrective drift required to realize the prescribed endpoint distributions.

Methodology
The formalism is rooted in the theory of Schrödinger bridges, interpreted through the lens of stochastic optimal control and large deviations theory.

Inference-Control Duality: The problem is framed as finding the most likely path evolution consistent with observed marginals, which is equivalent to minimizing the relative entropy (Kullback-Leibler divergence) between the controlled path law and the uncontrolled prior. By Girsanov's theorem, this is recast as a stochastic optimal control problem minimizing the expected control energy.
Mean-Field Limit: The authors consider the limit where the number of agents approaches infinity. The collective density evolves according to a nonlinear Vlasov–Fokker–Planck integro-partial differential equation (PDE).
Optimality Systems:
- For Cucker–Smale kinetics, the authors derive a system of nonlinear, coupled integro-PDEs. Using a Cole–Hopf (logarithmic) transformation, the optimal control is expressed as a gradient feedback $u^*_t = \sigma^2 \nabla_v \log \phi_t$ . The system consists of a forward equation for $\hat{\phi}_t$ and a backward equation for $\phi_t$ , coupled nonlocally through the interaction forces and the density $\mu_t = \phi_t \hat{\phi}_t$ .
- For Morse-potential driven kinetics, a similar optimality system is derived. Notably, the interaction force depends only on the spatial marginal, simplifying the coupling structure slightly compared to the Cucker–Smale case.
- For partial information (spatial marginals only), the formulation includes an additional relative entropy term $H(\mu_0, \nu_0)$ to account for the uncertainty in the initial velocity distribution. The boundary conditions for the forward-backward system are modified to reflect that only spatial marginals are fixed, while the terminal control vanishes ( $u^*_T = 0$ ) due to the lack of velocity constraints.
Numerical Solution: Due to the nonlinearity and nonlocal coupling, the authors propose nested fixed-point iterative schemes. These schemes utilize delayed evaluations to handle the nonlinearities, alternating between solving the forward and backward equations until convergence.

Key Contributions

Extension of Schrödinger Bridges: The work extends the classical Schrödinger bridge framework to interacting kinetic systems (second-order dynamics) under both complete (phase-space) and partial (spatial marginal) endpoint observations.
Derivation of Optimality Conditions: The authors derive the necessary optimality systems for both Cucker–Smale and Morse interaction models. These systems generalize the classical Schrödinger system to include nonlocal mean-field interactions and inertial effects.
Handling Partial Observations: A specific formulation is provided for scenarios where only spatial marginals are known, inferring the corresponding velocity distributions and the necessary corrective drift.
Numerical Algorithms: Novel nested iterative schemes are proposed to compute solutions for these coupled, nonlinear systems.

Results
Numerical examples in one spatial and one velocity dimension demonstrate the efficacy of the proposed framework:

Cucker–Smale Steering (Full Phase-Space): The controller successfully steers a disordered swarm with two subpopulations moving in opposite directions to a state of spatial splitting with synchronized velocities. The results show that the optimal controller dynamically exploits the Cucker–Smale alignment interaction when it aids the steering task (reducing control effort) and counteracts it when necessary. The accumulated cost of the interacting controller was significantly lower (approx. 50% and 62% of the cost) compared to controllers that ignore interactions or cancel them out.
Cucker–Smale Steering (Partial Information): When only spatial marginals were prescribed, the controller successfully inferred a velocity distribution consistent with the target spatial configuration. The control effort was again minimized by leveraging the interaction dynamics.
Morse-Potential Steering: The controller steered a population subject to short-range repulsion and long-range attraction toward a more concentrated, cohesive flock. In this case, the controller largely counteracted the repulsive interaction to achieve the target. The interacting controller still achieved a lower cost than a baseline controller that simply canceled interactions, though the cost was higher than the non-interacting benchmark due to the adversarial nature of the repulsion.

Significance and Claims
The paper claims to bridge the gap between kinetic mean-field limits and Schrödinger bridge formulations for interacting systems. It establishes that the inference-control connection, previously established for weakly interacting inertial systems, can be extended to endpoint-constrained problems with nonlocal interactions.

The authors emphasize that their framework provides a mechanism for collective steering where the control strategy adapts to the interaction forces: it exploits favorable interactions to reduce energy expenditure and counteracts adversarial ones to ensure the target is met. The work is positioned as a tool for applications requiring timely pattern formation or reconfiguration, such as targeted therapy with microrobotic swarms or emergency evacuation, where decentralized self-organization alone may be insufficient. The authors note that the optimization problems are nonconvex, meaning the derived solutions are stationary points (first-order necessary conditions) rather than guaranteed global minima, and suggest future work on second-order conditions and multi-marginal formulations.

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