Workshop on Stochastic Planning & Control of Dynamical Systems

Density Control of Gaussian Mixtures Models: From Controlling Large Populations to Generative AI. 

George Rapakoulias

We present a novel framework for controlling the state distribution of linear dynamical systems, in the special case that this can be expressed as a Gaussian Mixture Model. The central object is the approximation of the solution to the Schrödinger Bridge problem with GMM marginal distributions, an optimal diffusion process transporting one Gaussian mixture into another at minimal cost. Classical approaches to solve SBs often rely on spatial discretizations or costly neural network training, requiring considerable computation even in low dimensional problems. Building on recent advances in the fields of computational optimal transport and distributional control, we introduce a closed-form parameterization of a feasible set of solutions to the SB with GMM boundary distributions and approximate the solution within this feasible set by solving a linear program. This yields numerically efficient, interpretable, and constraint-aware control laws, enabling applications from steering interacting multi-agent populations to lightweight generative AI models.

Julia Toolbox for Stochastic Barrier Functions in Safety Verification of Stochastic Systems

Rayan Mazouz

We introduce StochasticBarrier.jl, an open-source Julia-based toolbox for generating Stochastic Barrier Functions to verify safety in discrete-time stochastic systems with additive Gaussian noise. The framework supports linear, polynomial, and piecewise affine dynamics, enabling verification of general nonlinear systems. The tool implements both sum-of-squares optimization and piecewise constant approaches, the latter offering three engines: two based on (dual) linear programming and one on gradient descent. Benchmarking across many case studies demonstrates that the tool outperforms state-of-the-art methods in computation time, scalability, and conservativeness of safety probability bounds.

Safe Dual Control through MPC

Tren M.J.T. Baltussen

Dual control addresses the fundamental trade-off between regulation and information acquisition in uncertain dynamical systems. While Model Predictive Control (MPC) provides a principled framework for constrained and safety-critical control, existing robust and stochastic MPC schemes often neglect the dependence of the posterior distribution of the state, i.e. the hyperstate, on the control policy itself. This separation limits their ability to capture the dual control effect, which is of particular interest in interactive environments. Our research develops MPC methods that explicitly incorporate the dual control effect, e.g., in Gaussian Process-based MPC. While effective as a heuristic, safety guarantees of GP-MPC are hindered by modeling errors. To address this, we present a multi-horizon, contingency MPC framework that systematically integrates learning-based models with robust constraint satisfaction. This approach ensures recursive feasibility and safety while mitigating conservatism by exploiting the receding horizon of MPC. We demonstrate our method on use cases in automated driving. This work is a first step toward theoretically supported, dual model predictive control methods for safety-critical systems.

Optimal Control of Probabilistic Dynamics Models via Mean Hamiltonian Minimization

David Leeftink

A central question in learning-based control is how to plan effectively under learned models from limited data. In this talk, we explore this question by building a bridge from classical optimal control theory to modern deep reinforcement learning. We introduce a practical framework based on the principle of mean Hamiltonian minimization, where we plan over an entire ensemble of learned dynamics models simultaneously. This approach provides a structured and effective alternative to common trajectory optimization methods. I will present our method, show its effectiveness on several nonlinear control problems, and discuss its potential as a robust foundation for building reliable decision-making algorithms.

Distributional Robust Control

Riccardo Cescon

Distributionally Robust Optimization (DRO) has recently gained attention in control as a principled approach to handle uncertainty. Classical controllers’ performance often degrade when the noise distribution deviates from the nominal one, whereas DRO provides robustness by optimizing against the most adverse distribution within an ambiguity set. Within this framework, the Wasserstein distance is the most common choice to define such sets, but it presents key limitations: the worst-case distribution is finitely supported when the nominal one is, and tractability in control problems typically requires restrictive assumptions such as linear dynamics. This talk introduces the Sinkhorn divergence as an alternative for constructing ambiguity sets. We demonstrate its applicability to standard control problems, showing how it overcomes some limitations of Wasserstein-based approaches. Finally, we outline extensions toward nonlinear control, where gradient-based optimization techniques enable the design of distributionally robust policies beyond the linear setting.

Formal Uncertainty Propagation in Wasserstein Distance

Steven Adams and Eduardo Figueiredo Mota Diniz Costa

Modern control systems face uncertainty both in their dynamics and in the probabilistic models used to describe them. Propagating such uncertainty through nonlinear stochastic systems is typically intractable. We present a framework for formal uncertainty propagation that combines optimal transport, distributionally robust optimization, and quantization. By modeling "uncertainty of uncertainty" as Wasserstein balls of distributions, our method enables efficient and guaranteed propagation of probabilistic sets through nonlinear dynamics. We demonstrate our open-source implementation (https://github.com/sjladams/WassProp) on a benchmark example during the demo.

Chance-Constrained Stochastic Trajectory Optimization for Spacecraft in Nonlinear Dynamical Systems

Naoya Kumagai

As humanity's presence in space expands, current mission design and operation practices may not scale to more frequent and ambitious missions within limited budgets. The state of the practice requires iteration between separate software that each perform the task of trajectory optimization and uncertainty quantification. To pave the path towards more automated mission design, we leverage stochastic optimal control techniques in order to handle, within a unified framework, trajectory optimization and common uncertainties such as stochastic acceleration and navigation error. We demonstrate an example of a transfer in the near-moon region where the framework modifies the original trajectory in highly nonlinear phases by considering uncertainty models.

Sequential Convex Programming for Stochastic Multi-agent Systems with Connectivity Constraints

Aman Tiwary

In this work, we present an optimization-based motion planning framework to generate feasible trajectories for a stochastic multi-agent system (MAS), navigating through a cluttered environment. The objective is to ensure global connectivity among the agents while avoiding both inter-agent collisions and obstacles, with at least a desired likelihood. The traditional approach to trajectory planning for multiple agents in a deterministic setting is to embed connectivity constraints within a mixed integer programming (MIP) formulation. However, when the MAS is subject to stochastic disturbances, the trajectories must satisfy the constraints with a specified probability. A natural way to model this is through chance constraints, but this makes the problem considerably more complex. The primary challenges arise from the integer constraints induced by connectivity requirements, the chance constraints introduced by the stochastic nature of the problem, and the interaction between the two. To address these challenges, we introduce two key relaxations. First, we propose a continuous reformulation of the connectivity constraint via complementarity conditions, leading to a purely continuous formulation. Second, the chance constraints are handled through a scenario-based relaxation, where independent scenarios are sampled from the distribution and incorporated into the optimization problem to ensure probabilistic feasibility of the solution. The resulting optimization problem is then solved using a continuous-time successive convexification framework, which guarantees convergence to feasible trajectories and continuous time satisfaction of constraints.