arXiv subjects

Riccardo Bonalli

Publications and source records attributed to Riccardo Bonalli.

At least 19 recordsLinked to original sources

arXiv2026

Learning Controlled Stochastic Differential Equations

We study the problem of learning controlled stochastic differential equations (SDEs) \[ dX_t = b(t,X_t,u_t)\,dt + σ(t,X_t,u_t)\,dW_t, \] whose drift and diffusion depend nonlinearly on time, state, and control values. From trajectory data, we aim to estimate coefficients whose induced density flows reproduce those of the observed dynamics. The data consist of several controls sampled from a finite-dimensional family and, for each control, multiple independent trajectories observed at discrete times over a fixed horizon. The controls are observed inputs, not learner-selected decisions. We propose a kernel method for multidimensional nonlinear controlled SDEs. The method estimates the density flow for each sampled control, then fits the drift $b$ and diffusion matrix $a=σσ^\top$ by matching the estimated flows through the Fokker-Planck equation. Under Sobolev regularity assumptions, we prove finite-sample bounds on the error between the density flows of the learned and true SDEs, measured in $L^2$ over controls, time, and state. The bounds quantify how the error decreases with the number $K$ of sampled controls, with rates determined by the state and control-parametrization dimensions and Sobolev regularity. We further derive uniform-in-control guarantees and CVaR-type bounds for tail-sensitive quantities. Numerical experiments illustrate the method, and an open-source Python implementation is provided.

stat.ML↗

arXiv2026

Safe and Steerable Geometric Motion Policies for Robotic Dexterous Manipulation

Robotic dexterous manipulation requires continuously reconciling objectives and constraints defined on heterogeneous geometric spaces: a robot controlled on a $\mathbb{R}^7$ configuration manifold may need to track end effector poses on $\mathrm{SE}(3)$ while satisfying obstacle avoidance margins in $\mathbb{R}$. We present Safe Pullback Bundle Dynamical Systems (SafePBDS), a geometrically consistent framework that computes optimal, certifiably safe configuration manifold accelerations from objectives and safety requirements on arbitrary task manifolds. SafePBDS builds on prior work that combines predefined task manifold dynamical systems to produce autonomous motion. Its first innovation is a pullback control barrier function construction, which converts task manifold safety conditions into linear constraints on configuration manifold accelerations. The second innovation is a task manifold action interface that allows a high-level policy to inject low dimensional residual motions; zero input recovers the autonomous behavior, while safety is preserved under arbitrary inputs. This lets high-level policies efficiently steer exploration while leaving precise motion to the autonomous behavior. We validate SafePBDS in simulation and on a 23-DOF Franka Panda-Allegro Hand platform. On dexterous grasping, SafePBDS achieves a $92.5\%$ success rate across 20 household objects and 120 trials. Using the action interface, the method can exclude any one of the four fingers during grasping via a one-dimensional action, achieving $94.4\%$ 3-finger grasp success across 3 objects and 36 trials. The efficient planning and safety guarantee of SafePBDS also enables the first model-based, fully actuated palm-down in-hand reorientation, exceeding $360^\circ$ of yaw rotation in both directions under varying object weight and wrist motion. Demo video and details: https://tml.stanford.edu/safe-pbds

cs.RO↗

arXiv2026

Convex Hulls of Reachable Sets

We study the convex hulls of reachable sets of nonlinear systems with bounded disturbances and uncertain initial conditions. Reachable sets play a critical role in control, but remain notoriously challenging to compute, and existing over-approximation tools tend to be conservative or computationally expensive. In this work, we characterize the convex hulls of reachable sets as the convex hulls of solutions of an ordinary differential equation with initial conditions on the sphere. This finite-dimensional characterization unlocks an efficient sampling-based estimation algorithm to accurately over-approximate reachable sets. We also study the structure of the boundary of the reachable convex hulls and derive error bounds for the estimation algorithm. We give applications to neural feedback loop analysis and robust MPC.

math.OC↗

arXiv2026

Safely Learning Controlled Stochastic Dynamics

We address the problem of safely learning controlled stochastic dynamics from discrete-time trajectory observations, ensuring system trajectories remain within predefined safe regions during both training and deployment. Safety-critical constraints of this kind are crucial in applications such as autonomous robotics, finance, and biomedicine. We introduce a method that ensures safe exploration and efficient estimation of system dynamics by iteratively expanding an initial known safe control set using kernel-based confidence bounds. After training, the learned model enables predictions of the system's dynamics and permits safety verification of any given control. Our approach requires only mild smoothness assumptions and access to an initial safe control set, enabling broad applicability to complex real-world systems. We provide theoretical guarantees for safety and derive adaptive learning rates that improve with increasing Sobolev regularity of the true dynamics. Experimental evaluations demonstrate the practical effectiveness of our method in terms of safety, estimation accuracy, and computational efficiency.

stat.ML↗

arXiv2025

Spectral properties of magnetic fields on sub-Riemannian contact manifolds

Motivated by some recent studies of the magnetic Laplacian on Riemannian manifolds, we focus on the first eigenvalue of the magnetic horizontal Laplacian on contact manifolds. We characterize conditions for positive spectral shift, and provide some sharp upper bounds. In the Riemannian setting, a genus 1 assumption is known to force the underlying metric to be flat when equality holds in the sharp upper bounds. Interestingly, we show that the equivalent topological condition in the three--dimensional contact setting consists of having first Betti number equal to 2. In this case, equality in our upper bounds implies that the structure is that of a Heisenberg left--invariant nilmanifold. We conclude by showing that, in some specific three--dimensional contact settings, the knowledge of the first eigenvalue of the magnetic Laplacian uniquely determines the manifold Chern class, fully determining the topology of the underlying manifold.

math.DG↗

arXiv2025

A Characterization of Law-Invariant and Coherent Risk Measures through Optimal Transport

In this article, we propose a novel characterization of law-invariant and coherent risk measures, based on a generalized optimal transport problem in which the second marginal of the admissible plans is not fixed, but required to lie within a target set of probability measures. One of the main contributions of this work is a general representation formula for such risk measures, which is closely related to Kusuoka's theorem. When the aforementioned target set is convex, our representation result allows for the systematic derivation of general duality formulas. To illustrate our findings, we explicitly compute the target sets associated with several classical law-invariant coherent risk measures, including the prototypical conditional value at risk and higher moment measures.

math.OC↗

arXiv2025

A Gradient Method for Risk Averse Control of a PDE-SDE Interconnected System

In this paper, we design a risk-averse controller for an interconnected system composed of a linear Stochastic Differential Equation (SDE) actuated through a linear parabolic heat equation. These dynamics arise in various applications, such as coupled heat transfer systems and chemical reaction processes that are subject to disturbances. While existing optimal control methods for these systems focus on minimizing average performance, this risk-neutral perspective may allow rare but highly undesirable system behaviors. To account for such events, we instead minimize the cost within a coherent risk measure. Our approach reformulates the coupled dynamics as a stochastic PDE, approximates it by a finite-dimensional SDE system, and applies a gradient-based method to compute a riskaverse feedback controller. Numerical simulations show that the proposed controller substantially reduces the tail of the cost distribution, improving reliability with only a minor reduction in average performance.

math.OC↗

arXiv2025

Optimal Control of an Interconnected SDE -Parabolic PDE System

In this paper, we design a controller for an interconnected system where a linear Stochastic Differential Equation (SDE) is actuated through a linear parabolic heat equation. These dynamics arise in various applications, such as coupled heat transfer systems and chemical reaction processes that are subject to disturbances. Our goal is to develop a computational method for approximating the controller that minimizes a quadratic cost associated with the state of the SDE component. To achieve this, we first perform a change of variables to shift the actuation inside the PDE domain and reformulate the system as a linear Stochastic Partial Differential Equation (SPDE). We use a spectral approximation of the Laplacian operator to discretize the coupled dynamics into a finite-dimensional SDE and compute the optimal control for this approximated system. The resulting control serves as an approximation of the optimal control for the original system. We then establish the convergence of the approximated optimal control and the corresponding closed-loop dynamics to their infinite-dimensional counterparts. Numerical simulations are provided to illustrate the effectiveness of our approach.

math.AP↗

arXiv2025

Non-Parametric Learning of Stochastic Differential Equations with Non-asymptotic Fast Rates of Convergence

We propose a novel non-parametric learning paradigm for the identification of drift and diffusion coefficients of multi-dimensional non-linear stochastic differential equations, which relies upon discrete-time observations of the state. The key idea essentially consists of fitting a RKHS-based approximation of the corresponding Fokker-Planck equation to such observations, yielding theoretical estimates of non-asymptotic learning rates which, unlike previous works, become increasingly tighter when the regularity of the unknown drift and diffusion coefficients becomes higher. Our method being kernel-based, offline pre-processing may be profitably leveraged to enable efficient numerical implementation, offering excellent balance between precision and computational complexity.

cs.LG↗

arXiv2025

Stabilization and Optimal Control of an Interconnected $n + m$ Hetero-directional Hyperbolic PDE-SDE System

In this paper, we design a controller for an interconnected system composed of a linear Stochastic Differential Equation (SDE) controlled through a linear hetero-directional hyperbolic Partial Differential Equation (PDE). Our objective is to steer the coupled system to a desired final state on average, while keeping the variance-in-time as small as possible, improving robustness to disturbances. By employing backstepping techniques, we decouple the original PDE, reformulating the system as an input delayed SDE with a stochastic drift. We first establish a controllability result, shading light on lower bounds for the variance. This shows that the system can never improve variance below strict structural limits. Under standard controllability conditions, we then design a controller that drives the mean of the states while keeping the variance bounded. Finally, we analyze the optimal control problem of variance minimization along the entire trajectory. Under additional controllability assumptions, we prove that the optimal control can achieve any variance level above the fundamental structural limit.

math.OC↗

arXiv2024

Sample Average Approximation for Stochastic Programming with Equality Constraints

We revisit the sample average approximation (SAA) approach for non-convex stochastic programming. We show that applying the SAA approach to problems with expected value equality constraints does not necessarily result in asymptotic optimality guarantees as the sample size increases. To address this issue, we relax the equality constraints. Then, we prove the asymptotic optimality of the modified SAA approach under mild smoothness and boundedness conditions on the equality constraint functions. Our analysis uses random set theory and concentration inequalities to characterize the approximation error from the sampling procedure. We apply our approach and analysis to the problem of stochastic optimal control for nonlinear dynamical systems under external disturbances modeled by a Wiener process. Numerical results on relevant stochastic programs show the reliability of the proposed approach. Results on a rocket-powered descent problem show that our computed solutions allow for significant uncertainty reduction compared to a deterministic baseline.

math.OC↗

arXiv2024

Stabilization and Optimal Control of Interconnected SDE - Scalar PDE System

In this paper, we design a controller for an interconnected system consisting of a linear Stochastic Differential Equation (SDE) actuated through a linear hyperbolic Partial Differential Equation (PDE). Our approach aims to minimize the variance of the state of the SDE component. We leverage a backstepping technique to transform the original PDE into an uncoupled stochastic PDE. As such, we reformulate our initial problem as the control of a delayed SDE with a non-deterministic drift. Under standard controllability assumptions, we design a controller steering the mean of the states to zero while keeping its covariance bounded. As final step, we address the optimal control of the delayed SDE employing Artstein's transformation and Linear Quadratic stochastic control techniques.

math.OC↗

arXiv2024

Estimating the Convex Hull of the Image of a Set with Smooth Boundary: Error Bounds and Applications

We study the problem of estimating the convex hull of the image $f(X)\subset\mathbb{R}^n$ of a compact set $X\subset\mathbb{R}^m$ with smooth boundary through a smooth function $f:\mathbb{R}^m\to\mathbb{R}^n$. Assuming that $f$ is a submersion, we derive a new bound on the Hausdorff distance between the convex hull of $f(X)$ and the convex hull of the images $f(x_i)$ of $M$ sampled inputs $x_i$ on the boundary of $X$. When applied to the problem of geometric inference from a random sample, our results give error bounds that are tighter and more general than in previous work. We present applications to the problems of robust optimization, of reachability analysis of dynamical systems, and of robust trajectory optimization under bounded uncertainty.

math.OC↗

arXiv2023

Mean-Covariance Steering of a Linear Stochastic System with Input Delay and Additive Noise

In this paper, we introduce a novel approach to solve the (mean-covariance) steering problem for a fairly general class of linear continuous-time stochastic systems subject to input delays. Specifically, we aim at steering delayed linear stochastic differential equations to a final desired random variable with given mean and covariance. We first establish a controllability result for these control systems, revealing the existence of a lower bound under which the covariance of the control system can not be steered. This structural threshold covariance stems from a unique combined effect due to stochastic diffusions and delays. Next, we propose a numerically cheap approach to reach any neighbor of this threshold covariance in finite time. Via an optimal control-based strategy, we enhance the aforementioned approach to keep the system covariance small at will in the whole control horizon. Under some additional assumptions on the dynamics, we give theoretical guarantees on the efficiency of our method. Finally, numerical simulations are provided to ground our theoretical findings, showcasing the ability of our methods in optimally approaching the covariance threshold.

math.OC↗

arXiv2023

Characterization of Singular Arcs in Spacecraft Trajectory Optimization

Low-thrust engines for interplanetary spacecraft transfers allow cost-effective space missions with flexible launch and arrival dates. To find fuel-optimal trajectories, an optimal control problem is to be solved. Pontryagin's Maximum Principle shows that the structure of the optimal control is bang-bang with the possibility of singular arcs. Even though the latter have been heuristically shown to rarely appear in practical applications, a full theoretical characterization does not exist in the literature. As a growing number of missions are expected to adopt low-thrust engines in the near future, such study is required to have a comprehensive understanding of the problem. This work presents analytical necessary conditions for the existence of singular arcs that only depend on three physical variables and not on the costates. Moreover, it provides an analytical expression of the singular control that depends on a limited set of physical variables. This is a fundamental feature, as simple evaluation of the necessary condition and of the singular control can be performed. Finally, it provides insightful information on the reasons why singular arcs are rare and it quantifies the possibility of their occurrence.

math.OC↗

arXiv2023

Risk-Averse Trajectory Optimization via Sample Average Approximation

Trajectory optimization under uncertainty underpins a wide range of applications in robotics. However, existing methods are limited in terms of reasoning about sources of epistemic and aleatoric uncertainty, space and time correlations, nonlinear dynamics, and non-convex constraints. In this work, we first introduce a continuous-time planning formulation with an average-value-at-risk constraint over the entire planning horizon. Then, we propose a sample-based approximation that unlocks an efficient and general-purpose algorithm for risk-averse trajectory optimization. We prove that the method is asymptotically optimal and derive finite-sample error bounds. Simulations demonstrate the high speed and reliability of the approach on problems with stochasticity in nonlinear dynamics, obstacle fields, interactions, and terrain parameters.

cs.RO↗

arXiv2023

On the Accessibility and Controllability of Statistical Linearization for Stochastic Control: Algebraic Rank Conditions and their Genericity

Statistical linearization has recently seen a particular surge of interest as a numerically cheap method for robust control of stochastic differential equations. Although it has already been successfully applied to control complex stochastic systems, accessibility and controllability properties of statistical linearization, which are key to make the robust control problem well-posed, have not been investigated yet. In this paper, we bridge this gap by providing sufficient conditions for the accessibility and controllability of statistical linearization. Specifically, we establish simple sufficient algebraic conditions for the accessibility and controllability of statistical linearization, which involve the rank of the Lie algebra generated by the drift only. In addition, we show these latter algebraic conditions are essentially sharp, by means of a counterexample, and that they are generic with respect to the drift and the initial condition.

math.OC↗

arXiv2023

A Gradient Descent-Ascent Method for Continuous-Time Risk-Averse Optimal Control

In this paper, we consider continuous-time stochastic optimal control problems where the cost is evaluated through a coherent risk measure. We provide an explicit gradient descent-ascent algorithm which applies to problems subject to non-linear stochastic differential equations. More specifically, we leverage duality properties of coherent risk measures to relax the problem via a smooth min-max reformulation which induces artificial strong concavity in the max subproblem. We then formulate necessary conditions of optimality for this relaxed problem which we leverage to prove convergence of the gradient descent-ascent algorithm to candidate solutions of the original problem. Finally, we showcase the efficiency of our algorithm through numerical simulations involving trajectory tracking problems and highlight the benefit of favoring risk measures over classical expectation.

math.OC↗