Impulsive mechanical systems exhibit discontinuous jumps in their state, and when such jumps are triggered by spatial events, the geometry of the impact surface carries information about the controllability of the hybrid dynamics. For mechanical systems defined on principal G-bundles, two qualitatively distinct types of impacts arise: interior impacts, associated with events on the shape space, and exterior impacts, associated with events on the fibers. A key distinction is that interior impacts preserve the mechanical connection, whereas exterior impacts generally do not. In this paper, we exploit this distinction by allowing actuation through exterior impacts. We study the pendulum-on-a-cart system, derive controlled reset laws induced by moving-wall impacts, and analyze the resulting periodic motions. Our results show that reset action alone does not provide a convincing stabilizing regime, whereas the addition of dissipation in the continuous flow yields exponentially stable periodic behavior for suitable feedback gains.

The nonholonomic integrator is a famous example in feedback design - although it is small-time locally controllable to the origin, no continuous feedback law exists. Therefore, any stabilizing feedback laws must be either time-varying or discontinuous. A previously studied discontinuous feedback law stabilizes initial conditions lying between two paraboloids and has a sliding mode on the xy-plane. We investigate the effect of introducing delays into this discontinuous feedback law. To a first-order analysis, the lag causes the sliding mode of the xy-plane to bifurcate into two switching regions where the resulting dynamics can be interpreted as a hybrid dynamical system with hysteresis. Counterintuitively, the presence of a delay can actually have a positive effect on both the size of the basin of attraction and the convergence rate of the controller. We also consider the natural generalization of the nonholonomic integrator to higher dimensions.

Nonholonomic systems are, so to speak, mechanical systems with a prescribed restriction on the velocities. A virtual nonholonomic constraint is a controlled invariant distribution associated with an affine connection mechanical control system. A Riemannian homogeneous space is, a Riemannian manifold that looks the same everywhere, as you move through it by the action of a Lie group. These Riemannian manifolds are not necessarily Lie groups themselves, but nonetheless possess certain symmetries and invariances that allow for similar results to be obtained. In this work, we introduce the notion of virtual constraint on Riemannian homogeneous spaces in a geometric framework which is a generalization of the classical controlled invariant distribution setting and we show the existence and uniqueness of a control law preserving the invariant distribution. Moreover we characterize the closed-loop dynamics obtained using the unique control law in terms of an affine connection. We illustrate the theory with new examples of nonholonomic control systems inspired by robotics applications. (c) 2025 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Data-driven modeling is playing an increasing role in robotics and control, yet standard learning methods typically ignore the geometric structure of nonholonomic systems. As a consequence, the learned dynamics may violate the nonholonomic constraints and produce physically inconsistent motions. In this paper, we introduce a structure-preserving Gaussian process (GP) framework for learning nonholonomic dynamics. Our main ingredient is a nonholonomic matrix-valued kernel that incorporates the constraint distribution directly into the GP prior. This construction ensures that the learned vector field satisfies the nonholonomic constraints for all inputs. We show that the proposed kernel is positive semidefinite, characterize its associated reproducing kernel Hilbert space as a space of admissible vector fields, and prove that the resulting estimator admits a coordinate representation adapted to the constraint distribution. We also establish the consistency of the learned model. Numerical simulations on a vertical rolling disk illustrate the effectiveness of the proposed approach.

This paper investigates the problem of identifying state-dependent switching systems, a class of hybrid dynamical systems that combine multiple linear or nonlinear modes. We propose two broad classes of switching systems: switching linear systems (SLSs) and switching polynomial systems (SPSs). We first formulate the joint estimation of the mode dynamics and switching rules as a mixed integer program. To solve its inherent scalability issue, we develop a hierarchy of convex relaxations and establish a bound and conditions under which these relaxations are tight. Building on these results, we propose a bilevel convex optimization framework that alternates between mode assignment and dynamics estimation, and we recover switching boundaries using margin-based polynomial classifiers. Numerical experiments on both linear and nonlinear oscillators demonstrate that the method accurately identifies mode dynamics and reconstructs switching surfaces from trajectory data. Our results provide a tractable optimization-based framework for switching system identification.

Stochastic hybrid systems combine continuous-time stochastic dynamics with discrete reset events, producing intrinsically non-Gaussian and often multimodal uncertainty. A consistent propagation law must also account for boundary-induced probability flux across guard sets, making direct density propagation through hybrid Fokker-Planck equations expensive. We develop a hybrid extension of the Max-Entropy Moment Kalman Filter (MEM-KF) that performs filtering from partial statistical information by propagating a finite collection of moments through stochastic hybrid dynamics and reconstructing beliefs using moment-constrained maximum-entropy distributions. The key step is a moment propagation rule derived from Dynkin's formula with a jump-sum, in which reset effects appear as a boundary-flux correction over the guard set. This yields tractable moment dynamics without solving the underlying hybrid PDE. In a stochastic bouncing-ball example, the proposed method captures reset-induced non-Gaussianity through corrected moment equations while retaining the MEM-KF's optimization-based maximum-entropy representation.

We introduce a mathematical technique based on modifying a given Riemannian metric and we investigate its applicability to controlling and stabilizing constrained mechanical systems. In essence our result is based on the construction of a complete Riemannian metric in the modified space where the constraint is included. In particular this can be applied to the controlled Lagrangians technique Bloch et al. [2000b, 2001] modifying its metric to additionally cover mechanical systems with configuration constraints via control. The technique used consists of approximating incomplete Riemannian metrics by complete ones, modifying the evolution near a boundary and finding a controller satisfying a given design criterion.

Foot slip is a major source of instability in bipedal locomotion on low-friction or uncertain terrain. Standard control approaches typically assume no-slip contact and therefore degrade when slip occurs. We propose a control framework that explicitly incorporates slip into the locomotion model through virtual nonholonomic constraints, which regulate the tangential stance-foot velocity while remaining compatible with the virtual holonomic constraints used to generate the walking gait. The resulting closed-loop system is formulated as a hybrid dynamical system with continuous swing dynamics and discrete impact events. A nonlinear feedback law enforces both classes of constraints and yields a slip-compatible hybrid zero dynamics manifold for the reduced-order locomotion dynamics. Stability of periodic walking gaits is characterized through the associated Poincaré map, and numerical results illustrate stabilization under slip conditions.

A central obstacle in nonlinear Bayesian filtering is representing the belief distribution. Moment-based filters address this by propagating polynomial moments and reconstructing a density from them. Recent work completes the predict-update loop via the maximum-entropy (MaxEnt) principle, but each step requires the partition function and its gradient, both n-dimensional integrals whose cost scales exponentially, restricting the demonstrated MaxEnt moment filtering to n ≤ 4. We avoid the partition function entirely by combining score matching with Stein's identity. In our setting, score matching reduces the density fit to a single linear solve whose coefficients are assembled directly from the propagated moments. The same parameters then drive Stein's identity to close the moment hierarchy during prediction and to recover posterior moments after each Bayesian update, keeping the full predict-update loop free of partition function evaluation. The resulting Score Kalman Filter (SKF) reduces to the classical information-form Kalman filter as a special case and performs every step through linear algebra. On nonlinear coupled-oscillator networks, the SKF runs through n=20 and reports lower RMSE than the EKF, UKF, EnKF, and particle-filter baselines on the tested synthetic benchmarks.

Discrete variational methods show excellent performance in numerical simulations of mechanical systems. In this paper, we adapt discrete variational integrators for the case of mechanical systems with double-bracket dissipation. In particular, we will work with forced Euler-Poincaré and forced Lie-Poisson systems, and the case of interest for us will be when the coadjoint orbits remain invariant, but the energy is decreasing along the orbit. This particular kind of dissipative system appears in various physical systems such as satellites with dampers, geophysical fluids, plasma physics and stellar dynamics. The proposed geometric integrator preserves the coadjoint orbits exactly. We highlight the advantages of this feature by comparing it with other general-purpose methods (including higher-order ones) across different numerical simulations.

Advances in methods of biological data collection are driving the rapid growth of comprehensive datasets across clinical and research settings. These datasets provide the opportunity to monitor biological systems in greater depth and at finer time steps than was achievable in the past. Classically, biomarkers are used to represent and track key aspects of a biological system. Biomarkers retain utility even with the availability of large datasets, since monitoring and interpreting changes in a vast number of molecules remains impractical. However, given the large number of molecules in these datasets, a major challenge is identifying the best biomarkers for a particular setting Here, we apply principles of observability theory to establish a general methodology for biomarker selection. We demonstrate that observability measures effectively identify biologically meaningful sensors in a range of time series transcriptomics data. Motivated by the practical considerations of biological systems, we introduce the method of dynamic sensor selection (DSS) to maximize observability over time, thus enabling observability over regimes where system dynamics themselves are subject to change. This observability framework is flexible, capable of modeling gene expression dynamics and using auxiliary data, including chromosome conformation, to select biomarkers. Additionally, we demonstrate the applicability of this approach beyond genomics by evaluating the observability of neural activity These applications demonstrate the utility of observability-guided biomarker selection for across a wide range of biological systems, from agriculture and biomanufacturing to neural applications and beyond.

This paper introduces a convex optimization framework for identifying switched network systems, in which both the node dynamics and the underlying graph topology switch between a finite number of configurations. Building on our recent convex identification method for general switching systems, we extend the formulation to structured network systems where each mode corresponds to a distinct adjacency matrix. We show that both the continuous node dynamics and binary network topologies can be identified from sampled state-velocity data by solving a sequence of convex programs. The proposed framework provides a unified and scalable way to recover piecewise network structures from data without a prior knowledge of mode labels at each state. Numerical results on diffusively coupled oscillators demonstrate accurate recovery of both mode dynamics and switching graphs.

This work generalizes the classical metriplectic formalism to model Hamiltonian systems with nonconservative dissipation. Classical metriplectic representations allow for the description of energy conservation and production of entropy via a suitable selection of an entropy function and a bilinear symmetric metric. By relaxing the Casimir invariance requirement of the entropy function, this paper shows that the generalized formalism induces the free energy analogous to thermodynamics. The monotonic change of free energy can serve as a more precise criterion than mechanical energy or entropy alone. This paper provides examples of the generalized metriplectic system in a 2-dimensional Hamiltonian system and SO(3). This paper also provides a bilevel convex optimization approach for the identification of the metriplectic system given measurements of the system.

First, we analyze the discrete Monge–Kantorovich problem, linking it with the minimization problem of linear functionals over adjoint orbits. Second, we consider its generalization to the setting of area preserving diffeomorphisms of the annulus. In both cases, we show how the problem can be linked to permutohedra, majorization, and to gradient flows with respect to a suitable metric.

In this paper, we provide a geometric characterization of virtual nonlinear nonholonomic constraints from a symplectic perspective. Under a transversality assumption, there is a unique control law making the trajectories of the associated closed-loop system satisfy the virtual nonlinear nonholonomic constraints. We characterize them in terms of the almost-tangent and a symplectic structure on TQ. In particular, we show that the closed-loop vector field satisfies a geometric equation of Chetaev type. Moreover, the closed-loop dynamics is obtained as the projection of the uncontrolled dynamics to the tangent bundle of the constraint submanifold defined by the virtual constraints.

Cosmic inflation provides a compelling framework for explaining several observed features of our Universe, but its viability depends on an efficient reheating phase that converts the inflaton's energy into Standard Model particles. This conversion often proceeds through non-perturbative mechanisms such as parametric resonance, which is described by Hill's equation. In this work, we investigate how stochastic fluctuations in the parameters of Hill's equation can influence particle production during reheating. We show that such fluctuations can arise from couplings to light scalar fields, and can significantly alter the stability bands in the resonance structure, thereby enhancing the growth of fluctuations and broadening the region of efficient energy transfer. Using random matrix theory and stochastic differential equations, we decompose the particle growth rate into deterministic and noise-induced components and demonstrate analytically and numerically that even modest noise leads to substantial particle production in otherwise stable regimes. These results suggest that stochastic effects can robustly enhance the efficacy of reheating across a wide swath of parameter space, with implications for early Universe cosmology, UV completions involving multiple scalar fields, and the resolution of the cosmological moduli problem.

Nonholonomic systems are mechanical systems with ideal velocity constraints that are not derivable from position constraints and with dynamics identified by the Lagrange – d’Alembert principle. This paper reviews infinite-dimensional and field-theoretic nonholonomic systems as well as Hamel’s formalism for these settings.

Virtual constraints are relations imposed on a control system that become invariant via feedback control, as opposed to physical constraints acting on the system. Nonholonomic systems are mechanical systems with non-integrable constraints on the velocities. In this work, we introduce the notion of virtual nonlinear nonholonomic constraints in a geometric framework which is a controlled invariant submanifold and we show the existence and uniqueness of a control law preserving this submanifold. We illustrate the theory with various examples and present simulation results for an application.

In this paper, we compare the performance of different numerical schemes in approximating Pontryagin’s Maximum Principle’s necessary conditions for the optimal control of fully actuated nonholonomic systems. Retraction maps are used as a seed to construct geometric integrators for the corresponding Hamilton’s equations. First, we obtain an intrinsic formulation of a discretization map on a distribution. Then, we illustrate this construction on a particular example for which the performance of different symplectic integrators is examined and compared with non-symplectic integrators.

In this paper, we give sufficient conditions for and deduce a control law under which a mechanical control system converges exponentially fast to a virtual linear nonholonomic constraint that is control invariant via the same feedback control. Virtual constraints are relations imposed on a control system that become invariant via feedback control, as opposed to physical constraints acting on the system. Virtual nonholonomic constraints, similarly to mechanical nonholonomic constraints, are a class of virtual constraints that depend on velocities rather than only on the configurations of the system.