Let a finite set of interacting particles be given, together with a symmetry Lie group $G$. Here we describe all possible dynamics that are jointly equivariant with respect to the action of $G$. This is relevant e.g., when one aims to describe collective dynamics that are independent of any coordinate change or external influence. We particularize the results to some key examples, i.e. for the most basic low dimensional symmetries that appear in collective dynamics on manifolds.

The market of Unmanned Aerial Vehicles (UAVs) for civil applications is extensively growing. Indeed, these airplanes are nowadays widely used in applications such as data gathering, agriculture monitoring and rescue. The UAVs are required to track a fixed or moving object, thus tracking control algorithms that ensure the system stability and that have a quick time response are developed. This paper tackles the problem of supervising a fixed target using a ϐixed wing UAV flying at a constant altitude and a constant speed. For that purpose, three control algorithms are developed. In all of the algorithms, the UAV is expected to hover around the target in a circular trajectory. Moreover, the three approaches are based upon a Lyapunov‑LaSalle stabilization method. The first tracking algorithm ensures the UAV to circle around the target however the path that the UAV will follow in order to join this pattern is not studied. In the second and third approach, two different techniques that allow the UAV to intercept its final circular pattern in the quickest possible time and thus follow the tangent to the circular pattern are presented. Simulation results that show and compare the performances of the proposed methods are presented.

The dynamic output stabilization problem is a central and important topic in control theory. A compelling example is the control of a drone where the only measurable output is its distance to a target. This scenario presents unique challenges, as the drone’s coordinates are not always observable, even after symmetry reductions. From a practical point of view, minimizing the time required to reach the target—closely tied to fuel efficiency—is critical, particularly in military applications. Interestingly, the optimal trajectories that achieve minimum time, typically straight line paths, coincide with trajectories where the system is unobservable. This paper introduces a novel methodology to reconcile these seemingly conflicting objectives: ensuring observability while minimizing travel time.

Fragmentation processes arise in many physical systems, including droplet breakup, aerosols, sprays, comminution, and granular media. A central difficulty in fragmentation modeling is the identification of the breakup law from observed particle-size distributions. In this work, we propose a physics-constrained neural framework for the inverse identification of fragmentation laws. Starting from a size-structured binary fragmentation equation, the kernel is decomposed into a breakup rate $B\left(s\right)$, depending on the parent size s, and a daughter-size distribution $\kappa \left(z\right)$, depending on the normalized daughter-size fraction z. The unknown functions B and $\kappa$ are represented by neural networks designed to satisfy physical constraints, including non-negativity of the breakup rate, non-negativity and normalization of the daughter distribution, and first-moment conservation in the symmetric binary setting. The direct fragmentation equation is solved using a first-moment-conserving quadrature discretization, and the inverse problem is formulated as a regularized optimization problem constrained by the fragmentation dynamics. Synthetic experiments, generated independently of the inverse solver, show that the proposed method can recover B and $\kappa$ from simulated particle-size distributions, with accurate recovery in the unimodal case and reasonable performance in the more challenging bimodal case. Robustness tests indicate stability with respect to moderate observational noise and highlight the importance of multiple observation times. For experimental droplet-breakup measurements without time-resolved particle-size distributions, the same constrained neural daughter-density representation is used to estimate effective marginal daughter-volume-fraction distributions from normalized daughter-to-parent volume fractions. The resulting distributions distinguish rim and node fragments and different breakup modes, while stratified cluster-bootstrap confidence bands quantify their sampling uncertainty. The experimental analysis is therefore interpreted as constrained density estimation from final fragment measurements rather than as full validation of the PDE-constrained inverse recovery.