A Mathematical Review of Reduced Aeroelastic Models, Multiagent Dynamics, and Control Allocation in UAV Systems

Unmanned Aerial Vehicles (UAVs) are complex nonlinear systems characterized by high dimensionality. They are prone to aerodynamic effects, structural dynamics, actuation constraints, and networked interactions, requiring advanced mathematical models and precise control. Their governing equations involve nonlinear rigid-body dynamics coupled with fluid and elasticity models, while modern architectures introduce redundancy that creates constrained mappings between generalized forces and actuator inputs. Coordinated UAV teams add another layer of mathematical structure through graph-based interaction models that determine consensus, formation keeping, and distributed stability. These characteristics give rise to several interconnected challenges. High-fidelity aerodynamic and aeroelastic solvers provide accurate results; however, these are computationally intensive, motivating the development of reduced-order models and data-driven approximations that preserve dominant physical behavior. Methods for quantifying uncertainty support robustness assessments by characterizing the effects of parametric variation and model form error. At the actuation level, control allocation problems rely on constrained linear algebra, convex optimization, and dynamic formulations to ensure feasible and stable realization of command forces and moments. In multi-agent systems, the spectral properties of adjacency and Laplacian matrices govern convergence and cooperative behavior. This article reviews the state of the art in these areas, highlights the mathematical foundations that relate them, and provides a coherent perspective on the methods that enable reliable modeling and control of modern UAV systems.