Practical finite-time formation-containment control of multiple amphibious USVs for moving-object surveillance under asynchronous water–air switching

Persistent surveillance of a moving maritime object requires unmanned systems to provide both long-endurance operation and rapid spatial reconfiguration. Amphibious unmanned surface vehicles (AUSVs) can meet these requirements through surface sailing and aerial flight; however, follower-specific asynchronous water–air switching simultaneously changes the system dynamics and the cooperative-task dimension. The surveillance task is formulated as a practical finite-time formation-containment problem for multiple AUSVs, in which virtual leaders generate a time-varying surveillance region around the moving object and prescribed offsets define the internal follower formation. To unify the two-dimensional surface task and the three-dimensional flight task, a mode-dependent active-coordinate framework is established, and the corresponding active-coordinate error system is constructed. Based on this framework, an active-coordinate neural-adaptive finite-time controller is developed to address mode-dependent nonlinear uncertainties and bounded residual effects. Lyapunov analysis first establishes that the active-coordinate errors enter and subsequently remain within practical neighborhoods in finite time under each fixed operating mode. This property is then extended to asynchronous water–air switching subject to follower-specific average dwell-time conditions. Simulation results demonstrate that the followers can enter the moving surveillance region and establish the prescribed mixed-mode internal formation.