We investigate the asymptotic behavior of nearly unstable Hawkes processes whose regression kernel has $L^1$ norm strictly greater than 1 and close to 1 as time goes to infinity. We find that the scaling size determines the scaling behavior of the processes as in Jaisson and Rosenbaum (2015). Specifically, after a suitable rescale of (Formula presented), the limit of the sequence of Hawkes processes is deterministic. Also, with another appropriate rescaling of 1/T² , the sequence converges in law to an integrated Cox–Ingersoll–Ross-like process. This theoretical result may apply to model the recent COVID-19 outbreak in epidemiology and phenomena in social networks.

Underactuated autonomous surface vehicles (ASVs) have stringent requirements on automatically tracking a predefined path. This paper proposes a model predictive control (MPC) approach based on adaptive line-of-sight (LOS) guidance for path following of ASVs. For the controller, a second-order nonlinear Nomoto model with disturbances is proposed as the vessel dynamic motion model after reviewing and comparing different ship motion models applied for path following control. For the guidance system, a novel adaptive LOS guidance with a variable acceptance circle radius is proposed to improve the precision of reference path tracking. Specifically, the acceptance circle radius is adapted with the angle between two adjacent straight segments of a reference path. Simulation experiments illustrate that the LOS guidance system with a variable acceptance circle radius results in smaller tracking errors compared with the fixed acceptance circle radius. The proposed path following method can track reference paths well even in the face of disturbances.

Motion control in absence of human involvement is difficult to realize for autonomous vessels because there usually exist environmental disturbances and unmeasurable states at the same time. A discrete-time model predictive control (MPC) approach based on a state-compensation extended state observer (SCESO) is proposed to achieve more precise control performance with state estimations and disturbance rejections simultaneously. The main idea is that lumped disturbances encompassing nonlinear dynamics and external disturbances are handled as two parts, unlike the standard extended state observer (ESO). Particularly, the nonlinear terms are compensated by estimated states and the external disturbances are considered as extended states and attenuated by the traditional ESO strategy. Assuming that the lumped disturbances are constant over the prediction horizon, the prediction model is linearized to save computational time since after linearization the online MPC optimization problems are solved as quadratic programming problems instead of nonlinear programming problems. The convergence of the proposed SCESO estimation errors to zero is proved even when the disturbances keep variable. Two case studies involving a numerical example and ship heading control have been conducted to verify the effectiveness of the proposed control method.