This paper investigates the adaptive optimal tracking control (AOTC) for underactuated surface vessels (USVs). Compared to the majority of existing studies, the control strategy in this paper innovatively combines an extended state observer (ESO) with reinforcement learning (RL). The designed ESO has high estimation accuracy and robust disturbance rejection capabilities for the unmeasurable information for USVs. To obtain the AOTC, the actor–critic (AC) networks based on RL are constructed to solve the Hamilton–Jacobi–Bellman (HJB) equations. Due to the uncertainties, it is challenging to obtain the optimal controller by directly solving the HJB equations. To address this issue, this paper employs neural networks (NNs) to approximate the uncertainties and solves the optimal controller via AC-RL and ESO. In addition, the adaptive parameters of the optimal controller is trained in parallel with AC networks, which can ensure that the trained networks can further improve tracking performance. The boundedness of AOTC for USVs is shown by Lyapunov stability theorem. Finally, simulation results demonstrate the effectiveness of the proposed algorithm.

This paper investigates the exponential and prescribed finite-time stabilization with time-varying controller. First, the constraints of boundedness and differentiability on time delays are simultaneously relaxed, the Lipschitz condition for activation function is also relaxed. Second, different from the traditional Lyapunov function, two different time-varying Lyapunov functions are respectively constructed to achieve the exponential and prescribed finite-time stabilization. Significantly, the exponential convergence rate and the settling time are constants that can be given in advance and are not affected by system parameters and initial states. In addition, the time-varying controllers have good tolerance for disturbance caused by discontinuous functions and the disturbance is perfectly resolved and does not affect the control performance. Especially, the form of controllers is relatively simple and there is not necessary to design the fractional-order controllers for prescribed finite-time stabilization. Furthermore, the exponential and prescribed finite-time stabilization for FNNs without delay are respectively established via continuous time-varying state feedback control. Finally, examples show the effectiveness of the proposed control methods.