Abstract This study is devoted to the stabilization of a generalized Korteweg-de Vries–Burgers equation via Neumann boundary velocity feedback control. A design criterion for nonlinear boundary control is developed by imposing a slope-restricted condition. Under this feedback law, the well-posedness of the closed-loop system is obtained through nonlinear semigroup theory, supported by a Faedo-Galerkin approximation and the Banach fixed-point theorem. In addition, it is shown that the closed-loop system exhibits global exponential stability for a nonzero dissipation coefficient, and is locally exponentially stable when the dissipation coefficient is zero. Finally, numerical experiments are provided to validate the theoretical findings.
Abstract This paper delves into practical finite-time stability (PFTS) for a type of fractional-order non-linear systems (FONSs) with unknown state and non-linear functions by employing the state observer and sliding mode control (SMC) methods. Since the state in the FONS is unknown, a state observer is provided. Moreover, the fuzzy logic system (FLS), the adaptive control technique and the integral SMC method are adopted to tackle the unknown non-linear functions. Then, an observer and FLS-based adaptive integral SMC scheme is developed to ensure the PFTS for the FONS. Lastly, a practical example is provided to confirm the correctness of the derived results.
This paper considers the control design and stability analysis of a wave equation with velocity recirculation and an input delay. When both an in-domain nonlocal term involving the velocity at an intermediate point and an input delay are present, the instability of the wave system is enhanced, posing significant challenges to the control design and stability analysis. We first transform the input delay into a first-order transport equation, resulting in a wave-transport coupled system. Then, we design three novel transformations to convert the wave-transport coupled system into two intermediate auxiliary systems and a final stable target system. Simultaneously, a delay-compensated control is proposed for the wave system with velocity recirculation using the backstepping method. Finally, we prove that the closed-loop system is well-posed and exponentially stable. Furthermore, we provide some numerical simulations to demonstrate the validity of the control.
This paper introduces the hyperbolic controller canonical form (HCCF) for boundary-actuated heterodirectional hyperbolic systems with multiple input components. The HCCF is a special state representation that significantly simplifies the design of state feedback tracking controllers for stabilizing desired reference trajectories. The form exists if the system is exactly controllable and involves a number of transport equations with normalized transport velocities, which are coupled solely at the actuated boundary. Similar to finite-dimensional systems, the HCCF is derived from the parametrization of the system's solutions in terms of a flat output. In the case of boundary-coupled systems without in-domain coupling, it is shown that the use of a special flat output always allows for the stabilization via static state feedback. The HCCF-based control design is illustrated for an example system.
Abstract The existence and uniqueness of mild solutions for second-order neutral stochastic partial functional differential equations with infinite delay are obtained by utilizing the M$\ddot{o}$nch fixed theorem and the Kuratowski measure of non-compactness. Further, we establish a new Gronwall–Bellman-type integral inequality with infinite delay to prove the exponential stability, asymptotical stability and global attracting set in the $p$th moment of mild solutions. Certain restrictive conditions have been removed. Our results improve and generalize some corresponding results. Finally, an example is given to show our main results.
