Boundary Feedback Research Articles

Exponential Stability for Lam'e System with Fractional Time-Varying and Boundary Feedback

Exponential Stability for Lam'e System with Fractional Time-Varying and Boundary Feedback

Well-posedness and exponential stability of piezoelectric beam with thermal effect and boundary or internal distributed delay

In this work, we focus on a one-dimensional initial-boundary value problem of a thermoelastic piezoelectric beam in the presence of magnetic effects with distributed delay acting in the mechanical equation. The well-posedness of the system is initially demonstrated by applying semigroup theory (Hille-Yosida theorem). Also, based on the construction of a Lyapunov functional, which is equivalent to the energy functional of the problem through multiplier techniques, we show that a unique dissipation through frictional damping is strong enough to ensure exponential stabilization of the model. Next, the results are compared to those of the electrostatic or quasi-static approaches. Moreover, we consider the fully dynamic and electrostatic or quasi-static piezoelectric beams with thermal effects, boundary feedback controllers and boundary distributed delays and using the multipliers technique, we establish an exponential stability result of the solution. Our results are related to the distributed delay weights. Finally, we give some numerical tests to illustrate the theoretical result.

Sliding mode control to stabilization of coupled time fractional parabolic PDEs subject to disturbances

AbstractIn this article, the stabilization of coupled time fractional parabolic partial differential equations subject to external disturbances is investigated. By using sliding mode control method and backstepping approach, a boundary state feedback controller is designed to reject the matched disturbance and achieve the Mittag‐Leffler input‐to‐state stability of closed‐loop system. The existence of the generalized solution to the closed‐loop system is proven by Galerkin approximation scheme. Simulations are presented to illustrate the validity of our theoretical results.

Adaptive boundary feedback control of a rigid–flexible system under unknown time-varying disturbance

This paper studies the boundary feedback control of a rotating disk-cable-mass system (DCMS). The system consists of a flexible cable with a rigid disk connected at its upper end and a tip mass at the lower boundary. We assume that the disk and the tip mass are affected by unknown time-varying disturbances, while the cable is influenced by distributed disturbance. We design two boundary controls to suppress the vibration of the flexible cable and simultaneously adjust the system’s rotating speed to a desired value. The first controller is designed assuming the system’s parameters are accurately known. The second controller is conceived under circumstances where some system parameters are unknown, for which an adaptive technique is employed to regulate the DCMS. For these two controls, we choose two Lyapunov functions to prove the stability of the closed-loop system. Theoretical results demonstrate that the states of the closed-loop system are ultimately uniformly bounded. At the end of the paper, we validate two boundary controls through numerical tests.

Neural Operator Approximations for Boundary Stabilization of Cascaded Parabolic PDEs

ABSTRACTThis article proposes a novel method to accelerate the boundary feedback control design of cascaded parabolic difference equations (PDEs) through DeepONet. The backstepping method has been widely used in boundary control problems of PDE systems, but solving the backstepping kernel function can be time‐consuming. To address this, a neural operator (NO) learning scheme is leveraged for accelerating the control design of cascaded parabolic PDEs. DeepONet, a class of deep neural networks designed for approximating nonlinear operators, has shown potential for approximating PDE backstepping designs in recent studies. Specifically, we focus on approximating gain kernel PDEs for two cascaded parabolic PDEs. We utilize neural operators to map only two kernel functions, while the other two are computed using the analytical solution, thus simplifying the training process. We establish the continuity and boundedness of the kernels, and demonstrate the existence of arbitrarily close DeepONet approximations to the kernel PDEs. Furthermore, we demonstrate that the DeepONet approximation gain kernels ensure stability when replacing the exact backstepping gain kernels. Notably, DeepONet operator exhibits computation speeds two orders of magnitude faster than PDE solvers for such gain functions, and their theoretically proven stabilizing capability is validated through simulations.

BFT‐Net: A transformer‐based boundary feedback network for kidney tumour segmentation

AbstractKidney tumours are among the top ten most common tumours, the automatic segmentation of medical images can help locate tumour locations. However, the segmentation of kidney tumour images still faces several challenges: firstly, there is a lack of renal tumour endoscopic datasets and no segmentation techniques for renal tumour endoscopic images; secondly, the intra‐class inconsistency of tumours caused by variations in size, location, and shape of renal tumours; thirdly, difficulty in semantic fusion during decoding; and finally, the issue of boundary blurring in the localization of lesions. To address the aforementioned issues, a new dataset called Re‐TMRS is proposed, and for this dataset, the transformer‐based boundary feedback network for kidney tumour segmentation (BFT‐Net) is proposed. This network incorporates an adaptive context extract module (ACE) to emphasize local contextual information, reduces the semantic gap through the mixed feature capture module (MFC), and ultimately improves boundary extraction capability through end‐to‐end optimization learning in the boundary assist module (BA). Through numerous experiments, it is demonstrated that the proposed model exhibits excellent segmentation ability and generalization performance. The mDice and mIoU on the Re‐TMRS dataset reach 91.1% and 91.8%, respectively.

Contributions of core, mantle and climatological processes to Earth’s polar motion

Earth’s spin axis slowly moves relative to the crust over time. A 120-year-long record of this polar motion from astronomical and more modern geodetic measurements displays interannual and multidecadal fluctuations of 20 to 40 milliarcseconds superimposed on a secular trend of about 3 milliarcseconds per year. Earth’s polar motion is thought to be driven by various surface and interior processes, but how these processes operate and interact to produce the observed signal remains enigmatic. Here we show that predictions made by an ensemble of physics-informed neural networks trained on measurements to capture geophysical processes can explain the main features of the observed polar motion. We find that glacial isostatic adjustment and mantle convection primarily account for the secular trend. Mass redistribution on the Earth’s surface—for example, ice melting and global changes in water storage—yields a relatively weak trend but explains about 90% of the interannual and multidecadal variations. We also find that core processes contribute to both the secular trend and fluctuations in polar motion, either due to variations in torque at the core–mantle boundary or dynamical feedback of the core in response to surface mass changes. Our findings provide constraints on core–mantle interactions for which observations are rare and global ice mass balance over the past century and suggest feedback operating between climate-related surface processes and core dynamics.

Stabilization of linear KdV equation with boundary time-delay feedback and internal saturation

This paper investigates the question of stabilization of linear Korteweg-de Vries equation with time-delay on the boundary feedback in presence of saturated source term. Under some conditions the well-posedness is proved. The exponential stability result is demonstrated, using an appropriate Lyapunov functional.

Adaptive sliding mode boundary control of a perturbed diffusion process

SummaryThis paper proposes a sliding‐mode‐based adaptive boundary control law for stabilizing a class of uncertain diffusion processes affected by a matched disturbance. The matched disturbance is assumed to be uniformly bounded along with its time derivative, whereas the corresponding upper bounding constants are not known. This motivates the use of adaptive control strategies. In addition, the spatially‐varying diffusion coefficient is also uncertain. To achieve asymptotic stability of the plant origin in the ‐sense in the presence of the disturbance, a discontinuous boundary feedback law is proposed where the gain of the discontinuous control term is adjusted according to a gradient‐based adaptation law. A constructive Lyapunov analysis supports the stability properties of the considered closed‐loop system, yielding sufficient convergence conditions in terms of suitable inequalities involving the controllers' tuning parameters. Simulation results are presented to corroborate the theoretical findings.

A note on boundary feedback stabilization for degenerate parabolic equations in multi-dimensional domains

Abstract In this paper, we are concerned with the problem of stabilization of a degenerate parabolic equation with a Dirichlet control, evolving in bounded domain 𝓞 ⊂ ℝ d , d ≥ 2. We apply the proportional control design technique based on the spectrum of the linear operator which governs the evolution equation. The stabilizing feedback control, we design here, is linear, of finite-dimensional structure, easily manageable from the computational point of view.

General Decay Rates of the Solution Energy in a Viscoelastic Wave Equation With Boundary Feedback and a Nonlinear Source

In a bounded domain, we consider a viscoelastic equation with a nonlinear feedback localized on a part of the boundary and certain initial data. We establish an explicit and general decay rate result, using some properties of the convex functions. Our new results substantially improve several earlier related results in the literature. Keywords: General decay, nonlinear source, viscoelastic, wave equation, relaxation function.

Existence Result and Exponential Stabilization of a Moving String with Time Dependent Delay in the Boundary Feedback

Existence Result and Exponential Stabilization of a Moving String with Time Dependent Delay in the Boundary Feedback

Boundary feedback control of Navier–Stokes equations around arbitrary bluff body with prescribed drag and lift coefficients

This paper presents a boundary feedback control method for the two- and three-dimensional Navier–Stokes equations around an arbitrary bluff body, with prescribed drag and lift coefficients. In order to determine the feedback control law, we consider an extended system coupling the equations governing the Navier–Stokes problem with an equation satisfied by the control on the bluff body, which is a part of the domain boundary. We utilize the artificial compressibility method, which approximates the Navier–Stokes equations in the limit, along with a priori estimation techniques, to establish a boundary control. Such a control law ensures that the prescriptions of the drag and lift coefficients are attained. Subsequently, a compactness result allows to let the compressibility parameter tend to zero. The application of the Faedo-Galerkin method is essential in reaching this result.

Saturated Boundary Feedback Stabilization for a Nonuniform Timoshenko Beam With Disturbances and Uncertainties

Saturated Boundary Feedback Stabilization for a Nonuniform Timoshenko Beam With Disturbances and Uncertainties

Stabilization of coupled delayed nonlinear time fractional reaction diffusion systems using sampled‐in‐space sensing and actuation

AbstractThis paper is considered with the asymptotic stabilization of coupled delayed nonlinear time fractional reaction diffusion systems (FRDSs) governed by fractional parabolic partial differential equations (PDEs) with space‐dependent coefficients under sampled‐data in space control. It is assumed that state measurements can be averaged measurements (AMs) or point measurements (PMs), and a finite number of sensing and actuation devices are located in a spaced manner along the spatial domain of the interest. With the proposed sampled‐data in space controller, the closed‐loop stability is obtained. Tuning rules of system parameters and control parameters are derived using the fractional Halanay's inequality and the fractional Lyapunov method. Subsequently, the dual problem of observer design is formulated. Fractional examples are used to valid the theoretical result. Discussions on the extension of sampled‐data boundary feedback stabilization are provided finally.

On the boundary stabilization of the KdV–KdV system with time-dependent delay

The boundary stabilization problem of the Boussinesq KdV–KdV type system is investigated in this paper. An appropriate boundary feedback law consisting of a linear combination of a damping mechanism and a delay term is designed. Then, considering time-varying delay feedback together with a smallness restriction on the length of the spatial domain and the initial data, we show that the problem under consideration is well-posed. The proof combines Kato’s approach and the fixed-point argument. Last but not least, we prove that the energy of the linearized KdV–KdV system decays exponentially by employing the Lyapunov method.

Iterative learning control of an Euler-Bernoulli beam with time-varying boundary disturbance

This paper investigates the stabilization of Euler-Bernoulli beam systems modeled by a partial differential equation (PDE) with unknown time-varying disturbance. An iterative learning controller is designed using only boundary state feedback to realize the vibration control subject to unknown boundary disturbance. The well-posedness for the closed-loop system is given by the operator semigroup theory. Furthermore, the exponentially stable for the closed-loop system is proved by the Lyapunov method. The comparisons with existing results are made to demonstrate the effectiveness and advantages of the proposed boundary iterative learning control method.

Sliding mode control for the stabilization of fractional heat equations subject to boundary uncertainty

By adopting the sliding mode control (SMC) and the generalized Lyapunov method, the boundary feedback stabilization issue is studied for the fractional diffusion system subject to boundary control matched disturbance. The classical sliding surface and the fractional integral sliding function are constructed and sliding mode controllers are designed respectively to realize the Mittag-Leffler (M-L) stabilization of the considered system. The controller based on the newly-introduced fractional integral sliding function not only helps to relax the constraints on the disturbance but also realizes the same stabilization effect as that of the classical one. The well-posedness result of the solution is also obtained for discontinuous fractional heat equations. Besides, a numerical experiment validates the theoretical outcomes.

Controllability and stabilization of a degenerate/singular Schrödinger equation

The aim of this paper is to prove controllability and stabilization properties for a degenerate and singular Schrödinger equation with degeneracy and singularity occurring at the boundary of the spatial domain. We first address the boundary control problem. In particular, by combining multiplier techniques and compactness-uniqueness argument, we prove direct and inverse inequalities for the associated adjoint system. Consequently, via the Hilbert Uniqueness Method, we deduce exact boundary controllability for the control system under consideration in any time T>0. Moreover, we investigate the stabilization problem for this class of equations in the range of subcritical coefficients of the singular potential. By introducing a suitable linear boundary feedback, we prove that the solution decays exponentially in an appropriate energy space.

Stability of the Transmission Schrödinger Equation with a Delay Term in the Boundary Feedback

Stability of the Transmission Schrödinger Equation with a Delay Term in the Boundary Feedback