Nonlinear Boundary Feedback Research Articles

Growth and blow-up of solutions for a viscoelastic wave equation with logarithmic source, fractional conditions, and non-linear boundary feedback

Growth and blow-up of solutions for a viscoelastic wave equation with logarithmic source, fractional conditions, and non-linear boundary feedback

Existence and exponential stability for the wave equation with nonlinear interior source and localized viscoelastic boundary feedback

In this work, we aim to investigate an integro-differential model that involves localized viscoelastic effects at the boundary of the domain under the history framework. We have established that the equation is well-posed and exhibits exponential stability when a localized admissible kernel is applied, along with the δ -condition.

General Decay Results for a Viscoelastic Euler–Bernoulli Equation with Logarithmic Nonlinearity Source and a Nonlinear Boundary Feedback

General Decay Results for a Viscoelastic Euler–Bernoulli Equation with Logarithmic Nonlinearity Source and a Nonlinear Boundary Feedback

Finite element approximation for Maxwell’s equations with Debye memory under a nonlinear boundary feedback with delay

Finite element approximation for Maxwell’s equations with Debye memory under a nonlinear boundary feedback with delay

Energy Decay Estimates of the Axially Moving Kirchhoff-type Beam

Energy Decay Estimates of the Axially Moving Kirchhoff-type Beam

Long time dynamics and upper semi-continuity of attractors for piezoelectric beams with nonlinear boundary feedback

A system of boundary-controlled piezoelectric beam equations, accounting for the interactions between mechanical vibrations and the fully-dynamic electromagnetic fields, is considered. Even though electrostatic and quasi-static electromagnetic field approximations of Maxwell’s equations are sufficient for most models of piezoelectric systems, where the magnetic permeability is completely discarded, the PDE model considered here retains the pronounced wave behavior of electromagnetic fields to accurately describe the dynamics for the most piezoelectric acoustic devices. It is also crucial to investigate whether the closed-loop dynamics of the fully-dynamic piezoelectric beam equations, with nonlinear state feedback and nonlinear external sources, is close to the one described by the electrostatic/quasi-static equations, when the magnetic permeability μ is small. Therefore, the asymptotic behavior is analyzed for the fully-dynamic model at first. The existence of global attractors with finite fractal dimension and the existence of exponential attractors are proved. Finally, the upper-semicontinuity of attractors with respect to magnetic permeability to the ones of the electrostatic/quasi-static beam equations is shown.

Sector stability criteria for a nonlinear axial motion string system

The paper investigates the exponential stability criterion for an axially moving string system driven by a nonlinear partial differential equation with nonlinear boundary feedback. The control criterion based on a sector condition contains a large class of nonlinearities, which is a negative feedback of the velocity at the right boundary of the moving string. By invoking nonlinear semigroup theory, the well‐posedness result of the closed‐loop system is verified under the sector criteria. Furthermore, a novel energy like function is constructed to establish the exponential stability of the closed‐loop system by using a integral‐type multiplier method and the generalized Gronwall‐type integral inequality.

Absolute boundary stabilization for an axially moving Kirchhoff beam

Absolute boundary stabilization for an axially moving Kirchhoff beam

Stabilization of the wave equation with a nonlinear delay term in the boundary conditions

Abstract A wave equation in a bounded and smooth domain of ℝ n {\mathbb{R}^{n}} with a delay term in the nonlinear boundary feedback is considered. Under suitable assumptions, global existence and uniform decay rates for the solutions are established. The proof of existence of solutions relies on a construction of suitable approximating problems for which the existence of the unique solution will be established using nonlinear semigroup theory and then passage to the limit gives the existence of solutions to the original problem. The uniform decay rates for the solutions are obtained by proving certain integral inequalities for the energy function and by establishing a comparison theorem which relates the asymptotic behavior of the energy and of the solutions to an appropriate dissipative ordinary differential equation.

Decay Results for a Viscoelastic Problem with Nonlinear Boundary Feedback and Logarithmic Source Term

The main goal of this work is to investigate the long-time behavior of a viscoelastic equation with a logarithmic source term and a nonlinear feedback localized on a part of the boundary. In the framework of potential well, we first show the global existence. Then, we discuss the asymptotic behavior of the problem with a very general assumption on the behavior of the relaxation function g, namely, $g^{\prime }(t)\le -\xi (t) G(g(t))$ . We establish explicit and general decay results from which we can recover the well-known exponential and polynomial rates when G(s) = sp and p covers the full admissible range [1,2). Our results are obtained without imposing any restrictive growth assumption on the boundary damping term. This work generalizes and improves many earlier results in the literature.

Global Stabilization of Two Dimensional Viscous Burgers\u2019 Equation by Nonlinear Neumann Boundary Feedback Control and Its Finite Element Analysis

In this article, global stabilization results for the two dimensional viscous Burgers’ equation, that is, convergence of unsteady solution to its constant steady state solution with any initial data, are established using a nonlinear Neumann boundary feedback control law. Then, applying C^0-conforming finite element method in spatial direction, optimal error estimates in L^infty (L^2) and in L^infty (H^1)-norms for the state variable and convergence result for the boundary feedback control law are derived. All the results preserve exponential stabilization property. Finally, several numerical experiments are conducted to confirm our theoretical findings.

Nonlinear and memory boundary feedback stabilization for Schr¨odinger equations with variable coefficients

Nonlinear and memory boundary feedback stabilization for Schr¨odinger equations with variable coefficients

Uniform stability of transmission of wave-plate equations with source on Riemannian manifold

Uniform stability of transmission of wave-plate equations with source on Riemannian manifold

A dynamic compensator for in-domain control of Burgers' equation

This paper addresses the problem of output regulation control of a Burgers' equation under pointwise in-domain actuation. A dynamic compensator that generates the in-domain control is developed for output regulation of the considered system. A nonlinear boundary feedback control is used to ensure the closed-loop stability of the Burgers' equation. The proposed control scheme is implemented by means of Adomian decomposition method and flatness-based trajectory planning. A numerical simulation study is carried out, and the obtained results demonstrate the efficiency and effectivity of the proposed approach.

Global Stabilization of BBM\u2013Burgers\u2019 Type Equations by Nonlinear Boundary Feedback Control Laws: Theory and Finite Element Error Analysis

In this article, global stabilization results for the Benjamin–Bona–Mahony–Burgers’ (BBM–B) type equations are obtained using nonlinear Neumann boundary feedback control laws. Based on the C^0-conforming finite element method, global stabilization results for the semidiscrete solution are also discussed. Optimal error estimates in L^infty (L^2), L^infty (H^1) and L^infty (L^infty )-norms for the state variable are derived, which preserve exponential stabilization property. Moreover, for the first time in the literature, superconvergence results for the boundary feedback control laws are established. Finally, several numerical experiments are conducted to confirm our theoretical findings.

Lyapunov-based boundary feedback design for parabolic PDEs

This paper presents a methodology for the construction of simple Control Lyapunov Functionals (CLFs) for boundary controlled parabolic Partial Differential Equations (PDEs). The proposed methodology provides functionals that contain only simple (and not double or triple) integrals of the state. Moreover, the constructed CLF is ‘almost diagonal’ in the sense that it contains only a finite number of cross-products of the (generalised) Fourier coefficients of the state. The methodology for the construction of a CLF is combined with a novel methodology for boundary feedback design in parabolic PDEs. The proposed feedback design methodology is Lyapunov-based and the feedback controller is an ‘integral’ controller with internal dynamics. It is also shown that the obtained simple CLFs can provide nonlinear boundary feedback laws which achieve global exponential stabilisation of semilinear parabolic PDEs with nonlinearities that satisfy a linear growth condition.

Global well-posedness and exponential stability for heterogeneous anisotropic Maxwell's equations under a nonlinear boundary feedback with delay

Global well-posedness and exponential stability for heterogeneous anisotropic Maxwell's equations under a nonlinear boundary feedback with delay

Well-Posedness and Stability of Infinite-Dimensional Linear Port-Hamiltonian Systems with Nonlinear Boundary Feedback

Boundary feedback stabilisation of linear port-Hamiltonian systems on an interval is considered. Generation and stability results already known for linear feedback are extended to nonlinear dissipative feedback, both to static feedback control and dynamic control via an (exponentially stabilising) nonlinear controller. A design method for nonlinear controllers of linear port-Hamiltonian systems is introduced. As a special case the Euler-Bernoulli beam is considered.

General and Optimal Decay Result for a Viscoelastic Problem with Nonlinear Boundary Feedback

In this paper, we consider a viscoleastic equation with a nonlinear feedback localized on a part of the boundary and a relaxation function satisfying g′(t) ≤−ξ(t)G(g(t)). We establish an explicit and general decay rate results, using the multiplier method and some properties of the convex functions. Our results are obtained without imposing any restrictive growth assumption on the damping term. This work generalizes and improves earlier results in the literature, in particular those of Messaoudi (Topological Methods in Nonlinear Analysis 51(2):413–427, 2018), Messaoudi and Mustafa (Nonlinear Analysis: Theory Methods & Applications 72(9–10):3602–3611, 2010), Mustafa (Mathematical Methods in the Applied Sciences 41(1): 192–204, 2018) and Wu (Zeitschrift fur angewandte Mathematik und Physik 63(1):65–106, 2012).

General and optimal decay for a viscoelastic equation with boundary feedback

We establish a general decay rate for a viscoelastic problem with a nonlinear boundary feedback and a relaxation function satisfying $g^{\prime}(t) \leq - \xi(t) g^{p}(t)$, $t\geq0$, $ 1\leq p < {3}/{2}$. This work generalizes and improves earlier results in the literature. In particular those of \cite{Caval5}, \cite{Messaoudi1} and \cite{Messaoudi6}.