On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback

The uniform stabilization of an originally regarded nondissipative system described by a semilinear wave equation with variable coefficients under the nonlinear boundary feedback is considered. The existence of both weak and strong solutions to the system is proven by the Galerkin method. The exponential stability of the system is obtained by introducing an equivalent energy function and using the energy multiplier method on the Riemannian manifold. This equivalent energy function shows particularly that the system is essentially a dissipative system. This result not only generalizes the result from constant coefficients to variable coefficients for these kinds of semilinear wave equations but also simplifies significantly the proof for constant coefficients case considered in [A. Guesmia, A new approach of stabilization of nondissipative distributed systems, SIAM J. Control Optim. 42 (2003) 24-52] where the system is claimed to be nondissipative.     

On Boundary Stability of Wave Equations with Variable Coefficients

In this paper, we consider the boundary stabilization of the wave equation with variable coefficients by Riemmannian geometry method subject to a different geometric condition which is motivated by the geometric multiplier identities. Several (multiplier) identities (inequalities) which have been built for constant wave equation by Kormornik and Zuazua are generalized to the variable coefficient case by some computational techniques in Riemmannian geometry, so that the precise estimates on the exponential decay rate are derived from those inequalitities. Also, the exponential decay for the solutions of semilinear wave equation with variable coefficients is obtained under natural growth and sign assumptions on the nonlinearity. Our method is rather general and can be adapted to other evolution systems with variable coefficients (e.g. elasticity plates) as well.     

A note on geometric conditions for boundary control of wave equations with variable coefficients

The analytical condition given by Wyler for boundary stabilization of wave equations with variable coefficients is compared with the geometrical condition derived by Yao in terms of the Riemannian geometry method for exact controllability of wave equations with variable coefficients. It is shown that these two conditions are equivalent.     

Stabilization of a one‐dimensional wave equation with variable coefficient under non‐collocated control and delayed observation

In this paper, we consider stabilization of a 1‐dimensional wave equation with variable coefficient where non‐collocated boundary observation suffers from an arbitrary time delay. Since input and output are non‐collocated with each other, it is more complex to design the observer system. After showing well‐posedness of the open‐loop system, the observer and predictor systems are constructed to give the estimated state feedback controller. Different from the partial differential equation with constant coefficients, the variable coefficient causes mathematical difficulties of the stabilization problem. By the approach of Riesz basis property, it is shown that the closed‐loop system is stable exponentially. Numerical simulations demonstrate the effect of the stable controller. This paper is devoted to the wave equation with variable coefficients generalized of that with constant coefficients for delayed observation and non‐collocated control.     

Event-triggered boundary stabilization for coupled semilinear reaction–diffusion systems with spatially varying coefficients

This paper aims to address the event-triggered Robin boundary control problem for exponential stabilization of the coupled semilinear reaction–diffusion systems with spatially varying coefficients. The main used method is the backstepping, which allows us to explicitly give the boundary control formulae. More precisely, we first explore the existence and uniqueness of classical solutions for the considered problem. After this, we propose an event-triggered boundary feedback control law to exponentially stabilize the system under consideration with the Zeno phenomenon being excluded. A numerical result is finally included to illustrate the efficiency of our designed controller.     

THE EXPONENTIAL STABILIZATION FOR A SEMILINEAR WAVE EQUATION WITH LOCALLY DISTRIBUTED FEEDBACK

This paper considers the exponential decay of the solution to a damped semilinear wave equation with variable coefficients in the principal part by Riemannian multiplier method. A differential geometric condition that ensures the exponential decay is obtained.     

Local uniform stability for the semilinear wave equation in inhomogeneous media with locally distributed Kelvin–Voigt damping

We consider the semilinear wave equation posed in an inhomogeneous medium Ω with smooth boundary subject to a local viscoelastic damping distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. We show that the energy of the wave equation goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase‐space. As far as we know, this is the first stabilization result for a semilinear wave equation with localized Kelvin–Voigt damping.     

Strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions

We consider a wave equation with semilinear porous acoustic boundary conditions. This is a coupled system of second and first order in time partial differential equations, with possibly semilinear boundary conditions on the interface. The results obtained are (i) strong stability for the linear model, (ii) exponential decay rates for the energy of the linear model, and (iii) local exponential decay rates for the energy of the semilinear model. This work builds on a previous result showing generation of a well-posed dynamical system. The main tools used in the proofs are (i) the Stability Theorem of Arendt-Batty, (ii) energy methods used in the study of a wave equation with boundary damping, and (iii) an abstract result of I. Lasiecka applicable to hyperbolic-like systems with nonlinearly perturbed boundary conditions.     

Optimal Control Problem Governed by Semilinear Parabolic Equation and its Algorithm

In this paper, an optimal control problem governed by semilinear parabolic equation which involves the control variable acting on forcing term and coefficients appearing in the higher order derivative terms is formulated and analyzed. The strong variation method, due originally to Mayne et al to solve the optimal control problem of a lumped parameter system, is extended to solve an optimal control problem governed by semilinear parabolic equation, a necessary condition is obtained, the strong variation algorithm for this optimal control problem is presented, and the corresponding convergence result of the algorithm is verified.     

Blow-up of solutions to semilinear wave equations with variable coefficients and boundary

This paper is devoted to studying initial-boundary value problems for semilinear wave equations and derivative semilinear wave equations with variable coefficients on exterior domain with subcritical exponents in n space dimensions. We will establish blow-up results for the initial-boundary value problems. It is proved that there can be no global solutions no matter how small the initial data are, and also we give the life span estimate of solutions for the problems.     

Uniform decay of the solution to a wave equation with memory conditions on the boundary

A wave equation with variable coefficients in principal part and memory conditions on the boundary is considered. The Riemannian geometry method is applied to prove the exponential decay of the energy provided the relaxation function also decays exponentially.     

Low‐gain adaptive stabilization of semilinear second‐order hyperbolic systems

In this paper low‐gain adaptive stabilization of undamped semilinear second‐order hyperbolic systems is considered in the case where the input and output operators are collocated. The linearized systems have an infinite number of poles and zeros on the imaginary axis. The adaptive stabilizer is constructed by a low‐gain adaptive velocity feedback. The closed‐loop system is governed by a non‐linear evolution equation. First, the well‐posedness of the closed‐loop system is shown. Next, an energy‐like function and a multiplier function are introduced and the exponential stability of the closed‐loop system is analysed. Some examples are given to illustrate the theory. Copyright © 2004 John Wiley & Sons, Ltd.     

On the nonlocal stabilization by starting control of the normal equation generated from the Helmholtz system

We consider the problem of stabilization near zero of semilinear normal parabolic equations connected with the 3D Helmholtz system with periodic boundary conditions and arbitrary initial datum. This problem was previously studied in Fursikov and Shatina (2018). As it was recently revealed, the control function suggested in that work contains a term impeding transferring the stabilization construction on the 3D Helmholtz system. The main concern of this paper is to prove that this term is not necessary for the stabilization result, and therefore the control function can be changed by a proper way.     

RAPID EXACT CONTROLLABILITY OF THE SEMILINEAR WAVE EQUATION

1.IntroductionConsiderthefollowingsemilinearwaveequationwithalocallydistributedcofltroller:whereI~g,11~$,nCR"isaboundeddomainwithaboundaryoffECI)andforeachtE[0,co),G(t)isasubdomainoffi.Intheabove,y(t,x)isthestateandXG(t)(x)u(t,x)isthecolltrol.Thus,u(t,x)istheintensityofthecontrolactionandG(t)isthelocationandtheshapeofthecontroller.Wewillallowthelocationandtheshapeofthecontrollertochange.LetU=Lfo.(0'co;L'(fl))andletQbeafamilyofset--valuedfunctionsG(t)definedon[0,co)takingsubdomainsoffias…     

变系数波方程振动传递的边界镇定

本文讨论了变系数波方程振动传递的边界镇定,应用黎曼几何方法和迹的正则性得到了所讨论问题的能量一致衰减率.     

On Stationary Nonequilibrium Measures for Wave Equations

Doklady Mathematics - In the paper, the Cauchy problem for wave equations with constant and variable coefficients is considered. We assume that the initial data are a random function with finite...     

Blow Up of Solutions to One Dimensional Initial-boundary Value Problems for Semilinear Wave Equations with Variable Coefficients

This paper is devoted to studying the following initial-boundary value problemfor one-dimensional semilinearwave equationswith variable coefficients andwith subcritical exponent: $u_{tt}-∂_x(a(x)∂_xu)=|u|^p, x > 0, t > 0, n=1,$ where $u=u(x,t)$ is a real-valued scalar unknown function in $[0,+∞)×[0,+∞)$, here a(x) is a smooth real-valued function of the variable $x∈(0,+∞)$. The exponents p satisfies $1 < p < +∞$ in (0.1). It is well-known that the number $p_c(1)=+∞$ is the critical exponent of the semilinear wave equation (0.1) in one space dimension (see for e.g., [1]). We will establish a blowup result for the above initial-boundary value problem, it is proved that there can be no global solutions no matter how small the initial data are, and also we give the lifespan estimate of solutions for above problem.     

Optimality Conditions for Semilinear Hyperbolic Equations with Controls in Coefficients

An optimal control problem for semilinear hyperbolic partial differential equations is considered. The control variable appears in coefficients. Necessary conditions for optimal controls are established by method of two-scale convergence and homogenized spike variation. Results for problems with state constraints are also stated.     

Multidomain Legendre-Galerkin Least-Squares Method for Linear Differential Equations with Variable Coefficients

The multidomain Legendre-Galerkin least-squares method is developed for solving linear differential problems with variable coefficients. By introducing a flux, the original differential equation is rewritten into an equivalent first-order system, and the Legendre Galerkin is applied to the discrete form of the corresponding least squares function. The proposed scheme is based on the Legendre-Galerkin method, and the Legendre/Chebyshev-Gauss-Lobatto collocation method is used to deal with the variable coefficients and the right hand side terms. The coercivity and continuity of the method are proved and the optimal error estimate in $H^1$-norm is obtained. Numerical examples are given to validate the efficiency and spectral accuracy of our scheme. Our scheme is also applied to the numerical solutions of the parabolic problems with discontinuous coefficients and the two-dimensional elliptic problems with piecewise constant coefficients, respectively.     

变系数波动方程所决定的控制系统的最小能量控制和快速控制问题

本文研究变系数波动方程所决定的控制系统的最小能量控制和快速控制问题 .在所讨论的系统精确可控的前提下 ,得到了系统的最小能量控制和快速控制的解析表达式