Boundary stabilization of a hybrid Euler—Bernoulli beam

We consider a problem of boundary stabilization of small flexural vibrations of a flexible structure modeled by an Euler-Bernoulli beam which is held by a rigid hub at one end and totally free at the other. The hub dynamics leads to a hybrid system of equations. By incorporating a condition of small rate of change of the deflection with respect tox as well ast, over the length of the beam, for appropriate initial conditions, uniform exponential decay of energy is established when a viscous boundary damping is present at the hub end.     

Asymptotics of eigenfrequencies and eigenmodes of non‐homogeneous inextensible filament with an end load

We consider a class of non‐selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non‐conservative boundary conditions at one end and a heavy load at the other end. The filament has a non‐constant density and is subject to a viscous damping with a non‐constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. We derive the spectral asymptotics for the aforementioned two‐parameter family of non‐selfadjoint operators. In the forthcoming papers, based on the asymptotical results of the present paper, we will prove the Riesz basis property of the eigenfunctions. The spectral results obtained in the aforementioned papers will allow us to solve boundary and/or distributed controllability problems for the filament using the spectral decomposition method. Copyright © 2001 John Wiley & Sons, Ltd.     

Exact boundary controllability for a kind of second‐order quasilinear hyperbolic systems and its applications

In this paper, by means of a constructive method based on the existence and uniqueness of the semi‐global C² solution, we establish the local exact boundary controllability for a kind of second‐order quasilinear hyperbolic systems. As an application, we obtain the one‐sided local exact boundary controllability for the first‐order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative eigenvalues are decoupled. Copyright © 2009 John Wiley & Sons, Ltd.     

LOCAL EXACT BOUNDARY CONTROLLABILITY FOR A CLASS OF QUASILINEAR HYPERBOLIC SYSTEMS

InroductionLet us consider the following irst order quaslllnear hyperbolic     

Exact observability and controllability for a nonsimple elastic rod

In this paper, we prove exact observability, boundary and internal exact controllability results for a nonsimple elastic rod. By the spectral analysis method, it is shown that the considered problem has a sequence of eigenfunctions, which forms a Riesz basis for the state space. The exact observability is proved by application of a modified version of Ingham’s Theorem. Then, via the Hilbert Uniqueness method, we prove a boundary controllability result with only one controller and an internal controllability result.     

Exact boundary controllability of partial nodal profile for network of strings

Based on the theory of exact boundary controllability of nodal profile for hyperbolic systems, the authors propose the concept of exact boundary controllability of partial nodal profile to expand the scope of applications. With the new concept, we can shorten the controllability time, save the number of controls, and increase the number of charged nodes with given nodal profiles. Furthermore, we introduce the concept of in-situ controlled node to deal with a new situation that one node can be charged and controlled simultaneously. The minimum number of boundary controls on the entire tree-like network is determined by using the concept of ‘degree of freedom of charged nodes’ introduced. And the concept of ‘control path’ is introduced to appropriately divide the network, so that we can determine the infimum of controllability time. General frameworks of constructive proof are given on a single interval, a star-like network, a chain-like network and a planar tree-like network for linear wave equation(s) with Dirichlet, Neumann, Robin and dissipative boundary conditions to establish a complete theory on the exact boundary controllability of partial nodal profile.     

Exact Null Controllability of a Nonlinear Thermoelastic Contact Problem

We study the controllability properties of a nonlinear parabolic system that models the temperature evolution of a one-dimensional thermoelastic rod that may come into contact with a rigid obstacle. Basically the system dynamics is described by a one-dimensional nonlocal heat equation with a nonlinear and nonlocal boundary condition of Newmann type. We focus on the control problem and treat the case when the control is distributed over the whole space domain. In this case the system is proved to be exactly null controllable provided the parameters of the system are smooth. The proof is based on changing the control variable and using Aubins Compactness Lemma to obtain an invariant set for the linearized controllability map. Then, by proving that the found solution is sufficiently smooth, we get the null controllability for the original system.     

Controllability of classical solutions implies controllability of weak solutions for a coupled system of wave equations and its application

In this paper, for a coupled system of one‐dimensional wave equations with Dirichlet boundary controls, we show that the controllability of classical solutions implies the controllability of weak solutions. This conclusion can be applied in proving some results that are hardly obtained by a direct way in the framework of classical solutions. For instance, we strictly derive the necessary conditions for the exact boundary synchronization by two groups in the framework of classical solutions for the coupled system of wave equations. Copyright © 2015 John Wiley & Sons, Ltd.     

Stabilization of a hybrid system of elasticity by feedback boundary damping

Summary A hybrid control system is presented as consisting of an elastic beam linked to a rigid body, and the system is asymptotically stabilized through feedback boundary damping. Solutions of the hybrid system are constructed that decay towards zero at nonexponential, even arbitrarily slow, decay rates. This feedback control analysis complements the authors' earlier report on the open-loop controllability of this same hybrid system, which is a simplified model of a space-structure.This research was partially supported by NSF Grant DMS 86-07687 and AFOSR-ISSA-860088, and the second author also received support from SERC.     

Exact Boundary Controllability for the Spatial Vibration of String with Dynamical Boundary Conditions

This paper deals with the spatial vibration of an elastic string with masses at the endpoints. The authors derive the corresponding quasilinear wave equation with dynamical boundary conditions, and prove the exact boundary controllability of this system by means of a constructive method with modular structure.     

具有复杂边界条件的杆的振动分析

本文研究一端带有集中质量并支以弹簧另一端作支承运动的杆的纵向振动.由于这个问题的边界条件比较复杂,且要考虑阻尼,因此本文只求稳态周期解.首先分析线性系统;然后考虑材料非线性,用摄动法求具有非线性边界条件的非线性方程的近似解析解.     

Local exact controllability of Schr\"{o}dinger equation with Sturm- Liouville boundary value problems

In this paper, we investigate the controllability of 1D bilinear Schr\"{o}dinger equation with Sturm-Liouville boundary value condition. The system represents a quantumn particle controlled by an electric field. K. Beauchard and C. Laurent have proved local controllability of 1D bilinear Schr\"{o}dinger equation with Dirichlet boundary value condition in some suitable Sobolev space based on the classical inverse mapping theorem. Using a similar method, we extend this result to Sturm-Liouville boundary value proplems.     

Exact boundary controllability for 1‐D quasilinear wave equations with dynamical boundary conditions

By equivalently replacing the dynamical boundary condition by a kind of nonlocal boundary conditions, and noting a hidden regularity of solution on the boundary with a dynamical boundary condition, a constructive method with modular structure is used to get the local exact boundary controllability for 1‐D quasilinear wave equations with dynamical boundary conditions. Copyright © 2016 John Wiley & Sons, Ltd.     

Exact boundary controllability for non‐autonomous quasilinear wave equations

In this paper, we first show that quite different from the autonomous case, the exact boundary controllability for non‐autonomous wave equations possesses various possibilities. Then we adopt a constructive method to establish the exact boundary controllability for one‐dimensional non‐autonomous quasilinear wave equations with various types of boundary conditions. Finally, we apply the results to multi‐dimensional quasilinear wave equation with rotation invariance. Copyright © 2007 John Wiley & Sons, Ltd.     

Exact controllability for some degenerate wave equations

In this paper, we study one-dimensional linear degenerate wave equations with a distributed controller. We establish observability inequalities for degenerate wave equation by multiplier method. We also deduce the exact controllability for degenerate wave equation by Hilbert uniqueness method when the control acts on the nondegenerate boundary. Moreover, an explicit expression for the controllability time is given.     

Contrôlabilité exacte frontière d’un système hybride en élasticité

We consider the exact controllability of a hybrid system consisting of an elastic beam, clamped at one end and attached at the other end to a rigid antenna. Such a system is governed by one partial differential equation and two ordinary differential equations. Using the HUM method, we prove that the hybrid system is exactly controllable in an arbitrarily short time in the usual energy space.     

Controllability results for the two-dimensional heat equation with mixed boundary conditions using Carleman inequalities: a linear and a semilinear case

In this paper, we prove controllability results for a two-dimensional semilinear heat equation with mixed boundary conditions. It is well-known that mixed boundary conditions can present a singular behaviour of the solution. First, we will prove global Carleman estimates then we will use these inequalities to obtain controllability results.     

Asymptotic and spectral analysis of non‐selfadjoint operators generated by a filament model with a critical value of a boundary parameter

We consider a class of non‐selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non‐conservative boundary conditions at one end and a heavy load at the other end. The filament has a non‐constant density and is subject to a viscous damping with a non‐constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. In our previous paper (Mathematical Methods in the Applied Sciences 2001; 24 (15) : 1139–1169), we have derived the asymptotic approximations for the eigenvalues and eigenfunctions of the aforementioned non‐selfadjoint operators when the boundary parameters were arbitrary complex numbers except for one specific value of one of the parameters. We call this value the critical value of the boundary parameter. It has been shown (in Mathematical Methods in the Applied Sciences 2001; 24 (15) : 1139–1169) that the entire set of the eigenvalues is located in a strip parallel to the real axis. The latter property is crucial for the proof of the fact that the set of the root vectors of the operator forms a Riesz basis in the state space of the system. In the present paper, we derive the asymptotics of the spectrum exactly in the case of the critical value of the boundary parameter. We show that in this case, the asymptotics of the eigenvalues is totally different, i.e. both the imaginary and real parts of eigenvalues tend to ∞as the number of an eigenvalue increases. We will show in our next paper, that as an indirect consequence of such a behaviour of the eigenvalues, the set of the root vectors of the corresponding operator is not uniformly minimal (let alone the Riesz basis property). Copyright © 2003 John Wiley & Sons, Ltd.     

Exact Boundary Controllability of Weak Solutions for a Kind of First Order Hyperbolic System — the HUM Method*

The exact boundary controllability and the exact boundary observability for the 1-D first order linear hyperbolic system were studied by the constructive method in the framework of weak solutions in the work [Lu, X. and Li, T. T., Exact boundary controllability of weak solutions for a kind of first order hyperbolic system — the constructive method, Chin. Ann. Math. Ser. B, 42(5), 2021, 643–676]. In this paper, in order to study these problems from the viewpoint of duality, the authors establish ...     

一维绝热流方程组的精确边界能控性

利用一阶拟线性双曲组混合初边值问题的精确能控性理论,通过对边界速度或压强的控制,实现了一维绝热流方程组的精确边界能控性.