Spectrum-determined Growth Condition Research Articles

Approximation of infinite-dimensional observer-based state feedback for systems with boundary control and observation

Infinite-dimensional linear systems with unbounded input and output operators are considered. For the purpose of finite-dimensional observer-based state feedback, an observer approximation scheme will be developed which can be directly combined with existing late-lumping controllers and observer output injection gains. It relies on a decomposition of the feedback gain, resp. observer output injection gain, into a bounded and an unbounded part. Based on a perturbation result, the spectrum-determined growth condition is established, for the closed loop.

Spectral and numerical analysis for a thermoelastic problem with double porosity and second sound

We study some spectral and numerical properties of the solutions to a thermoelastic problem with double porosity. The model includes Cattaneo-type evolution law for the heat flux to remove the physical paradox of infinite propagation speed of the classical Fourier’s law. Firstly, we prove that the operator determined by the considered problem has compact resolvent and generates a C 0 -semigroup in an appropriate Hilbert space. We also show that there is a sequence of generalized eigenfunctions of the linear operator that forms a Riesz basis. By a detailed spectral analysis, we obtain the expressions of the spectrum and we deduce that the spectrum determined growth condition holds. Therefore we prove that the energy of the considered problem decays exponentially to a rate determined explicitly by the physical parameters. Finally, some numerical simulations based on Chebyshev spectral method for spatial discretization are given to confirm the exponential stability result and to show the distribution of the eigenvalues and the variables of the problem.

Boundary stabilization for a star-shaped network of variable coefficients strings linked by a point mass

<p style='text-indent:20px;'>This study is concerned with the pointwise stabilization for a star-shaped network of <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula> variable coefficients strings connected at the common node by a point mass and subject to boundary feedback dampings at all extreme nodes. It is shown that the closed-loop system has a sequence of generalized eigenfunctions which forms a Riesz basis for the state Hilbert space. As a consequence, the spectrum-determined growth condition fulfills. In the meanwhile, the asymptotic expression of the spectrum is presented, and the exponential stability of the system is obtained by giving the optimal decay rate. We prove also that a phenomenon of lack of uniform stability occurs in the absence of damper at one extreme node. This paper reconfirmed the main stability results given by Hansen and Zuazua [SIAM J. Control Optim., <b>33</b> (1995), 1357-1391] in a very particular case.</p>

Exponential stability of a pendulum in dynamic boundary feedback with a viscous damped wave equation

In this paper, we continue the earlier work [Lu, L., & Wang, D. L. (2017). Dynamic boundary feedback of a pendulum coupled with a viscous damped wave equation. In Proceedings of the 36th Chinese Control Conference(CCC) (pp. 1676–1680)] on study the stability of a pendulum coupled with a viscous damped wave equation model. This time we get the exponential stability result which is much better than the previous strong stability. By a detailed spectral analysis and operator separation, we establish the Riesz basis property as well as the spectrum determined growth condition for the system. Finally, the exponential stability of the system is achieved.

Riesz Basis Property and Exponential Stability for One-Dimensional Thermoelastic System with Variable Coefficients

In this paper, we study Riesz basis property and stability for a nonuniform thermoelastic system with Dirichlet-Dirichlet boundary condition, where the heat subsystem is considered as a control to the whole coupled system. By means of the matrix operator pencil method, we obtain the asymptotic expressions of the eigenpairs, which are exactly coincident to the constant coefficients case. We then show that there exists a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state space and the spectrum determined growth condition is therefore proved. As a consequence, the exponential stability of the system is concluded.

Exponential stability of two strings under joint damping with variable coefficients

We study the stabilization of two vibrating strings with variable physical coefficients joined by a feedback control at a common endpoint and subject to non-symmetrical boundary conditions. We prove that this system is exponentially stable under sufficient conditions on the physical coefficients. For this result, we show that the system has a sequence of the generalized eigenvectors which forms a Riesz basis with parentheses for the state Hilbert space, and as a consequence the spectrum-determined growth condition holds.

On the Stabilization of an Overhead Crane System With Dynamic and Delayed Boundary Conditions

This article is dedicated to the investigation of long-time behavior of an overhead crane with input delays in the boundary control. Contrary to previous works, the compensating terms in the proposed feedback control law are not of the same type as the delayed term. Next, the closed-loop system is shown to be well-posed in the sense of semigroups theory. Furthermore, the asymptotic convergence for the solutions of the system to a stationary position, which depends on the initial data, is obtained by applying LaSalle's invariance principle. Then, the convergence rate is shown to be exponential by means of the frequency-domain method. The asymptotic distribution of the eigenvalues, as well as the eigenfunctions of the system, are also provided, based on which, the differentiability of its semigroup is proved, and hence, the spectrum-determined growth condition holds. Finally, the outcomes are illustrated by a set of numerical simulations.

Riesz basis of exponential family for a hyperbolic system

Abstract This paper studies a linear hyperbolic system with boundary conditions that was first studied under some weaker conditions in [8, 11]. Problems on the expansion of a semigroup and a criterion for being a Riesz basis are discussed in the present paper. It is shown that the associated linear system is the infinitesimal generator of a C 0 {C_{0}} -semigroup; its spectrum consists of zeros of a sine-type function, and its exponential system { e λ n ⁢ t } n ≥ 1 {\{e^{\lambda_{n}t}\}_{n\geq 1}} constitutes a Riesz basis in L 2 ⁢ [ 0 , T ] {L^{2}[0,T]} . Furthermore, by the spectral analysis method, it is also shown that the linear system has a sequence of eigenvectors, which form a Riesz basis in Hilbert space, and hence the spectrum-determined growth condition is deduced.

Stabilization of Variable Coefficients Euler-Bernoulli Beam with Viscous Damping under a Force Control in Rotation and Velocity Rotation

This paper investigates the problem of exponential stability for a damped Euler-Bernoulli beam with variable coefficients clamped at one end and subjected to a force control in rotation and velocity rotation. We adopt the Riesz basis approach for show that the closed-loop system is a Riesz spectral system. Therefore, the exponential stability and the spectrum-determined growth condition are obtained.

Stabilisation of Schrödinger equation in dynamic boundary feedback with a memory-typed heat equation

ABSTRACTIn this work, we study the dynamic behaviour for a heat equation with exponential polynomial kernel memory to be a controller for a Schrödinger system. By introducing some new variables, the time-variant system is transformed into a time-invariant one. Remarkably, the resolvent of the closed-loop system operator is not compact anymore. The residual spectrum is shown to be empty and the continuous spectrum consisting of finite isolated points are obtained. It is shown that the sequence of generalised eigenfunctions forms a Riesz basis for the Hilbert state space. This deduces the spectrum-determined growth condition for the C0-semigroup, and the exponential stability is then established.

Spectral analysis of thermoelastic systems under nonclassical thermal models

ABSTRACTWe study some spectral properties of the solutions to generalized thermoelastic systems under Lord–Shulman, Green–Lindsay, and Green–Naghdi of type-II models. First, we prove that the linear operator of each model has compact resolvent and generates a C0−semigroup in an appropriate Hilbert space. We also show that there is a sequence of generalized eigenfunctions of the linear operator that forms a Riesz basis. By a detailed spectral analysis, we obtain the expressions of the spectrum and we deduce that the spectrum-determined growth condition holds. Therefore, if the imaginary axis is not an asymptote of the spectrum, we prove that the energy of each model decays exponentially to a rate determined explicitly by the physical parameters. Finally, some simulations are given for each model to support our results.

Exponential stability of an active constrained layer beam actuated by a voltage source without magnetic effects

We study the boundary stabilization of an active constrained layer (ACL) beam consisting of a stiff layer, a viscoelastic layer and a piezoelectric layer. The piezoelectric layer is actuated by a voltage source without magnetic effects. The system is modeled as a Rayleigh beam coupled with two wave equations. By using an asymptotic technique, we present the asymptotic expressions for the eigenpairs of the system. We show that the generalized eigenfunctions form a Riesz basis in the state space, and hence the spectrum determined growth condition holds. Finally, the exponential stability of the closed-loop system is established.

Asymptotic analysis of a nonsimple thermoelastic rod

The asymptotic analysis of a one-dimensional nonsimple thermoelastic problem is considered in this paper. By a detailed spectral analysis, the asymptotic expressions for eigenvalues and eigenfunctions of the considered system are developed. It is shown that the eigenfunctions form a Riesz basis on the Hilbert space and the eigenvalues asymptotically fall on two branches. One branch is along the negative horizontal axis in the complex plane and the other branch is asymptotic to a vertical line that is parallel to the imaginary axis. This gives the spectrum-determined growth condition for the $C_0-$semigroup associated to the system, and consequently, the asymptotic and the exponential stability of the solutions are deduced. The approach developed in this paper confirms the already-existing results; furthermore, it can be extended to a larger field of applications such as coupled system of rod or beam with diffusion equation. The method will be illustrated by an example of thermoelastic beam equations with Dirichlet boundary conditions.

Spectral analysis and stabilization of a coupled wave-ODE system

In this paper, an interconnected wave-ODE system with K-V damping in the wave equation and unknown parameters in the ODE is considered. It is found that the spectrum of the system operator is composed of two parts: Point spectrum and continuous spectrum. The continuous spectrum consists of an isolated point \(- \tfrac{1} {d}\), and there are two branches of the asymptotic eigenvalues: The first branch is accumulating towards \(- \tfrac{1} {d}\), and the other branch tends to −∞. It is shown that there is a sequence of generalized eigenfunctions, which forms a Riesz basis for the Hilbert state space. As a consequence, the spectrum-determined growth condition and exponential stability of the system are concluded.

Dynamic Behavior of a One-Dimensional Wave Equation with Memory and Viscous Damping

We study the dynamic behavior of a one-dimensional wave equation with both exponential polynomial kernel memory and viscous damping under the Dirichlet boundary condition. By introducing some new variables, the time-variant system is changed into a time-invariant one. The detailed spectral analysis is presented. It is shown that all eigenvalues of the system approach a line that is parallel to the imaginary axis. The residual and continuous spectral sets are shown to be empty. The main result is the spectrum-determined growth condition that is one of the most difficult problems for infinite-dimensional systems. Consequently, an exponential stability is concluded.

Boundary feedback stabilization of a Schrödinger equation interconnected with a heat equation

In this paper, we study stabilization for a Schrodinger equation, which is interconnected with a heat equation via boundary coupling. A direct boundary feedback control is adopted. By a detailed spectral analysis, it is found that there are two branches of eigenvalues: one is along the negative real axis, and the other is approaching to a vertical line, which is parallel to the imagine axis. Moreover, it is shown that there is a set of generalized eigenfunctions, which forms a Riesz basis for the energy state space. Finally, the spectrum-determined growth condition is held and the exponential stability of the system is then concluded.

Spectral analysis and exponential stability of one-dimensional wave equation with viscoelastic damping

This paper presents the exponential stability of a one-dimensional wave equation with viscoelastic damping. Using the asymptotic analysis technique, we prove that the spectrum of the system operator consists of two parts: the point and continuous spectrum. The continuous spectrum is a set of N points which are the limits of the eigenvalues of the system, and the point spectrum is a set of three classes of eigenvalues: one is a subset of N isolated simple points, the second is approaching to a vertical line which parallels to the imagine axis, and the third class is distributed around the continuous spectrum. Moreover, the Riesz basis property of the generalized eigenfunctions of the system is verified. Consequently, the spectrum-determined growth condition holds true and the exponential stability of the system is then established.

Stabilization of an ODE–Schrödinger Cascade

We consider the problem of stabilization of a linear ODE with input dynamics governed by the linearized Schrödinger equation. The interconnection between the ODE and Schrödinger equation is bi-directional at a single point. We construct an explicit feedback law that compensates the Schrödinger dynamics at the inputs of the ODE and stabilizes the overall system. Our design is based on a two-step backstepping transformation by introducing an intermediate system and an intermediate control. By adopting the Riesz basis approach, the exponential stability of the closed-loop system is built with the pre-designed decay rate and the spectrum-determined growth condition is obtained. A numerical simulation is provided to illustrate the effectiveness of the proposed design.

Stabilization of the Euler–Bernoulli equation via boundary connection with heat equation

In this paper, we are concerned with the stabilization of a coupled system of Euler–Bernoulli beam or plate with heat equation, where the heat equation (or vice versa the beam equation) is considered as the controller of the whole system. The dissipative damping is produced in the heat equation via the boundary connections only. The one-dimensional problem is thoroughly studied by Riesz basis approach: The closed-loop system is showed to be a Riesz spectral system and the spectrum-determined growth condition holds. As the consequences, the boundary connections with dissipation only in heat equation can stabilize exponentially the whole system, and the solution of the system has the Gevrey regularity. The exponential stability is proved for a two dimensional system with additional dissipation in the boundary of the plate part. The study gives rise to a different design in control of distributed parameter systems through weak connections with subsystems where the controls are imposed.

Exponential stability and spectral analysis of the inverted pendulum system under two delayed position feedbacks

In this paper, we examine the stability of a linearized inverted pendulum system with two delayed position feedbacks. The semigroup approach is adopted in investigation for the well-posedness of the closed loop system. We prove that the spectrum of the system is located in the left complex half-plane and its real part tends to???? when the feedback gains satisfy some additional conditions. The asymptotic eigenvalues of the system is presented. By estimating the norm of the Riesz spectrum projection of the system operator that does not have the uniformly upper bound, we show that the eigenfunctions of the system do not form a basis in the state Hilbert space. Furthermore, the spectrum determined growth condition of the system is concluded and the exponential stability of the system is then established. Finally, numerical simulation is presented by applying the MATLAB software.