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Baozhu Guo - One of the best experts on this subject based on the ideXlab platform.
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Riesz Basis Property and Exponential Stability for One-Dimensional Thermoelastic System with Variable Coefficients
'EDP Sciences', 2021Co-Authors: Baozhu Guo, Han-jing RenAbstract: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
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Riesz Basis generation dual Basis approach
2019Co-Authors: Baozhu Guo, Junmin WangAbstract:This chapter is devoted to the Riesz Basis property for wave-like equations. It follows the routs from concrete example to abstract framework. This demonstrates the overall technique. In this chapter, the adjoint Basis approach in various situations is introduced, which makes use of the Riesz Basis property of exponential families, in particular, the Basis property of generalized divided difference (GDD). The chapter starts with two connected strings with span point dissipative feedbacks. An N-connected string system is discussed for its Riesz Basis property. The Riesz Basis property for an abstract hyperbolic system is also discussed. It also presents three tree-shaped string systems with joint’s feedbacks. The last section discusses the stability of wave equation with delayed output feedback control by the Riesz Basis approach.
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Riesz Basis generation green function approach
2019Co-Authors: Baozhu Guo, Junmin WangAbstract:In this chapter, we discuss another approach for Riesz Basis the so-called Green function approach. This happens for beam equations where the boundary feedback control is of the same order as the original but we cannot apply the approach in Chapter 4 for this situation because the exponential family is usually not Riesz Basis for the spectrum not being the strip paralleling to the imaginary axis. It starts with a rotating beam with shear force feedback. The system is not dissipative in the traditional sense. Section 5.2 discusses the beam equation with the conjugate variables appearing at the same boundary. The Green function method is applied to develop the Riesz Basis property and the well-posedness of the system is concluded as a consequence. Section 5.3 presents a one-link flexible manipulator with rotational inertia.
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Riesz Basis generation comparison method
2019Co-Authors: Baozhu Guo, Junmin WangAbstract:This chapter provides a panoramic view on the comparison method. It discusses systematically how the comparison method is used to derive the Riesz Basis generation for systems described by partial differential equations. A basic assumption for comparison method to be working is that the feedback can be considered as a perturbation of the system itself, that is, the order of the feedback is lower than the order of original system, which is clear from the spectrum or transfer function point of view. It starts with a constant beam equation with collocated boundary feedback control, and then the beam equation with variable coefficients. A beam equation with span pointwise control is presented to show its Riesz Basis and exponential stability. A one-dimensional thermoelastic system is fully discussed. The Riesz Basis property has been developed for these systems, which implies particularly that the dynamics of the system is completely determined by vibration frequencies. Mathematically, all the operators are of compact resolvent. In the last section, however, an example of the Boltzmann integral model is presented where the resolvent is not compact and the continuous spectrum exists. Two different types of Boltzmann integrals for the dynamics of vibrating systems are discussed and the Riesz Basis property has been developed.
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on spectrum and Riesz Basis property for one dimensional wave equation with boltzmann damping
ESAIM: Control Optimisation and Calculus of Variations, 2012Co-Authors: Baozhu Guo, Guodong ZhangAbstract:In this paper, we study the one-dimensional wave equation with Boltzmann damping. Two different Boltzmann integrals that represent the memory of materials are considered. The spectral properties for both cases are thoroughly analyzed. It is found that when the memory of system is counted from the infinity, the spectrum of system contains a left half complex plane, which is sharp contrast to the most results in elastic vibration systems that the vibrating dynamics can be considered from the vibration frequency point of view. This suggests us to investigate the system with memory counted from the vibrating starting moment. In the latter case, it is shown that the spectrum of system determines completely the dynamic behavior of the vibration: there is a sequence of generalized eigenfunctions of the system, which forms a Riesz Basis for the state space. As the consequences, the spectrum-determined growth condition and exponential stability are concluded. The results of this paper expositorily demonstrate the proper modeling the elastic systems with Boltzmann damping.
Junmin Wang - One of the best experts on this subject based on the ideXlab platform.
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Riesz Basis generation dual Basis approach
2019Co-Authors: Baozhu Guo, Junmin WangAbstract:This chapter is devoted to the Riesz Basis property for wave-like equations. It follows the routs from concrete example to abstract framework. This demonstrates the overall technique. In this chapter, the adjoint Basis approach in various situations is introduced, which makes use of the Riesz Basis property of exponential families, in particular, the Basis property of generalized divided difference (GDD). The chapter starts with two connected strings with span point dissipative feedbacks. An N-connected string system is discussed for its Riesz Basis property. The Riesz Basis property for an abstract hyperbolic system is also discussed. It also presents three tree-shaped string systems with joint’s feedbacks. The last section discusses the stability of wave equation with delayed output feedback control by the Riesz Basis approach.
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Riesz Basis generation comparison method
2019Co-Authors: Baozhu Guo, Junmin WangAbstract:This chapter provides a panoramic view on the comparison method. It discusses systematically how the comparison method is used to derive the Riesz Basis generation for systems described by partial differential equations. A basic assumption for comparison method to be working is that the feedback can be considered as a perturbation of the system itself, that is, the order of the feedback is lower than the order of original system, which is clear from the spectrum or transfer function point of view. It starts with a constant beam equation with collocated boundary feedback control, and then the beam equation with variable coefficients. A beam equation with span pointwise control is presented to show its Riesz Basis and exponential stability. A one-dimensional thermoelastic system is fully discussed. The Riesz Basis property has been developed for these systems, which implies particularly that the dynamics of the system is completely determined by vibration frequencies. Mathematically, all the operators are of compact resolvent. In the last section, however, an example of the Boltzmann integral model is presented where the resolvent is not compact and the continuous spectrum exists. Two different types of Boltzmann integrals for the dynamics of vibrating systems are discussed and the Riesz Basis property has been developed.
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Riesz Basis generation green function approach
2019Co-Authors: Baozhu Guo, Junmin WangAbstract:In this chapter, we discuss another approach for Riesz Basis the so-called Green function approach. This happens for beam equations where the boundary feedback control is of the same order as the original but we cannot apply the approach in Chapter 4 for this situation because the exponential family is usually not Riesz Basis for the spectrum not being the strip paralleling to the imaginary axis. It starts with a rotating beam with shear force feedback. The system is not dissipative in the traditional sense. Section 5.2 discusses the beam equation with the conjugate variables appearing at the same boundary. The Green function method is applied to develop the Riesz Basis property and the well-posedness of the system is concluded as a consequence. Section 5.3 presents a one-link flexible manipulator with rotational inertia.
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Riesz Basis approach to feedback stabilization for a cantilever beam system
Chinese Control Conference, 2017Co-Authors: Junmin Wang, Mengqing Xiong, Chao YangAbstract:In this paper, we study the stabilization of an elastic beam system with axial force and a tip mass. The system is modeled as a Rayleigh beam equation. We propose a boundary feedback control moment to stabilize the closed-loop system. We first present the asymptotic expressions for the eigenpairs of the system and then show that the generalized eigenfunctions form a Riesz Basis in the state space. Finally, we prove the exponential stability of the closed-loop system.
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A Riesz Basis approach to exponential stability in thermoelasticity of type III
2013 9th Asian Control Conference (ASCC), 2013Co-Authors: Jing Wang, Junmin WangAbstract:Using a Riesz Basis approach, we investigate, in this paper, the exponential stability for a one-dimensional linear thermoelasticity of type III with Dirichlet-Dirichlet boundary conditions. A detailed spectral analysis gives that the spectrum of the system contains two parts: the point and continuous spectrum. It is shown that, by asymptotic analysis, there are three classes of eigenvalues: one is along the negative real axis approaching to - ∞, the second is approaching to a vertical line which parallels to the imagine axis, and the third class is distributed around the continuous spectrum which is an accumulation point of the last classes of eigenvalues. Moreover, it is pointed out that there is a sequence of generalized eigenfunctions, which forms a Riesz Basis for the energy state space. Finally, the spectrum-determined growth condition holds true and the exponential stability of the system is then established.
Marianna A Shubov - One of the best experts on this subject based on the ideXlab platform.
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spectral asymptotics instability and Riesz Basis property of root vectors for rayleigh beam model with non dissipative boundary conditions
Asymptotic Analysis, 2014Co-Authors: Marianna A ShubovAbstract:In this paper we investigate the Rayleigh beam model with non-dissipative boundary conditions recently considered in the literature. The beam is clamped at the left end and subject to a feedback control type boundary conditions at the right. The components of the 2-dim input vector are shear and moment at the right end and the components of the observation vector are time derivatives of displacement and slope at the right end. The input is related to the observation through a co-diagonal matrix depending on two non-negative control parameters. The paper contains two main results. First we derive the leading term, the second order term, and the remainder in the asymptotic representation of the eigenmodes of the system. It follows from spectral asymptotics that the system has an infinite alternating sequence of stable and unstable eigenmodes. This type of instability has been recently obtained in the literature by a different method. The instability result might be of interest for energy harvesting models where destabilization of a system is desirable. The second result is the fact that the generalized eigenvectors of the dynamics generator of the system form a Riesz Basis in the state space equipped with the energy metric. This result implies, in particular, that the system generates a C0-semigroup in the state space.
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Riesz Basis property of mode shapes for aircraft wing model subsonic case
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2006Co-Authors: Marianna A ShubovAbstract:The present paper is devoted to the Riesz Basis property of the mode shapes for an aircraft wing model in an inviscid subsonic airflow. The model has been developed in the Flight Systems Research Center of the University of California at Los Angeles in collaboration with NASA Dryden Flight Research Center. The model has been successfully tested in a series of flight experiments at Edwards Airforce Base, CA, and has been extensively studied numerically. The model is governed by a system of two coupled integro-differential equations and a two parameter family of boundary conditions modelling the action of the self-straining actuators. The system of equations of motion is equivalent to a single operator evolution–convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator, which is a finite—meromorphic operator—valued function of the spectral parameter. Its poles are precisely the aeroelastic modes. In the author’s previous works, it has been shown that the set of aeroelastic modes asymptotically splits into two disjoint subsets called the b-branch and the d-branch, and precise spectral asymptotics with respect to the eigenvalue number have been derived for both branches. The asymptotical approximations for the mode shapes have also been obtained. In the present work, the author proves that the set of the mode shapes forms an unconditional Basis (the Riesz Basis) in the Hilbert state space of the system. The results of this paper will be important for the reconstruction of the solution of the original initial boundary-value problem from its Laplace transform and for the analysis of the flutter phenomenon in the forthcoming work.
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Riesz Basis property of root vectors of non self adjoint operators generated by aircraft wing model in subsonic airflow
Mathematical Methods in The Applied Sciences, 2000Co-Authors: Marianna A ShubovAbstract:This paper is the third in a series of several works devoted to the asymptotic and spectral analysis of a model of an aircraft wing in a subsonic air flow. This model has been developed in the Flight Systems Research Center of UCLA and is presented in the works by Balakrishnan. The model is governed by a system of two coupled integro-differential equations and a two-parameter family of boundary conditions modeling the action of the self-straining actuators. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. The system of equations of motion is equivalent to a single operator evolution-convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator, which is an operator-valued function of the spectral parameter. This generalized resolvent operator is a finite-meromorphic function on the complex plane having the branch cut along the negative real semi-axis. Its poles are precisely the aeroelastic modes and the residues at these poles are the projectors on the generalized eigenspaces. In the first two papers (see [33, 34]) and in the present one, our main object of interest is the dynamics generator of the differential parts of the system. This generator is a non-self-adjoint operator in the energy space with a purely discrete spectrum. In the first paper, we have shown that the spectrum consists of two branches, and have derived their precise spectral asymptotics with respect to the eigenvalue number. In the second paper, we have derived the asymptotical approximations for the mode shapes. Based on the asymptotical results of the first two papers, in the present paper, we (a) prove that the set of the generalized eigenvectors of the aforementioned differential operator is complete in the energy space; (b) construct the set of vectors which is biorthogonal to the set of the generalized eigenvectors in the case when theremight be not only eigenvectors but associate vectors as well; and (c) prove that the set of the generalized eigenvectors forms a Riesz Basis in the energy space. To prove the main result of the paper, we made use of the Nagy-Foias functional model for non-self-adjoint operators. The results of all three papers will be important for the reconstruction of the solution of the original initial-boundary-value problem from its Laplace transform in the forthcoming papers.
S P Yung - One of the best experts on this subject based on the ideXlab platform.
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stability and Riesz Basis property of a star shaped network of euler bernoulli beams with joint damping
Networks and Heterogeneous Media, 2008Co-Authors: S P YungAbstract:In this paper we study a star-shaped network of Euler-Bernoulli beams, in which a new geometric condition at the common node is imposed. For the network, we give a method to assert whether or not the system is asymptotically stable. In addition, by spectral analysis of the system operator, we prove that there exists a sequence of its root vectors that forms a Riesz Basis with parentheses for the Hilbert state space.
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Riesz Basis property exponential stability of variable coefficient euler bernoulli beams with indefinite damping
Ima Journal of Applied Mathematics, 2005Co-Authors: Junmin Wang, S P YungAbstract:We study damped Euler–Bernoulli beams that have nonuniform thickness or density. These nonuniform features result in variable coefficient beam equations. We prove that despite the nonuniform features, the eigenfunctions of the beam form a Riesz Basis and asymptotic behaviour of the beam system can be deduced without any restrictions on the sign of the damping. We also provide an answer to the frequently asked question on damping: ‘how much more positive than negative should the damping be without disrupting the exponential stability?’, and result in a criterion condition which ensures that the system is exponentially stable.
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exponential stability of variable coefficients rayleigh beams under boundary feedback controls a Riesz Basis approach
Systems & Control Letters, 2004Co-Authors: Junmin Wang, S P YungAbstract:In this paper, we study the boundary stabilizing feedback control problem of Rayleigh beams that have non-homogeneous spatial parameters. We show that no matter how non-homogeneous the Rayleigh beam is, as long as it has positive mass density, stiffness and mass moment of inertia, it can always be exponentially stabilized when the control parameters are properly chosen. The main steps are a detail asymptotic analysis of the spectrum of the system and the proving of that the generalized eigenfunctions of the feedback control system form a Riesz Basis in the state Hilbert space. As a by-product, a conjecture in Guo (J. Optim. Theory Appl. 112(3) (2002) 529) is answered.
Andreas Fleige - One of the best experts on this subject based on the ideXlab platform.
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the Riesz Basis property of an indefinite sturm liouville problem with non separated boundary conditions
arXiv: Spectral Theory, 2013Co-Authors: Branko Curgus, Andreas Fleige, Aleksey KostenkoAbstract:We consider a regular indefinite Sturm-Liouville eigenvalue problem \{$-f" + q f = \lambda r f$} on $[a,b]$ subject to general self-adjoint boundary conditions and with a weight function $r$ which changes its sign at finitely many, so-called turning points. We give sufficient and in some cases necessary and sufficient conditions for the Riesz Basis property of this eigenvalue problem. In the case of separated boundary conditions we extend the class of weight functions $r$ for which the Riesz Basis property can be completely characterized in terms of the local behavior of $r$ in a neighborhood of the turning points. We identify a class of non-separated boundary conditions for which, in addition to the local behavior of $r$ in a neighborhood of the turning points, local conditions on $r$ near the boundary are needed for the Riesz Basis property. As an application, it is shown that the Riesz Basis property for the periodic boundary conditions is closely related to a regular HELP-type inequality without boundary conditions.
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a review of a Riesz Basis property for indefinite sturm liouville problems
Operators and Matrices, 2011Co-Authors: Paul Binding, Andreas FleigeAbstract:Consider the indefinite Sturm-Liouville problem−f ′′ = λrf, f(−1) = f(1) = 0 with an indefinite weight function r ∈ L[−1, 1] satisfying xr(x) > 0. A number of conditions for the so-called Riesz Basis property are reviewed, i.e. conditions such that the eigenfunctions form a Ries Basis of the Hilbert space L|r|[−1, 1]. Most of these conditions were previously known to be either necessary or sufficient (or both if r is odd). Now it is shown that all are equivalent to the Riesz Basis property if r is strongly odd dominated, i.e. the even part of r is dominated by the odd part in some sense. One of the equivalent conditions is the validity of the HELP-type inequality on [0, 1] (∫ 1
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conditions for an indefinite sturm liouville Riesz Basis property
2009Co-Authors: Paul Binding, Andreas FleigeAbstract:We consider the Sturm-Liouville problem −y″ = λry on [−1, 1] with Dirichlet boundary conditions and with an indefinite weight function r which changes sign at 0. We discuss several conditions known to be either necessary or sufficient for the eigenfunctions to form a Riesz Basis of the Hilbert space L 2,|r|(−1, 1). Assuming that the odd part of r dominates the even part in a certain sense, we show that the above conditions (and also some new ones) are in fact all equivalent to this Riesz Basis property.
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the Riesz Basis property of an indefinite sturm liouville problem with a non odd weight function
Integral Equations and Operator Theory, 2008Co-Authors: Andreas FleigeAbstract:For the Sturm-Liouville eigenvalue problem − f′′ = λrf on [−1, 1] with Dirichlet boundary conditions and with an indefinite weight function r changing its sign at 0 we discuss the question whether the eigenfunctions form a Riesz Basis of the Hilbert space L2|r|[− 1, 1]. So far a number of sufficient conditions on r for the Riesz Basis property are known. However, a sufficient and necessary condition is only known in the special case of an odd weight function r. We shall here give a generalization of this sufficient and necessary condition for certain generally non-odd weight functions satisfying an additional assumption.
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a necessary aspect of the generalized beals condition for the Riesz Basis property of indefinite sturm liouville problems
2007Co-Authors: Andreas FleigeAbstract:For the Sturm-Liouville eigenvalue problem −f″ = γrf [−1, 1] with Dirichlet boundary conditions and with an indefinite weight function r changing it’s sign at 0 we discuss the question whether the eigenfunctions form a Riesz Basis of the Hilbert space L |r| 2 [−1, 1]. In the nineties the sufficient so called generalized one hand Beals condition was found for this Riesz Basis property. Now using a new criterion of Parfyonov we show that already the old approach gives rise to a necessary and sufficient condition for the Riesz Basis property under certain additional assumptions.
