gen qi xu

gen qi xu
Tianjin University | tju · Department of Mathematics

PhD

About

246
Publications
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2,603
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August 1983 - August 2003
Shanxi University
Position
  • Professor (Full)

Publications

Publications (246)
Article
Full-text available
The longtime behavior of a kind of fully magnetic effected nonlinear multi-dimensional piezoelectric beam with viscoelastic infinite memory is considered. The well-posedness of this nonlinear coupled PDEs’ system is showed by means of the semigroup theories and Banach fixed-point theorem. Based on frequency-domain analysis, it is proved that the co...
Article
In this paper, we consider the stabilization issues of a reaction-diffusion equation with variable coefficients and boundary input delay. At first, we design an observer based on the system output to estimate the state of the system. Due to the present of time delay in control, we design a dynamic feedback controller based on the state information...
Article
We study the spectral problem associated with the equation governing the small transverse motions of a viscoelastic tube of finite length conveying an ideal fluid. The boundary conditions considered are of general form, accounting for a combination of elasticity and viscous damping acting on both the slopes and the displacements of the ends of the...
Preprint
Full-text available
In this paper we investigate a class of linear evolution process with memory in Banach space by a different approach. Suppose that the linear evolution process is well posed, we introduce a family pair of bounded linear operators , {(G(t), F (t)), t ≥ 0}, that is called the resolvent family for the linear evolution process with memory, the F (t) is...
Preprint
The long time behavior of a kind of fully magnetic effected nonlinear piezoelectric beam with viscoelastic infinite memory is considered. The well-posedness of this nonlinear coupled PDEs system is showed by mean of the semigroup theories and Banach fixed point theorem. Based on frequency domain analysis, it is proved that the corresponding coupled...
Preprint
Full-text available
In the present paper we study the stability problem for a stretched tube conveying an ideal fluid with boundary damping. The spectral problem concerns operator functions of the forms \begin{equation*} \mathcal{M}\left(\lambda\right)=\lambda^2G+\lambda D+C\quad\text{and}\quad\mathcal{P}\left(\lambda\right)=\lambda I-T \end{equation*} taking values i...
Preprint
Full-text available
In this paper, we investigate the stabilization of an Euler-Bernoulli beam with time delays in the boundary controller. The boundary velocity feedback law is applied to obtain the closed-loop system. It is shown that this system generates a C 0-semigroup of linear operators. Moreover, the stability of the closed-loop system is discussed for differe...
Article
In this paper, the stability of a linear Timoshenko beam system involved with infinite memory is considered. Different from the previous results on where the monotony of kernel is always fulfilled, the memory kernel under consideration is assumed to be non-monotonic. The well-posedness of the system is obtained by means of resolvent family theory a...
Article
This paper is a survey for development of linear distributed parameter system. At first we point out some questions existing in current study of control theory for the Lp linear system with an unbounded control operator and an unbounded observation operator, such as stabilization problem and observer theory that are closely relevant to state feedba...
Article
We consider the general networks of elastic strings with Neumann boundary feedbacks and collocated observations in this paper. By selecting an appropriate multiplier, we show that this system is input-output \begin{document}$ L^2 $\end{document}-well-posed. Moreover, we verify its regularity by calculating the input-output transfer function of syst...
Article
The uniform stabilization problem is addressed for a Heat-ODE cascaded system with boundary delayed control. A simple, direct and easily calculated controller is proposed, in which the known control law is sufficiently applied. With the controller the cascaded system with delayed control is exponentially stabilized. In particular, in the proof of s...
Preprint
Full-text available
We study the spectral problem associated with the equation governing the small transverse motions of a viscoelastic tube of finite length conveying an ideal fluid. The boundary conditions considered are of general form, accounting for a combination of elasticity and viscous damping or friction acting on both the slopes and the displacements of the...
Article
Full-text available
In this paper, we study the component configuration issue of the line-shaped wave networks which is made of two viscoelastic components and an elastic component and the viscoelastic parts produce the infinite memory and damping and distributed delay. The structural memory of viscoelastic component results in energy dissipative and the damping memor...
Article
Full-text available
In this paper, the stabilization for Schrödinger equation subject to internal damping and boundary disturbance at the control end is investigated. Due to its immeasurability, the nonlinear observer system is designed to obtain the state information, and the existence of weak solution and its convergence for the nonlinear observer system are proved....
Article
This article studies a parallel repairable degradation system with two similar components and a repairman who can take a single vacation. Suppose that the system consists of two components that cannot be repaired "as good as new" after failures; when the repairman has a single vacation, the fault component of system may not be repaired immediately,...
Preprint
Full-text available
We deal with the stability problem for a stretched Euler-Bernoulli beam on a star graph with three edges. The beams are hinged with respect to the boundary vertices. The inner vertex is capable of both translation and rotation, the latter of which is subject to a combination of elastic and frictional effects. We present detailed results on the prop...
Article
Full-text available
In this paper, we study the well-posedness and stability of a wave equation with infinitely structural memory, herein the memory kernel function does not satisfy the monotonicity. For the model, the history function space setting is a main difficulty because the usual space setting will lead the shift semigroup to be a unbounded semigroup. In the p...
Preprint
This paper is concerned with the controller design for distributed parameter systems with time-delays in the boundary. An implementable control law is presented in light of a state predictor and a feasible calculation scheme of this control law is provided based on the adjoint theory. The resulting closed-loop system is proved to be exponentially s...
Article
In this paper, we consider the uniform stabilization problem of a 1-d wave equation with variable coefficients, anti-damping and delayed boundary control. We design a new kind of state feedback controller to stabilize the system exponentially. The designed controller is taken as the integral form, whose kernel functions will be regarded as the sele...
Article
In the paper, we characterize the polynomial stability of C 0 semigroup T(t) with generator A on Hilbert space H. Let A have compact resolvent and there be a sequence of the eigenvectors of A that forms a Riesz basis for H. By the asymptotic relation of the real part and imaginary part of eigenvalues of A, we give the optimal decay rate of polynomi...
Article
In this paper, we introduce a new kind of feedback controller for an Euler‐Bernoulli beam with boundary delayed control. Different from the approach of the Smith predictor and dynamic feedback control, the new feedback controller is of the integral form in spatial variable, that is called the integral‐type full state feedback controller. Our goal o...
Article
In this paper, we consider the uniform stabilization problem for a coupled 1-d variable coefficient wave equation with anti-dampings in the interior domain and joint difference-type control. We use the integral-type feedback control to stabilize the system, in which the integral kernel functions are regarded as parameters. Our goal in the present p...
Preprint
In this paper, we consider the uniform stabilization of some high-dimensional wave equations with partial Dirichlet delayed control. Herein we design a parameterization feedback controller to stabilize the system. This is a new approach of controller design which overcomes the difficulty in stability analysis of the closed-loop system. The detailed...
Article
Full-text available
The stability of general tree-shaped wave networks with variable coefficients under boundary feedback controls is considered. Making full use of the tree-shaped structures, we present a detailed asymptotic spectral analysis of the networks. By proposing the from-root-to-leaf calculating technique, we deduce an explicit recursive expression for the...
Article
Full-text available
This paper is a survey for development of linear distributed parameter system. At first we point out some questions existing in current study of control theory for the Lp linear system with an unbounded control operator and an unbounded observation operator, such as stabilization problem and observer theory that are closely relevant to state feedba...
Article
We study the anti-disturbance problem of a 1-d wave equation with boundary control matched disturbance. In earlier literature, the authors always designed the controller such as the sliding mode control and the active disturbance rejection control to stabilize the system. However, most of the corresponding closed-loop systems are boundedly stable....
Article
In this paper, a parallel repairable system is investigated. The system consisting of two similar components and a repairman with single vacation, in which suppose that the life of component satisfies the exponentially distribution and the repair time of the component and the vacation time of the repairman both follow the general distribution while...
Article
In this paper, we study the solvability of a distribution-valued heat equation with nonlocal initial condition. Under proper assumption on parameters we get the explicit solution of the distribution-valued heat equation. As an application, we further consider the stabilization problem of heat equation with partial-delay in internal control. By the...
Article
This work is devoted to discussing that how the locations of the thermal and viscous damping affect the stability of the 1-d elastic systems. The spatial domain of the 1-d thermoelastic system under consideration can be divided into three sub-intervals, that is, the thermoelastic region of type II, the one of type II with viscous damping, and the o...
Article
Full-text available
In this paper, we study the spectrum of the Timoshenko beam with time delay in interior damping. At first, we prove the well-posedness of the delayed equations by the semigroup theory. To obtain the detailed spectral information of operator \(\mathcal {A}\) determined by the equations, we decompose \(\mathcal {A}\) into a sequence of unbounded oper...
Article
The stabilization problem for a joint string subject to pointwise input disturbance is concerned in this article. First, a method like sliding mode control is adopted to resist the disturbance. Observation blind point which leads to zero output is found in this non-linear system. Second, the existence and uniqueness of the solution for this non-lin...
Article
The stability of general tree-shaped wave networks with variable coefficients under boundary feedback controls is considered. Making full use of the tree-shaped structures, we present a detailed asymptotic spectral analysis of the networks. By proposing the from-root-to-leaf calculating technique, we deduce an explicit recursive expression for the...
Article
In this paper, we study boundary feedback stabilization of a linear wave equation by saturated linear or nonlinear Neumann control laws. Firstly we prove asymptotic stabilization of the closed-loop system when the feedback control law has a linear growth rate around zero. In particular, we study the effect of spatial dimension on the decay rate of...
Article
In this paper, the stabilization problem of one-dimensional Schrödinger equation with boundary disturbance is concerned. The variable structure control technique is adopted to design the nonlinear feedback controller. For the Schrödinger system, the observation blind point problem is discussed and the relationship between the initial data and obser...
Preprint
In this study, the stabilization problem for Schrödinger equation with distributed input time delay is considered. The main idea of solving the stabilization problem is transformation. The original time delay system is firstly transformed into the undelayed system, and then the feedback control law which can stabilize the undelayed system is found....
Article
Full-text available
For a distributed parameter system with an input delay in the boundary, a feedback control law is presented by means of the backstepping method. The square integrability of input signal is verified based on the target system. Then, the boundedness and invertibility of the corresponding backstepping transformation are proved under the regularity of...
Article
In this paper, we studied the necessary condition of distributed parameter system with exact controllability. Let $\Phi^t_0$ be the control mapping. We introduce new a class of control operators that is called the I-class control which satisfy $\mathcal{R}(\Phi^t_0)\cap \mathcal{R}(T(t))$ is closed set for $t>0$. If the system is exactly controllab...
Article
Full-text available
This article considers the stabilization problem of a rotating disk-beam system with localized thermal effect and torque control. Assume that the disk rotates with nonuniform angular velocity. A subdomain of the elastic beam is with thermoelastic damping, which is a kind of intrinsic one since thermoelasticity exists in almost all materials. Using...
Article
In this paper, we consider the exponential stabilization problem of a Timoshenko beam with interior local controls with input delays. In the past, most of the stabilization results for the Timoshenko beam were on the boundary control with input delays. In the present paper we shall extend the method treating the boundary control with delays to the...
Article
Design of controller subject to a constraint for a Schrödinger equation is considered based on the energy functional of the system. Thus, the resulting closed-loop system is nonlinear and its well-posedness is proven by the nonlinear monotone operator theory and a complex form of the nonlinear Lax-Milgram theorem. The asymptotic stability and expon...
Article
In this paper, we are concerned with the controller design for multi-dimensional Schrödinger equation with the internal delay control. We introduce a new approach to design the feedback control law based on the system equivalence. First, we construct a target system with the desired exponential stability. Second, we select a proper transformation a...
Article
The anti-disturbance stabilization problem of a kind of serially connected hybrid PDE–ODE system is considered. The system is constituted by n wave equations with n point masses at joint nodes and the free end, where the unknown disturbances exist. We stabilize the system by designing the nonlinear pointwise feedback control laws, in which the tech...
Article
The large-time behavior of a 1-d coupled string-beam system is considered. Combining a detailed spectral analysis with resolvent estimation, we obtain two kinds of energy decay rates of the string-beam system with different locations of the frictional damping. On one hand, if the frictional damping is only actuated in the beam part, the system lack...
Article
Full-text available
In this paper, we consider the stabilization problem of a wave equation with a tip mass, which undergoes the external disturbances at the tip mass end. Here, the disturbance may be exponentially increasing. For such a model, the usual sliding mode control method cannot be applied. Therefore, we employ the active disturbance rejection control (ADRC)...
Article
In this paper, the stability problem of 1-d wave equation with the boundary delay and the interior control is considered. The well-posedness of the closed-loop system is investigated by the linear operator. Based on the idea of Lyapunov functional technology, we give the condition on the relationship between the control parameter α and the delay pa...
Article
The boundary control problem of a cantilever Euler-Bernoulli is considered in this paper. If the control at the right end of the beam is of the form wxxx(1, t) = u(t − τ) + r(t), where τ > 0 is the input time-delay and r(t) is an unknown external disturbance, a dynamic feedback control strategy based on the methods of partial state predictor and ac...
Article
In this paper we consider the feedback stabilization of an Euler-Bernoulli beam with the boundary time-delay disturbance. Due to unknown time-delay coefficient, the system might be exponentially increasing at the lack of control. We design the feedback control law based on Lyapunov function method. Different from usual use of Lyapunov function meth...
Article
Full-text available
In this paper, the energy decay rate of a 1-d mixed type I and type II thermoelastic system is considered. The system consists of two kinds of thermoelastic components. One is the classical thermoelasticity (so-called type I), another one is nonclassical thermoelasticity without dissipation (named type II). These two components are coupled at the i...
Article
This paper studies the stabilization problem of an Euler-Bernoulli beam with a tip mass, which undergoes unknown but uniform bounded disturbance at tip mass. Here the nonlinear feedback control law is used to cancel the effects of the external disturbances. For the controlled nonlinear system, the authors prove the well-posedness by the maximal mon...
Article
In this paper, we are concerned with rapid stabilization of an Euler-Bernoulli beam with internal delayed control. Herein we introduce a new approach of the feedback control design from the system equivalence point of view. The design approach can be divided into several steps. First, we construct a target system of the desired stability. Second, w...
Article
In this paper, we consider the exponential stabilization for Timoshenko beam with different delays in the boundary control. Suppose that the controller outputs are of the form α1u1(t) + β1u1(t - t1) + γ1u1 (t - t2) and α2u2(t) + β2u2(t - t1) + γ2u2(t - t2), where u1(t) and u2(t) are the inputs of boundary controllers.
Article
Full-text available
The boundary stabilization problem of a thermoelastic system of type II with a tip-mass is considered with the assumption that there is an external non-uniform bounded disturbance at the control end. In order to estimate this kind of disturbance, a time-varying high-gain estimator is designed, where the idea of active disturbance rejection control...
Article
In this paper, we study the exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls. For the networks, there are some results on the exponential stability, but no result on estimate of the decay rate. The present work mainly estimates the decay rate for these systems, including signal wave equation, serially con...
Article
This paper considers the stabilization of a wave equation with interior input delay: �\mu_1u(x; t) + �\mu_2u(x; t -\tau ), where u(x; t) is the control input. A new dynamic feedback control law is obtained to stabilize the closedloop system exponentially for any time delay \tau> 0 provided that j�\mu_1\ne j\mu_�2j. Moreover, some su�cient condition...
Article
In this paper, we investigate the exponential stability of Timoshenko beam system with interior damping and boundary delays. At first, applying the semigroup theory of bounded linear operators we prove the well-posedness of the system. And then we give the exponential stability analysis of the system by constructing an appropriate Lyapunov function...
Article
We consider the stabilization problem of an Euler-Bernoulli beam with tip mass, which undergoes non-uniform bounded disturbance. We employ the idea of active disturbance rejection control to design a disturbance estimator that has a time-varying gain of exponential-type, and design a feedback controller, in which the estimate of disturbance is used...
Article
In this paper, we investigate the exponential stability of a Timoshenko beam with interior time delays and boundary damping. At first we show that the system is well-posed by the semigroup method. Next we construct an appropriate functional to study the exponential stability. We transform the exponential stability into the solvability of inequality...
Article
In this paper, we consider the exponential stabilization issue of Timoshenko beam with input and output delays. By using the Luenberger observer and Smith predictor we obtain an estimate of the state of the system, and by the partial state predictor we transform the delayed system into a without delay system, and then by the collocated feedback of...
Article
In this paper, we consider the exponential stabilization issue of Timoshenko beam with input and output delays. By using the Luenberger observer and Smith predictor we obtain an estimate of the state of the system, and by the partial state predictor we transform the delayed system into a without delay system, and then by the collocated feedback of...
Article
This paper concerns the existence of a solution and the stabilization of a one-dimensional Schrödinger equation with internal disturbance. Here, a non-linear controller is designed to stabilize the system. The existence of the solution for the system is verified by variation method, which can be regarded as an extended version of the Lions–Lax–Milg...
Article
Full-text available
We study the exponential stability of Euler-Bernoulli beam with interior time delays and boundary damping. At first, we prove the well-posedness of the system by the C 0 semigroup theory. Next we study the exponential stability of the system by constructing appropriate Lyapunov functionals. We transform the exponential stability issue into the s...
Article
In this paper, we consider the stabilization problem of the conservation systems with interior disturbance. Employing the idea of sliding-mode control, we design a nonlinear distributed feedback controller. We prove the solvability of the resulted closed-loop system by the maximal monotone operator theory. Further we prove the exponential stability...
Article
In this paper, a transmission problem between elastic and thermoelastic material is considered. Assume that these two materials are connected by a vibrating concentrated mass. By a detailed spectral analysis, the asymptotic expressions of the eigenvalues of the system are obtained, and based on which, the Riesz basis property of the eigenvectors is...
Article
In this paper, the stability of a one-dimensional thermoelastic system with boundary damping is considered. The theory of thermoelasticity under consideration is developed by Green and Naghdi, which is named as "thermoelasticity of type II". This system consists of two strongly coupled wave equations. By the frequency domain method, we prove that t...
Article
In this paper, we consider the output-feedback exponential stabilization of Timoshenko beam with the boundary control and input distributed delay. Suppose that the outputs of controllers are of the forms α1u1(t) + β1u1(t − τ) + � 0 −τ g1(η)u1(t + η)dη and α2u2(t) + β2u2(t − τ) + � 0 −τ g2(η)u2(t + η)dη respectively, where u1(t) and u2(t) are the in...
Article
The stabilization problem of a 1D Schrödinger equation subject to boundary control is concerned in this paper. The control input is suffered from time delay. A “partial state” predictor is designed for the system and undelayed system is deduced. Based on the undelayed system, a feedback control strategy is designed to stabilize the original system....
Article
This paper considers the controllability and Riesz basis generation property of linear infinite dimensional systems with C0-group generators and one-dimensional admissible input operators. The corresponding results of [Advances in Mathematical Systems Theory, 2000, pp. 221-242] under the assumption of algebraic simplicity for eigenvalues of the gen...
Article
This paper is concerned with the boundary stabilization of a Timoshenko beam with a tip payload under the boundary external disturbances. Nonlinear feedback control laws are designed to reduce the effects of the external disturbances. Since the controlled system is nonlinear, the well-posedness of the nonlinear closed-loop system is investigated us...
Article
We studied the exponential stabilization problem of a compounded system composed of a flow equation and an Euler-Bernoulli beam, which is equivalent to a cantilever Euler-Bernoulli beam with a delay controller. We designed a dynamic feedback controller that stabilizes exponentially the system provided that the eigenvalues of the free system are not...
Article
Full-text available
This paper concerns with the stabilization of a Timoshenko beam with bounded constraints on boundary feedback controls. Since the resulting controlled system is nonlinear, the weak well-posedness is proven by theories of the nonlinear monotone operators and the optimization. Then, the asymptotical stability of the controlled beam is analyzed by the...
Article
In this paper, we consider the energy decay rate of a thermoelastic Bresse system with variable coefficients. Assume that the thermo-propagation in the system satisfies the Cattaneo's law, which can eliminate the paradox of infinite speed of thermal propagation in the assumption of the Fourier's law in the classical theory of thermoelasticity. Mean...
Article
We consider the system with two different kinds of switch controls. The system is described by the wave equations with the structure of binary tree, of which one end is fixed and the other two ends are damped by the velocities feedback controls α(t)u2t(x, t); β(t)u3t(x, t) respectively, where the α(t), β(t) are both Heaviside-type functions of peri...
Article
In this paper, we consider the initial-boundary value problem of a binary bifurcation model of the human arterial system. Firstly, we obtain a new pressure coupling condition at the junction based on the mass and energy conservation law. Then, we prove that the linearization system is interior well-posed and L2 well-posed by using the semigroup the...
Conference Paper
feedback control design and exponential stabilization of the systems with input delay–A survey Genqi Xu Contents Controller Delay History about control delay Answer to Questions Idea and Method Dynamic feedback control design and exponential stabilization of the systems with input delay–A survey Genqi Xu Tianjin University, China 8/4/2014, 3th Conf...
Article
A new method used to prove the global convergence of the nonlinear conjugate gradient methods, the spectral method, is presented in this paper, and it is applied to a new conjugate gradient algorithm with sufficiently descent property. By analyzing the descent property, several concrete forms of this algorithm are suggested. Under standard Wolfe li...
Article
In this paper, we consider a serially connected thermoelastic rods of type II, where the heat conduction is described by the theories of Green and Naghdi. Using the frequency domain method, together with multiplier techniques, we show that the energy of this multi-connected system decays to zero exponentially in the presence of boundary and joint d...
Article
In this paper, we explore an approach to check the exponential stability of the distributed parameter system with non-collocated feedback. We regard the characteristic determinant as an entire function with parameters that usually are the feedback gains. By checking that there is no zero of this entire function on the imaginary axis for some parame...
Article
Full-text available
We introduce a new approach to investigate the stability of controlled tree-shaped wave networks and subtrees of complex wave networks. It is motivated by regarding the network as branching out from a single edge. We present the recursive relations of the Laplacian transforms of adjacent edges of the system in its branching order, which form the ch...
Article
Full-text available
Two exact formulae for the eigenvalues of one-dimensional wave equations on general feedback controlled networks are presented. Especially, by them together with the exponential polynomial theory and the graph theory, it is shown that the oscillation of tree-shaped networks with one fixed vertex can rest in finite time with appropriate dampers. In...
Article
In this paper, two optimal problems for a system composed of a main unit and a standby with common cause failure and critical human error are considered. The optimal repair rates for the optimal reliability and the optimal balance between system availability and repair cost are respectively discussed by constructing suitable objective functions and...
Article
We study the stability of a robot system composed of two Euler-Bernoulli beams with non-collocated controllers. By the detailed spectral analysis, we prove that the asymptotical spectra of the system are distributed in the complex left-half plane and there is a sequence of the generalized eigenfunctions that forms a Riesz basis in the energy space....
Article
Full-text available
By the theory of Sturm–Liouville eigenvalue problems, it is shown that the stability of one-dimensional wave equations with variable coefficients coupled with an ordinary differential equation (ODE) system on general tree-shaped networks is equivalent to that of its subsystem (called the base system). Thus, it is proved that the coupled system can...
Article
We study the problem of state reconstruction for exactly observable systems. Here we present two approaches for state reconstruction, the Riesz basis approach and the dual Hilbert uniqueness method (HUM) based on an iteration algorithm. The Riesz basis approach gives an explicit representation of the initial state of the system, whereas the HUM dua...