In this paper we investigate the relation between discrete and continuous operators. More precise... more In this paper we investigate the relation between discrete and continuous operators. More precisely, we investigate the properties of the semigroup generated by A, and the sequence A n d , n ∈ N, where A d = (I + A)(I − A) −1. We show that if A and A −1 generate a uniformly bounded, strongly continuous semigroup on a Hilbert space, then A d is power bounded. For analytic semigroups we can prove stronger results. If A is the infinitesimal generator of an analytic semigroup, then power boundedness of A d is equivalent to the uniform boundedness of the semigroup generated by A.
A vibrating system with some kind of internal damping represents a distributed or passive control... more A vibrating system with some kind of internal damping represents a distributed or passive control. In this article, a wave equation with clamped boundary conditions and internal Kelvin-Voigt damping is considered. It is shown that the spectrum of the system operator is composed of two parts: point spectrum and continuous spectrum. The point spectrum consists of isolated eigenvalues of finite algebraic multiplicity, and the continuous spectrum that is identical to the essential spectrum is an interval on the left real axis. The asymptotic behavior of eigenvalues is presented.
Abstract. We design a stabilizing linear boundary feedback control for a one-link flexible manipu... more Abstract. We design a stabilizing linear boundary feedback control for a one-link flexible manipulator with rotational inertia. The system is modelled as a Rayleigh beam rotating around one endpoint, with the torque at this endpoint as the control input. The closed-loop system is nondissipative, so that its well posedness is not easy to establish. We study the asymptotic properties of
A bs t ract : This paper is concerned with an optimal control problem of an ablation-transpiratio... more A bs t ract : This paper is concerned with an optimal control problem of an ablation-transpiration cooling control system with Stefan-Signorini boundary condition. As the continuation of the authors' previous paper , the Dubovits Rii-Milyutin functional ap2 proach is again adopted in investigation of the Pontryagin' s maximun principle of the system. The necessary optimality condition is presented for the problem with free final horizon and phase constraints. Ke ywor ds : Ablation-transpiration cooling process ; Stefan-Signorini problem ; Optimal control ; Maximum principle In Section 2 , we first formulate an optimal control problem for an ablation-transpiration system. Following the steps of the Dubovitskii and Milyutin approach in subsections , we derive the corresponding Pontryagin' s maximum principle . The main improvement , when compared with [ 1 ] , is that the following restrictions are removed. The first restriction is that the admissible control
Proceedings of the 44th IEEE Conference on Decision and Control, 2005
This article addressed the stabilization of a system of 1D swelling porous elastic soils with flu... more This article addressed the stabilization of a system of 1D swelling porous elastic soils with fluid saturation. The system is described by strongly coupled vibrating fluid and solid elastic materials. Using Riesz basis approach, we show that the whole system can be exponentially stabilized by only one internal viscous damping with variable feedback gain imposed in the fluid part, which is sharp contrast with the same effect by two dampings in existing literature. Moreover, the explicit asymptotic expressions of high eigenfrequencies exhibit clearly how this one damping can affect the another part of solid vibration.
This paper addresses the basis property of a linear hyperbolic system with dynamic boundary condi... more This paper addresses the basis property of a linear hyperbolic system with dynamic boundary condition in one space variable whose general form was first studied in . It is shown that under a regularity assumption, the spectrum of the system displays a distribution on the complex plane similar to zeros of a sine-type function and the generalized eigenfunctions of the system constitute a Riesz basis for its root subspace. The state space thereby decomposes into a topological direct sum of the root subspace with another invariant subspace in which the associated semigroup is supperstable: that is to say, the semigroup is identical to zero after a finite time. As a consequence, the spectrum-determined growth condition is established.
The identifiability of spatial variable coefficients for the vibrating string and Euler-Bernoulli... more The identifiability of spatial variable coefficients for the vibrating string and Euler-Bernoulli beam are considered. It is shown that the coefficients can be determined by means of boundary control and observation in a finite time duration. These results can be considered as the generalization of infinite-time coefficients identifiability through the application of the Ingham-Beurling theorem.
Proceedings of the IEEE Conference on Decision and Control
We are concerned with parameter estimation and stabilization of a one-dimensional wave equation w... more We are concerned with parameter estimation and stabilization of a one-dimensional wave equation with unknown constant signal disturbance suffered from boundary observation at one end and the non-collocated control at another end. An adaptive observer is designed in terms of measured velocity with unknown constant disturbance. The backstepping method is adopted in design of the feedback law. It is shown that the resulting closed-loop system is asymptotically stable. Meanwhile, the unknown constant is estimated.
An open-loop system of linear elasticity with Dirichlet boundary control and collocated observati... more An open-loop system of linear elasticity with Dirichlet boundary control and collocated observation is considered. The main result we obtained states that this system is well-posed in the sense of Salamon. The result deduces the exponential stability of the closed-loop system under proportional output feedback. Hence it answers positively an open question proposed by Liu and Krstić [IMA J. Appl. Math., 65 (2000), pp. 109-121]. Moreover, the well-posedness, together with the regularity of the open-loop system in the sense of Weiss that was obtained in our separate paper [S. G. Chai and B. Z. Guo, Feedthrough Operator for Linear Elasticity System with Boundary Control and Observation, Preprint, School of Computational and Applied Mathematics, University of the Witwatersrand, South Africa, 2008], makes this infinite-dimensional system parallel in many ways to a linear finite-dimensional system in the framework of well-posed and regular linear infinite-dimensional systems. 20.1.2 of [23]), we have
A new spectral analysis for the asymptotic locations ofeigenvalues of a constrained translating s... more A new spectral analysis for the asymptotic locations ofeigenvalues of a constrained translating string is presented. The constraint modeled by a spring-mass-dashpot is located at any position along the string. Asymptotic solutions for the eigenvalues are determined from the characteristic equation of the coupled system of contraint and string for all constraint parameters. Damping in the constraint dissipates vibration energy in all modes whenever its dimensionless location along the string is an irrational number. It is shown that although all eigenvalues have strictly negative real parts, an infinite number of them approach the imaginary axis. The analytical predictions for the distribution of eigenvalues are validated by numerical analyses.
In this paper, we are concerned with the following nonlinear parabolic equation with a gradient t... more In this paper, we are concerned with the following nonlinear parabolic equation with a gradient term and Neumann boundary condition:
In this paper we investigate the relation between discrete and continuous operators. More precise... more In this paper we investigate the relation between discrete and continuous operators. More precisely, we investigate the properties of the semigroup generated by A, and the sequence A n d , n ∈ N, where A d = (I + A)(I − A) −1. We show that if A and A −1 generate a uniformly bounded, strongly continuous semigroup on a Hilbert space, then A d is power bounded. For analytic semigroups we can prove stronger results. If A is the infinitesimal generator of an analytic semigroup, then power boundedness of A d is equivalent to the uniform boundedness of the semigroup generated by A.
A vibrating system with some kind of internal damping represents a distributed or passive control... more A vibrating system with some kind of internal damping represents a distributed or passive control. In this article, a wave equation with clamped boundary conditions and internal Kelvin-Voigt damping is considered. It is shown that the spectrum of the system operator is composed of two parts: point spectrum and continuous spectrum. The point spectrum consists of isolated eigenvalues of finite algebraic multiplicity, and the continuous spectrum that is identical to the essential spectrum is an interval on the left real axis. The asymptotic behavior of eigenvalues is presented.
Abstract. We design a stabilizing linear boundary feedback control for a one-link flexible manipu... more Abstract. We design a stabilizing linear boundary feedback control for a one-link flexible manipulator with rotational inertia. The system is modelled as a Rayleigh beam rotating around one endpoint, with the torque at this endpoint as the control input. The closed-loop system is nondissipative, so that its well posedness is not easy to establish. We study the asymptotic properties of
A bs t ract : This paper is concerned with an optimal control problem of an ablation-transpiratio... more A bs t ract : This paper is concerned with an optimal control problem of an ablation-transpiration cooling control system with Stefan-Signorini boundary condition. As the continuation of the authors' previous paper , the Dubovits Rii-Milyutin functional ap2 proach is again adopted in investigation of the Pontryagin' s maximun principle of the system. The necessary optimality condition is presented for the problem with free final horizon and phase constraints. Ke ywor ds : Ablation-transpiration cooling process ; Stefan-Signorini problem ; Optimal control ; Maximum principle In Section 2 , we first formulate an optimal control problem for an ablation-transpiration system. Following the steps of the Dubovitskii and Milyutin approach in subsections , we derive the corresponding Pontryagin' s maximum principle . The main improvement , when compared with [ 1 ] , is that the following restrictions are removed. The first restriction is that the admissible control
Proceedings of the 44th IEEE Conference on Decision and Control, 2005
This article addressed the stabilization of a system of 1D swelling porous elastic soils with flu... more This article addressed the stabilization of a system of 1D swelling porous elastic soils with fluid saturation. The system is described by strongly coupled vibrating fluid and solid elastic materials. Using Riesz basis approach, we show that the whole system can be exponentially stabilized by only one internal viscous damping with variable feedback gain imposed in the fluid part, which is sharp contrast with the same effect by two dampings in existing literature. Moreover, the explicit asymptotic expressions of high eigenfrequencies exhibit clearly how this one damping can affect the another part of solid vibration.
This paper addresses the basis property of a linear hyperbolic system with dynamic boundary condi... more This paper addresses the basis property of a linear hyperbolic system with dynamic boundary condition in one space variable whose general form was first studied in . It is shown that under a regularity assumption, the spectrum of the system displays a distribution on the complex plane similar to zeros of a sine-type function and the generalized eigenfunctions of the system constitute a Riesz basis for its root subspace. The state space thereby decomposes into a topological direct sum of the root subspace with another invariant subspace in which the associated semigroup is supperstable: that is to say, the semigroup is identical to zero after a finite time. As a consequence, the spectrum-determined growth condition is established.
The identifiability of spatial variable coefficients for the vibrating string and Euler-Bernoulli... more The identifiability of spatial variable coefficients for the vibrating string and Euler-Bernoulli beam are considered. It is shown that the coefficients can be determined by means of boundary control and observation in a finite time duration. These results can be considered as the generalization of infinite-time coefficients identifiability through the application of the Ingham-Beurling theorem.
Proceedings of the IEEE Conference on Decision and Control
We are concerned with parameter estimation and stabilization of a one-dimensional wave equation w... more We are concerned with parameter estimation and stabilization of a one-dimensional wave equation with unknown constant signal disturbance suffered from boundary observation at one end and the non-collocated control at another end. An adaptive observer is designed in terms of measured velocity with unknown constant disturbance. The backstepping method is adopted in design of the feedback law. It is shown that the resulting closed-loop system is asymptotically stable. Meanwhile, the unknown constant is estimated.
An open-loop system of linear elasticity with Dirichlet boundary control and collocated observati... more An open-loop system of linear elasticity with Dirichlet boundary control and collocated observation is considered. The main result we obtained states that this system is well-posed in the sense of Salamon. The result deduces the exponential stability of the closed-loop system under proportional output feedback. Hence it answers positively an open question proposed by Liu and Krstić [IMA J. Appl. Math., 65 (2000), pp. 109-121]. Moreover, the well-posedness, together with the regularity of the open-loop system in the sense of Weiss that was obtained in our separate paper [S. G. Chai and B. Z. Guo, Feedthrough Operator for Linear Elasticity System with Boundary Control and Observation, Preprint, School of Computational and Applied Mathematics, University of the Witwatersrand, South Africa, 2008], makes this infinite-dimensional system parallel in many ways to a linear finite-dimensional system in the framework of well-posed and regular linear infinite-dimensional systems. 20.1.2 of [23]), we have
A new spectral analysis for the asymptotic locations ofeigenvalues of a constrained translating s... more A new spectral analysis for the asymptotic locations ofeigenvalues of a constrained translating string is presented. The constraint modeled by a spring-mass-dashpot is located at any position along the string. Asymptotic solutions for the eigenvalues are determined from the characteristic equation of the coupled system of contraint and string for all constraint parameters. Damping in the constraint dissipates vibration energy in all modes whenever its dimensionless location along the string is an irrational number. It is shown that although all eigenvalues have strictly negative real parts, an infinite number of them approach the imaginary axis. The analytical predictions for the distribution of eigenvalues are validated by numerical analyses.
In this paper, we are concerned with the following nonlinear parabolic equation with a gradient t... more In this paper, we are concerned with the following nonlinear parabolic equation with a gradient term and Neumann boundary condition:
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Papers by Bao-zhu Guo