On stability of a class of positive linear functional difference equations

We first give a sufficient condition for positivity of the solution semigroup of linear functional difference equations. Then, we obtain a Perron–Frobenius theorem for positive linear functional difference equations. Next, we offer a new explicit criterion for exponential stability of a wide class of positive equations. Finally, we study stability radii of positive linear functional difference equations. It is proved that complex, real and positive stability radius of positive equations under structured perturbations (or affine perturbations) coincide and can be computed by explicit formulae. Pham Huu Anh Ngoc and Toshiki Naito are supported by the Japan Society for Promotion of Science (JSPS) ID No. P 05049.     

On a class of positive linear differential equations with infinite delay

We give an explicit criterion for positivity of the solution semigroup of linear differential equations with infinite delay and a Perron-Frobenius type theorem for positive equations. Furthermore, a novel criterion for the exponential asymptotic stability of positive equations is presented. Finally, we provide a sufficient condition for the exponential asymptotic stability of positive equations subjected to structured perturbations. A simple example is given to illustrate the obtained results.     

Perturbation of strongly and polynomially stable Riesz-spectral operators

In this paper we consider bounded and relatively bounded finite rank perturbations of a Riesz-spectral operator generating a polynomially stable semigroup of linear operators on a Hilbert space. We concentrate on a commonly encountered situation where the spectrum of the unperturbed operator is contained in the open left half-plane of the complex plane and approaches the imaginary axis asymptotically. We present conditions on the perturbing operator such that the spectrum of the perturbed operator is contained in the open left half-plane of the complex plane and additional conditions for the strong and polynomial stabilities of the perturbed semigroup. We consider two applications of the perturbation results. In the first example we apply the results to the perturbation of a polynomially stabilized one-dimensional wave equation. In the second example we consider the perturbation of a closed-loop system consisting of a distributed parameter system and an observer-based feedback controller solving the robust output regulation problem related to an infinite-dimensional signal generator.     

Strong stability radii of positive linear time‐delay systems

In this paper, we study robustness of the strong delay‐independent stability of linear time‐delay systems under multi‐perturbation and affine perturbation of coefficient matrices via the concept of strong delay‐independent stability radius (shortly, strong stability radius). We prove that for class of positive time‐delay systems, complex and real strong stability radii of positive linear time‐delay systems under multi‐perturbations (or affine perturbations) coincide and they are computed via simple formulae. Apart from that, we derive solution of a global optimization problem associated with the problem of computing of the strong stability radii of a positive linear time‐delay system. An example is given to illustrate the obtained results. Copyright © 2005 John Wiley & Sons, Ltd.     

具有边界控制的线性Timoshenko型系统的指数稳定性

本文研究多孔弹性材料在实际应用中的镇定问题.多孔物体的动力学行为由线性Timoshenko型方程描述,这样的系统一般只是渐近稳定但不指数稳定.假定系统两端都是自由的,在自由端对系统施加边界速度反馈控制,本文讨论闭环系统的适定性和指数稳定性.首先,利用有界线性算子半群理论得到了系统的适定性.进一步对系统算子的本征值的渐近值估计,得到算子谱分布在一个带域,相互分离的,模充分大的本征值都是简单本征值.通过引入一个辅助算子,利用它的谱性质以及有界线性算子的扰动理论,得到系统的广义本征向量的完整性以及Riesz基性质.最后利用Riesz基性质和谱分布得到闭环系统的指数稳定性.     

The Popov criterion for strongly stable distributed parameter systems

In this article we generalize the Popov criterion to the class of strongly stable infinite-dimensional linear systems; the semigroup is strongly stable and the input to state, state to output and input to output maps are all bounded on the infinite-time interval. One application is to show that integral control can be used to track constant reference signals for positive-real strongly stable systems in the presence of sectorial non-linearities. A second application is to show the robustness of asymptotic stability of positive-real strongly stable systems to a large class of non-linear perturbations. Systems satisfying the assumptions in this paper include dissipative systems with collocated actuators and sensors.     

Stability and robustness analysis of boundary control of a flexible space structure

We consider the question of the robustness of boundary control for a simplified SCOLE (spacecraft control laboratory experiment) model. A continuum model consisting of a long flexible mast joining two rigid bodies, one of which represents the space shuttle orbiter, the other an antenna reflector, is developed. The dynamics of the system are represented by a coupled set of ordinary and partial differential equations, and formulated as a wave equation in a Hilbert space by using a semigroup approach. For a continuum, distributed controls have often been proposed to stabilize the perturbed system. However, distributed controls for the continuous system are not practical to implement. Hence, for easy implementation we employ the boundary controls, as a special class of localized controls, that are control forces or torques applied only at the boundary. In this paper it is shown that the flexible system subject to a certain class of perturbations remains stabilizable by the same boundary feedback control as that for the nominal system. The results of numerical simulations for the simplified model are also presented to demonstrate the effective strong stabilization and robustness of the boundary controller with respect to the various perturbations.     

Compactness and nuclearity of the Hankel operator and internal stability of infinite-dimensional state linear systems

We examine the relationships between the exponential (or strong) stability of certain classes of regular linear systems and the compactness and nuclearity properties of the Hankel operator. New sufficient conditions for nuclearity are given for exponentially stable regular linear systems with an analytic semigroup.     

Stability and robust stability of positive linear Volterra difference equations

We first introduce a class of positive linear Volterra difference equations. Then, we offer explicit criteria for uniform asymptotic stability of positive equations. Furthermore, we get a new Perron–Frobenius theorem for positive linear Volterra difference equations. Finally, we study robust stability of positive equations under structured perturbations and affine perturbations. Two explicit stability bounds with respect to these perturbations are given. Copyright © 2008 John Wiley & Sons, Ltd.     

Robust stability of perturbed switching systems by matrix measure

The robust stability of switching systems is a very difficult problem because of the difficulty in separating the perturbations of the models and the reset maps. In this paper, we apply matrix measure to separate the perturbations of the models and the reset maps. Then the conditions for robust stability of perturbed switching systems with structured perturbations can be readily derived. Sufficient conditions for robust stability of switching systems are proposed, and bounds for structured perturbations based on nonnegative matrices that still ensure stability are derived.     

负载Cartesian柔性臂非线性反馈控制

本文考虑ｘ和ｙ方向水平运动的Ｃａｒｔｅｓｉａｎ机器人控制系统。对末端具有负载的单杆柔性机械臂，我们给出了一个连续的分布参数模型，并提出了一类传感器和控制器并置在根部的非线性反馈控制。运用抽象Ｈｉｌｂｅｒｔ空间非线性算子半群理论，获得整个闭环系统的强渐近稳定。     

Robustness analysis and synthesis of SISO systems under both plant and controller perturbations

In this paper, we consider the robustness analysis and synthesis problem for a class of control systems under both plant and controller perturbations. Necessary and sufficient conditions are given for robust stability of SISO systems subject to additive norm-bounded perturbations. Linear matrix inequality (LMI)-based controller design method is presented. Robust stability of systems with both plant and controller perturbations, and stability analysis of a class of multilinear disc polynomials are unified in the same framework, and can be solved by computing the largest norm of a second-order matrix along the imaginary axis.     

Exponential stability criteria for discrete time‐delay systems

In this paper, we study the exponential stability of linear discrete time‐delay systems with slowly varying coefficients and nonlinear perturbations. We establish the robustness of the exponential stability in Hilbert spaces, in the sense that the exponential stability for a given linear equation persists under sufficiently small perturbations. As an application of the main results, we discuss the exponential stability of a general nonlinear system. The main novelty of this work is that we always consider the exponential behavior of solutions with respect to an specific ball. Copyright © 2013 John Wiley & Sons, Ltd.     

Hybrid Ishikawa iterative methods for a nonexpansive semigroup in Hilbert space

In this paper, on the base of the Ishikawa iteration method and the hybrid method in mathematical programming, we give two new strong convergence methods for finding a point in the common fixed point set of a nonexpansive semigroup in Hilbert space.     

Necessary and sufficient stability conditions for the case of dependent additive perturbations

In this paper we examine the stability properties of linear multivariable feedback systems in the presence of structured additive perturbations characterized by dependencies occurring in the perturbation matrix. Such dependencies may take the form of dependent phases between elements of element-by-element bounded perturbations and/or repeated blocks in block-structured perturbations. By using non-diagonal and possibly complex transformation matrices, it is shown how stability conditions that are both necessary and sufficient may be obtained for this class of perturbation.     

Lack of exponential decay in viscoelastic materials with voids

We study the stability of solutions to a coupled evolution system associated with an isotropic porous and centrosymmetric viscoelastic solid with porous dissipation (a porous elastic system with history). With the help of the method of the semigroup theory and some novel observations, we prove successfully that the condition of equal wave-speed propagation is still necessary for exponential stability of the system in the case of Dirichlet–Dirichlet boundary conditions.     

Robustness of stability of time-varying index-1 DAEs

We study exponential stability and its robustness for time-varying linear index-1 differential-algebraic equations. The effect of perturbations on the leading coefficient matrix is investigated. An appropriate class of allowable perturbations is introduced. Robustness of exponential stability with respect to a certain class of perturbations is proved in terms of the Bohl exponent and perturbation operator. Finally, a stability radius involving these perturbations is introduced and investigated. In particular, a lower bound for the stability radius is derived. The results are presented by means of illustrative examples.     

Strong convergence for a minimization problem on solutions of systems of equilibrium problems and common fixed points of an infinite family and semigroup of nonexpansive mappings

In this paper, we introduce a suitable Mann type algorithm for finding a common element of the set of solutions of systems of equilibrium problems and the set of common fixed points of an infinite family and left amenable semigroup of nonexpansive mappings in Hilbert spaces. Then we prove the strong convergence of the proposed iterative scheme to the unique solution of the minimization problem on the solution of systems of equilibrium problems and the common fixed points of an infinite family and left amenable semigroup of nonexpansive mappings. Our results extend and improve the recent result of Colao and Marino [V. Colao and G. Marino, Strong convergence for a minimization problem on points of equilibrium and common fixed points of an infinite family of nonexpansive mappings, Nonlinear Anal. 73 (2010) 3513–3524] and many others.     

时滞切换不确定神经网络系统的指数稳定性

本文研究了具有无穷时滞切换不确定细胞神经网络(UCNNs)系统任意切换下的指数稳定性.利用同胚映射和M-矩阵理论,得到UCNNs系统平衡点存在性,唯一性和指数稳定性的充分条件;利用Lyapunov泛函方法,研究了时滞切换UCNNs系统任意切换下的鲁棒指数稳定性,并得到确保系统全局指数稳定的充分条件.     

Stability radii of infinite-dimensional systems subjected to unbounded stochastic perturbations

In this paper, we use the framework of stability radii to study the robust stability of linear deterministic systems on real Hilbert spaces which are subjected to unbounded stochastic perturbations. First, we establish an existence and uniqueness theorem of the solution of the abstract equation describing the system. Then we characterize the stability radius in terms of a Lyapunov equation or equivalently in terms of the norm of an input-output operator.