FINITE ELEMENTS WITH LOCAL PROJECTION STABILIZATION FOR INCOMPRESSIBLE FLOW PROBLEMS

In this paper we review recent developments in the analysis of finite element methods for incompressible flow problems with local projection stabilization （LPS）. These methods preserve the favourable stability and approximation properties of classical residual-based stabilization （RBS） techniques but avoid the strong coupling of velocity and pressure in the stabilization terms. LPS-methods belong to the class of symmetric stabilization techniques and may be characterized as variational multiscale methods. In this work we summarize the most important a priori estimates of this class of stabilization schemes developed in the past 6 years. We consider the Stokes equations, the Oseen linearization and the NavierStokes equations. Furthermore, we apply it to optimal control problems with linear（ized） flow problems, since the symmetry of the stabilization leads to the nice feature that the operations ＂discretize＂ and ＂optimize＂ commute.     

UNIFORM STABILITY AND ERROR ANALYSIS FOR SOME DISCONTINUOUS GALERKIN METHODS

In this paper,we provide a number of new estimates on the stability and convergence of both hybrid discontinuous Galerkin(HDG)and weak Galerkin(WG)methods.By using the standard Brezzi theory on mixed methods,we carefully define appropriate norms for the various discretization variables and then establish that the stability and error estimates hold uniformly with respect to stabilization and discretization parameters.As a result,by taking appropriate limit of the stabilization parameters,we show that the HDG method converges to a primal conforming method and the WG method converges to a mixed conforming method.     

Uniform analysis of a stabilized hybrid finite element method for Reissner-Mindlin plates

This paper presents a low order stabilized hybrid quadrilateral finite element method for ReissnerMindlin plates based on Hellinger-Reissner variational principle,which includes variables of displacements,shear stresses and bending moments.The approach uses continuous piecewise isoparametric bilinear interpolations for the approximations of the transverse displacement and rotation.The stabilization achieved by adding a stabilization term of least-squares to the original hybrid scheme,allows independent approximations of the stresses and moments.The stress approximation adopts a piecewise independent 4-parameter mode satisfying an accuracy-enhanced condition.The approximation of moments employs a piecewise-independent 5-parameter mode.This method can be viewed as a stabilized version of the hybrid finite element scheme proposed in [Carstensen C,Xie X,Yu G,et al.A priori and a posteriori analysis for a locking-free low order quadrilateral hybrid finite element for Reissner-Mindlin plates.Comput Methods Appl Mech Engrg,2011,200:1161-1175],where the approximations of stresses and moments are required to satisfy an equilibrium criterion.A priori error analysis shows that the method is uniform with respect to the plate thickness t.Numerical experiments confirm the theoretical results.     

STABILIZATION OF MARKOVIAN JUMP LINEAR SYSTEMS VIA RANDOM COMMUNICATION TIME DELAYED AND IMPULSIVE CONTROLLERS

This paper deals with the stabilization problems for a networked control system via time-delayed and impulsive controllers. The random communication delays in the model signal are modeled as a Markov chain. First, we introduce a hybrid controller with delay and impulses for the networked control systems. Then, some sufficient conditions are proposed for the design of a hybrid controller such that the closed-loop system is stochastically stable.     

UNIFORM SUPERCONVERGENCE OF A FINITE ELEMENT METHOD WITH EDGE STABILIZATION FOR CONVECTION-DIFFUSION PROBLEMS

In the present paper the edge stabilization technique is applied to a convection-diffusion problem with exponential boundary layers on the unit square, using a Shishkin mesh with bilinear finite elements in the layer regions and linear elements on the coarse part of the mesh. An error bound is proved for ‖πu-u^h‖Е, where πu is some interpolant of the solution u and uh the discrete solution. This supercloseness result implies an optimal error estimate with respect to the L2 norm and opens the door to the application of postprocessing for improving the discrete solution.     

Stabilization of fractional bilinear systems with multiple inputs

In this paper, we study in a constructive way the stabilization problem of fractional bilinear systems with multiple inputs. Using the quadratic Lyapunov functions and some additional hypotheses on the unit sphere, we construct stabilizing feedback laws for the considered fractional bilinear system. A numerical example is given to illustrate the efficiency of the obtained result.     

Boundary Feedback Stabilization of Naghdi＇s Model

We consider the stabilization of Naghdi‘s model by boundary feedbacks where the model has a middle surface of any shape. First, applying the semigroup approach and the regularity of elliptic boundary value problems, we obtain the existence, the uniqueness, and the properties of solutions to Naghdi‘s model. Finally, we establish the exponential decay rates for Naghdi‘s model under some checkable geometric conditions on the middle surface.     

STABILIZED FEM FOR CONVECTION-DIFFUSION PROBLEMS ON LAYER-ADAPTED MESHES

The application of a standard Galerkin finite element method for convection-diffusion problems leads to oscillations in the discrete solution, therefore stabilization seems to be necessary. We discuss several recent stabilization methods, especially its combination with a Galerkin method on layer-adapted meshes. Supercloseness results obtained allow an improvement of the discrete solution using recovery techniques.     

ON DECENTRALIZED OBSERVATION AND DECENTRALIZED CONTROL OF LARGE SCALE SYSTEMS

In this paper, the problem of decentralized observation and decentralized control of linear, time-invariant large scale systems is studied. Based on conventional observer theory, the necessary and sufficient conditions for existence of completely decentralized observer are obtained. Moreover, the necessary and sufficient condition is given for stabilization of large scale systems by local output with state of the subsystem estimated through the completely decentralized observer.     

左三角范畴的稳定化和幂等完备

Let (CΩ△) be a left triangulated category with a fully faithful endofunctor.We show a triangle-equivalence (S(C),△) ～=(S(C),△),where (S(C)△,) denotes the stabilization of the idempotent completion of (C,△) and (S(C),△) denotes the idempotent completion of the stabilization of (C,△).     

纯量反馈系统同时强镇定的充分条件

一个稳定的补偿器可同时镇定n个对象(同时强镇定)等价于一个补偿器(不一定稳定)同时镇定n 1个对象(同时镇定)．两个以上对象的同时强镇定和三个以上对象的伺时镇定是线性系统中一个急待解决的公开问题．文中所作的基本假定是所有的对象具有相同的简单不稳定零点，在此条件下给出了n个对象同时强镇定的一个充分条件．当仅有一个不稳定零点时．容易检验是否同时强镇定，否则仅需确定n个对象的不稳定零点并且判定由不稳定零点导出一个相应矩阵是正定的，就能判定n个对象同时强镇定．因此是一个易于检验的充分条件．文章同时给出了n个对象同时强镇定的算法，丰富了同时强镇定的充分条件．     

Stabilization and set stabilization of switched Boolean control networks via flipping mechanism

Stabilization and set stabilization of switched Boolean control networks are investigated by using flipping mechanism in this paper. Firstly, with the help of Warshall algorithm, an explicit criterion for the stabilization of switched Boolean control networks is derived. Secondly, the necessary and sufficient condition for the solvability of stabilization of switched Boolean control networks, by flipping some elements of perturbation set once, is presented. Thirdly, a search algorithm is proposed to calculate the minimum number of stabilization flipped nodes and what exactly they are. Furthermore, a necessary and sufficient condition is established for the solvability of set stabilization of switched Boolean control networks by flipping some elements of perturbation set once. Analogously, an algorithm is given to find the minimum number of set stabilization flipped nodes. Finally, examples are shown to demonstrate the feasibility of the above results.     

Vibrational stabilization and the Brockett problem

The present paper deals with the exposition of methods for solving the Brockett problem on the stabilization of linear control systems by a nonstationary feedback. The paper consists of two parts. We consider continuous linear control systems in the first part and discrete systems in the second part. In the first part, we consider two approaches to the solution of the Brockett problem. The first approach permits one to obtain low-frequency stabilization, and the second part deals with high-frequency stabilization. Both approaches permit one to derive necessary and sufficient stabilization conditions for two-dimensional (and three-dimensional, for the first approach) linear systems with scalar inputs and outputs. In the second part, we consider an analog of the Brockett problem for discrete linear control systems. Sufficient conditions for low-frequency stabilization of linear discrete systems are obtained with the use of a piecewise constant periodic feedback with sufficiently large period. We obtain necessary and sufficient conditions for the stabilization of two-dimensional discrete systems. In the second part, we also consider the control problem for the spectrum (the pole assignment problem) of the monodromy matrix for discrete systems with a periodic feedback.     

Gyroscopic stabilization of nonconservative systems

The problem is considered of the stabilization of a mechanical system having only nonconservative positional forces by adding gyroscopic forces. The gyroscopic stabilization is proved to be always realizable in the case of a degenerate matrix of nonconservative forces and even number of coordinates. If the matrix of nonconservative forces is nonsingular then a possibility of the gyroscopic stabilization is established for all systems whose number of coordinates is divisible by four. For a nonautonomous systemwith nonconservative positional forces and dissipative forces with complete dissipation, some sufficient conditions are obtained for stabilization up to the exponential stability by addition of gyroscopic forces.     

Optimal Control and Stabilization for Some Fisher-Like Models

This article concerns optimal control and stabilization for some Fisher-like models with control acting in a subdomain ω. We investigate the optimal position of ω for some optimal harvesting problems. First, we refer to a logistic model with diffusion. We remember the necessary optimality conditions, and then obtain an iterative method to improve the position of ω for the optimal harvesting effort (for a simplified model without logistic term). Next, we consider the null stabilization for a controlled Fisher model and obtain a descent method to improve the position of ω in order to get a faster stabilization to zero. Numerical tests illustrating the effect of the last method are given. We also studied the null stabilization for a prey-predator system and have reduced it to the study of the null stabilizability for a related Fisher model.     

一类非线性系统的局部镇定

研究一类非线性系统的局部状态反馈镇定问题．基于中心流形理论,给出一类非线性系统渐近镇定的充分条件,并设计出镇定系统的反馈控制律．文中利用具有齐次导数的Lyapunov函数方法,特别研究了一类平面非线性系统及具有二重零特征值的一类非线性系统的渐近镇定问题,给出了系统镇定的若干充分条件,并构造出控制律．文中的例表明了所得结果的有效性．     

Stabilization of Unstable Periodic Orbits of One-Dimensional Chaotic Maps

Methods have been recently developed for stabilization of periodic orbits that become unstable when the parameter crosses a certain value. However, in a typical case, one-parameter families of nonlinear maps produce periodic orbits that are unstable from the start. In this article, we propose a method for stabilization of such orbits in the one-dimensional case. Results are reported for stabilization of periodic orbits of a quadratic map.     

Stabilization of functions and its application

The concepts of polynomial stabilization, strong polynomial stabilization, and strong stabilization are introduced for a fundamental system of solutions of linear differential equations. Some criteria of such kinds of stabilizations and applications to the theory of existence and uniqueness of solutions of ordinary differential equations are given. An abstract scheme of the obtained results is presented for Banach spaces.     

Controllability and stabilization of periodic switched Boolean control networks with application to asynchronous updating

This paper investigates the periodic switching point controllability and stabilization of periodic switched Boolean control networks (PSBCNs), and applies the obtained results to the stabilization of deterministic asynchronous Boolean control networks (DABCNs). Firstly, using the algebraic state space representation of PSBCNs, a kind of periodic switching point controllability matrix is constructed, based on which, a necessary and sufficient condition is presented for the periodic switching point reachability and controllability of PSBCNs. Secondly, using the reachable set of PSBCNs, a constructive procedure is proposed to design time-variant state feedback controllers for the periodic switching point stabilization of PSBCNs. Finally, by converting the dynamics of DABCNs into the form of PSBCNs, the time-variant state feedback stabilization problem of DABCNs is solved.     

Stability and Stabilization Criteria for a Class of Uncertain Neutral Systems with Time-Varying Delays

In this paper, the robust asymptotic stability and stabilization for a class of uncertain neutral system with time-varying delays are considered. Based on the Lyapunov-Krasovskii functional theory, some stability and stabilization criteria are derived. Delay-dependent and delay-independent criteria are proposed for the stability and stabilization of the considered systems. State and output feedbacks are considered to stabilize the uncertain neutral system. A linear matrix inequality approach and a genetic algorithm are used to solve the stability and stabilization problems. Finally, some numerical examples are shown to illustrate the use of the obtained results.The research reported here was supported by the National Science Council of Taiwan under Grant NSC 91-2213-E-214-016.Communicated by C. T. Leondes