Qualitative properties and stabilizability of a model for blood thrombin formation

We perform a qualitative analysis of a differential equation that was originally introduced by Stortelder, Hemker and Hemker to model the formation of thrombin, and discuss issues of controllability and stabilizability. Results include a general proof of convergence to equilibrium, and of local exponential stabilizability.     

Exponential stabilization of nonlinear discrete-time systems

In this paper, we give sufficient conditions for the exponential stabilizability of a class of perturbed non-autonomous difference equations with slowly varying coefficients. Under appropriate growth conditions on the perturbations, we establish explicit results concerning the feedback exponential stabilizability.     

基于开关控制的线性奇异系统的二次状态反馈镇定问题

主要讨论基于开关控制的线性奇异系统的二次状态反馈镇定问题.利用二次反馈镇定的概念,给出了线性奇异系统基于异步开关控制的二次状态反馈镇定问题可解的两个充分条件.进一步,对于带有范数有界的不确定项的奇异线性系统,给出了其可以基于异步开关控制的二次状态反馈鲁棒镇定的可解性条件.     

Feedback stabilization of 2D Navier–Stokes equations with Navier slip boundary conditions

We study the local exponential stabilizability with internally distributed feedback controllers for the incompressible 2D-Navier–Stokes equations with Navier slip boundary conditions. These controllers are localized in a subdomain and take values in a finite-dimensional space.     

Robust exponential stability and stabilizability of linear parameter dependent systems with delays

The robust exponential stability and stabilizability problems are addressed in this paper for a class of linear parameter dependent systems with interval time-varying and constant delays. In this paper, restrictions on the derivative of the time-varying delay is not required which allows the time-delay to be a fast time-varying function. Based on the Lyapunov-Krasovskii theory, we derive delay-dependent exponential stability and stabilizability conditions in terms of linear matrix inequalities (LMIs) which can be solved by various available algorithms. Numerical examples are given to illustrate the effectiveness of our theoretical results.     

Set stabilizability of switched Boolean control networks via Ledley antecedence solution

This paper studies two kinds of set stabilizability issues of switched Boolean control networks (SBCNs) by Ledley antecedence solution, that is, pointwise set stabilizability and set stabilizability under arbitrary switching signals. Firstly, based on the state transition matrix of SBCNs, the mode-dependent truth matrix is defined. Secondly, using the mode-dependent truth matrix in every step, a switching signal and the corresponding Ledley antecedence solutions are determined. Furthermore, a state feedback switching signal and a state feedback control are obtained for the pointwise set stabilizability. Thirdly, with the help of all mode-dependent truth matrices, the Ledley antecedence solutions are derived for a set of Boolean inclusions, which admits a state feedback control for the set stabilizability under arbitrary switching signals. Finally, an example is given to show the effectiveness of the proposed results.     

Stabilizability of a class of stochastic bilinear hybrid systems

The problem of the stabilizability of stochastic nonlinear hybrid systems with a Markovian or any switching rule is considered. Using the Lyapunov technique sufficient conditions for the asymptotic stabilizability in probability by a smooth controller in every structure are found. In particular, the asymptotic stabilizability in probability problem of stochastic bilinear hybrid systems with a Markovian or any switching rule is discussed and a closed-loop controller is found. Also the sufficient conditions for the exponential mean-square stabilizability for bilinear hybrid systems with any switching based on the Lie algebra approach are formulated and an open-loop controller is designed. The obtained results are illustrated by examples and simulations.     

Exponential stabilization of distributed semilinear systems by optimal control

This paper studies exponential stabilization of distributed semilinear systems. The paper (i) gives a constrained feedback control that ensure the exponential stabilizability and (ii) shows that this control is the unique solution of an appropriate minimization problem. Examples of hyperbolic partial equations are provided.     

On a class of strongly stabilizable systems of neutral type

We consider the strong stabilizability problem for a delayed system of neutral type. For simplicity the case of one delay in state is studied. We separate a class of such systems and give a constructive solution of the problem in this case, without the derivative of the localized delayed state. Our results are based on an abstract theorem on the strong stabilizability of contractive systems in Hilbert space. An illustrating example is also given.     

An operator theory of linear functional differential equations

An operator theory, based on convolution rings and modules, is developed for various classes of first-order vector functional differential equations of the retarded type with finite and infinite delays. The existence and construction of complete solutions is approached in a novel manner by incorporating initial data into the operator framework. Results are then obtained on exponential and asymptotic stability. The operator framework is also applied to the study of state feedback. Constructive results on spectrum assignability and stabilizability by feedback are given.     

Exponential decay for the fragmentation or cell-division equation

We consider a classical integro-differential equation that arises in various applications as a model for cell-division or fragmentation. In biology, it describes the evolution of the density of cells that grow and divide. We prove the existence of a stable steady distribution (first positive eigenvector) under general assumptions in the variable coefficients case. We also prove the exponential convergence, for large times, of solutions toward such a steady state.     

Boundary Observability and Stabilization for Westervelt Type Wave Equations without Interior Damping

In this paper we show boundary observability and boundary stabilizability by linear feedbacks for a class of nonlinear wave equations including the undamped Westervelt model used in nonlinear acoustics. We prove local existence for undamped generalized Westervelt equations with homogeneous Dirichlet boundary conditions as well as global existence and exponential decay with absorbing type boundary conditions.     

Stabilizability of linear switching systems

A complete characterization of stabilizability for linear switching systems is not available in the literature. In this paper, we show that the asymptotic stabilizability of linear switching systems is equivalent to the existence of a hybrid Lyapunov function for the controlled system, for a suitable control strategy. Further, we prove that asymptotic stabilizability of a switching system with minimum dwell time, is equivalent to Input to State Stability (ISS) of the controlled switching system, with a stabilizing control law. We then derive some structural reductions of the hybrid state space, which allow a decomposition of the original problem into simpler subproblems. The relationships between this approach and the well-known Kalman decomposition of linear dynamic control systems are explored.     

关于开或闭模型的指数稳定性

讨论了系统解的渐进稳定性和指数稳定性,证明了系统在Banach空间中生成正的C_0-半群以及系统算子0本征值的存在性,系统算子的谱点均为于复平面的左半平面且在虚轴上除0外无谱,并通过分析系统本质谱界经过扰动后的变化,进一步表明在一定条件下,系统的动态解以指数形式收敛于系统的稳态解.     

Stabilizability and a separation principle for periodic orbits

In this paper, we derive a result for stabilizability and a separation principle for periodic orbits. First, using degree theory, we derive a necessary condition for local asymptotic stabilizability of periodic orbits. This condition is similar to the famous Brockett's necessary condition (1983) for local asymptotic stabilizability for equilibria. Next, we derive a separation principle for periodic orbits. Our separation principle states that if a state feedback system defined in the neighborhood of a periodic orbit is asymptotically stabilizable and if an exponentially good state estimator is known, then the composite state feedback-state estimator scheme is locally orbitally asymptotically stable.     

Internal Stabilizability of Some Diffusive Models

We consider a single species population dynamics model with age dependence, spatial structure, and a nonlocal birth process arising as a boundary condition. We prove that under a suitable internal feedback control, one can improve the stabilizability results given in Kubo and Langlais [J. Math. Biol.29 (1991), 363-378]. This result is optimal.Our proof relies on an identical stabilizability result of independent interest for the heat equation, that we state and prove in Section 3.     

Quadratic stabilizability of uncertain linear systems containing both constant and time-varying uncertain parameters

This paper is concerned with the problem of stabilizing an uncertain linear system using state feedback control. The uncertain systems under consideration are described by state equations containing unknown but bounded uncertain parameters. The uncertain parameters are classified into two types: either constant or time-varying. Indeed, the main feature of this paper is that it allows one to exploit the fact that some of the uncertain parameters are constant. In order to investigate the question of stabilizability, quadratic Lyapunov functions are used. Hence, the paper deals with the notion of quadratic stabilizability. The main result of the paper is a necessary and sufficient condition for the quadratic stabilizability of the uncertain systems under consideration.     

On the Long-Time Behavior of the Quantum Fokker-Planck Equation

We analyze the long-time behavior of transport equations for a class of dissipative quantum systems with Fokker-planck type diffusion operator, subject to confining potentials of harmonic oscillator type. We establish the existence and uniqueness of a non-equilibrium steady state for the corresponding dynamics. Further, using a (classical) convex Sobolev inequality, we prove an optimal exponential rate of decay towards this state and additionally give precise dispersion estimates in those cases, where no stationary state exists.     

Feedback stabilization of a simplified 1d fluid–particle system

We consider the feedback stabilization of a simplified 1d model for a fluid–structure interaction system. The fluid equation is the viscous Burgers equation whereas the motion of the particle is given by the Newton's laws. We stabilize this system around a stationary state by using feedbacks located at the exterior boundary of the fluid domain. With one input, we obtain a local stabilizability of the system with an exponential decay rate of order σ<_σ0$\sigma <{\sigma }_0$. An arbitrary order for the exponential decay rate can be proved if a unique continuation result holds true or if two inputs are used to stabilize the system. Our method is based on general arguments for stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domains of the stationary state and of the stabilized solution are different.     

Exponential Behaviour of Stochastic 2D Navier-Stokes Equations Driven by Levy Noise

In this paper, some results on the pathwise exponential stability are established for the weak solutions of stochastic 2D Navier-Stokes equation driven by noise. Also, some results and comments concerning the stabilizability and stabilization of these equations are stated.