On a Quasilinear Degenerate Parabolic Equation from Prandtl Boundary Layer Theory

The equation arising from Prandtl boundary layer theory $$\frac{\partial u}{\partial t} -\frac{\partial }{\partial x_i}\left( a(u,x,t)\frac{\partial u}{\partial x_i}\right)-f_i(x)D_iu+c(x,t)u=g(x,t)$$ is considered. The existence of the entropy solution can be proved by BV estimate method. The interesting problem is that, since $a(\cdot,x,t)$ may be degenerate on the boundary, the usual boundary value condition may be overdetermined. Accordingly, only dependent on a partial boundary value condition, the stability of solutions can be expected. This expectation is turned to reality by Kružkov's bi-variables method, a reasonable partial boundary value condition matching up with the equation is found first time. Moreover, if $a_{x_i}(\cdot,x,t)\mid_{x\in \partial \Omega}=a(\cdot,x,t)\mid_{x\in \partial \Omega}=0$ and $f_i(x)\mid_{x\in \partial \Omega}=0$, the stability can be proved even without any boundary value condition.     

抛物型非线性边界控制系统解映射的正则性

该文利用半群理论等方法讨论了抛物型非线性边界控制系统解映射的连续性、可微性     

Robust static output feedback variable structure control for nonlinear systems

In this paper, a robust stabilization problem for nonlinearsystems is investigated using static output feedback. Matchedand mismatched uncertainties are considered and their boundingfunctions take general forms. Based on a Lyapunov analysis approachand sliding mode techniques, some conclusions are presented.Then, a robust nonlinear variable structure control scheme issynthesized to stabilize the system globally, and it is notnecessary for the nominal system to be linearizable globally.The conservatism is reduced by using uncertainty bounds in thecontrol design. Finally, a mass–spring system is employedto illustrate the effectiveness of the results.     

一类具有饱和因子的奇异时滞系统的鲁棒保性能控制

研究了具有饱和因子的非线性奇异时滞系统的鲁棒保性能控制问题.目的是设计一个鲁棒控制器和保成本控制器,通过线性矩阵不等式方法(LMI)得出了鲁棒控制器和保性能控制器存在的充分条件.当这些LMI方法是可解时,分别给出了鲁棒控制器和保性能控制器的解析表达式.     

Robust Stabilization of Linear Systems with Delayed State and Control

Robust stabilization of linear systems with delays on both the state and control input is studied in this paper. Using an improved Lyapunov-Krasovskii functional, we establish new criteria that ensure the robust stability of the closed-loop system with memoryless state feedback controls. The generalized conditions are derived in terms of linear matrix inequalities (LMIs), allowing us to compute simultaneously the two bounds that characterize the exponential stability rate of the solution and can be easily solved by numerical algorithms. This work was supported by the National Basic Program in Natural Sciences. The authors thank the anonymous referees for valuable comments to improve the paper.     

Energy Decay of Solutions to a Nondegenerate Wave Equation with a Fractional Boundary Control

In this paper, we study the energy decay rate for a one-dimensional nondegenerate wave equation under a fractional control applied at the boundary. We proved the polynomial decay result with an estimation of the decay rates. Our result is established using the frequency-domain method and Borichev-Tomilov theorem.     

Robust stabilization of linear systems with time-varying point delays via a delay-free dynamic controller

^** Email: wepdepam{at}lg.ehu.es This paper deals with the problem of robust closed-loop stabilizationagainst parametrical uncertainties of linear systems subjectto internal (i.e. in the state) and external (i.e. in the output),possibly time-varying and unbounded point delays of a boundedtime-derivative. The output-feedback linear stabilizing controlleris delay free and dynamic. It is assumed that the undelayedplant (i.e. the delay-free part of the plant) is stabilizableand detectable. The synthesis process of the stabilizing controllerinvolves three major actions. First, an augmented system isbuilt with the dynamic equations of both plant and controller.At this step, the controller structure is available but a particularstabilizing controller parametrization still remains undetermined.Subsequently, a Lyapunov matrix equation is ensured to be solvablefor the augmented closed-loop delay-free system so that sucha system is stable with a large stability abscissa related tothe amounts of uncertainties and delay contributions to thedynamics. At this stage, one takes the advantage that the augmentedsystem may be stabilized by an appropriate dynamic controllerof minimum order since the undelayed plant is stabilizable anddetectable. Finally, a complementary matrix equality is manipulatedto establish the closed-loop stability tolerance of the augmenteddelay system, related to that of the delay-free one, to thedelayed dynamics and parametrical uncertainties.     

On Vector Helmholtz Equation with a Coupling Boundary Condition

The Helmholtz equation is sometimes supplemented by conditions that include the specification of the boundary value of the divergence of the unknown.In this paper, we study the vector Helmholtz problem in domains of both C~(1,1)and Lipschitz.We es- tablish a rigorous variational analysis such as equivalence,existence and uniqueness. And we propose finite element approximations based on the uncoupled solutions.Fi- nally we present a convergence analysis and error estimates.     

Sufficient Conditions for Robust Stability of LQG Optimal Control Systems Including Delayed Perturbations

Linear-quadratic Gaussian (LQG) optimal control systems subject to time-varying delay and nonlinear state perturbations are considered. Some robust stability conditions are derived which result in several bounds on the delayed state perturbations so that the uncertain linear-quadratic Gaussian optimal control systems with time-varying delay can remain stable in the sense of uniform ultimate boundedness. The modified Lyapunov equation and the improved Razumikhin-type theorem are employed to investigate such robust stability conditions. Finally, a numerical example is given to demonstrate the validity of the results.     

具有转向点的一类奇摄动边值问题解的多重边界层现象

本文讨论如下边值问题：x=0是转向点（c（0）=0），而在x=－1处出现多重边界层现象．对不同层次采用不同的伸长变量，构造具有不同量级的边界层校正项，得到关于解的一致有效的渐近展开式和有关的余项估计．     

采用可变边界层法削弱一类非线性变结构系统的抖振

针对一类非线性系统,提出一种削弱变结构控制系统抖振的新方法-可变边界层法,推导出变结构控制系统的稳态误差与饱和特性宽度之间定量的数学关系,通过系统的稳态误差指标可以设计出边界层宽度变化的饱和特性,从而既削弱了系统的抖振又满足系统稳态误差指标,仿真结果验证了该方法的有效性.     

一类由抛物-双曲耦合组支配的边界最优控制问题

本文讨论了一类由抛物-双曲耦合组支配的第三边值最优控制问题.证明了该问题最优控制的存在性及其bang-bang性质.     

On Two-Point Boundary Conditions in Optimal Control Problems

Let x=g(t,x(t),u(t)) be the governing equation of an optimal control problem with two-point boundary conditions h ₀(x(a))+h ₁(x(b)) = 0, where x: [a,b] ⁿ is continuous, u: [a,b] ^k-n is piecewise continuous and left continuous, h₀,h₁: ^{n q} are continuously differentiable, and g:[a,b]× ^{k n} is continuous. The paper finds functions _i C¹([a,b]× ⁿ ) such that (x(t),u(t)) is a solution of the governing equation if and only if

多时滞非线性中立型反馈系统的鲁棒稳定性

The stabilization of a class of neutral systems with multiple time-delays is considered. To stabilize the neutral system with nonlinear uncertainty, a state feedback control law via compound memory and memoryless feedback is derived. by constructed Lyapunov functional, delay-independent stability criteria are proposed that are sufficient to ensure a uniform asymptotic stability property. Finally, two concise examples are provided to illustrate the feasibility of our results.     

Optimization Techniques for Solving Elliptic Control Problems with Control and State Constraints: Part 1. Boundary Control

We study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints. In a first part we consider boundary control problems with either Dirichlet or Neumann conditions. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. It is shown that a recently developed interior point method is able to solve these problems even for high discretizations. Several numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang-bang and singular controls. The necessary conditions of optimality are checked numerically in the presence of active control and state constraints.     

Boundary Control of Physical Fields in a Piezoceramic Layer Having a Tunnel Hole with a System of Electrodes

An approach to the optimal (from the viewpoint of minimum energy consumption) control of the characteristics of coupled fields is illustrated with the example of an antiplane dynamic problem of electroelasticity for a piezoceramic layer with a tunnel hole. The vibrations in the layer are excited by time-dependent harmonic loads of two types: by the difference of electric potentials applied to a pair of infinitely long electrodes located on the surface of the hole and by shear forces or electric charges continuously distributed on the given sections of the layer base, which are assumed as control actions. The inverse problem is solved by using the solution of the corresponding direct mixed antiplane problem of electroelasticity, with the help of which the optimization problem is reduced to the -problem of moments. The functions of optimal control allowing one, for example, to disconnect the external circuit of the generator of electric voltage are given.     

具有反馈控制的单种群对数模型稳定性

研究具有反馈控制的单种群对数模型．通过构造适当的Lyapunov函数．我们让得系统的正平衡点是无条件全局稳定的．所得结果补充和完善了已有的结果．     

Exact Boundary Synchronization for a Coupled System of Wave Equations with Neumann Boundary Controls

In this paper, for a coupled system of wave equations with Neumann boundary controls, the exact boundary synchronization is taken into consideration. Results are then extended to the case of synchronization by groups. Moreover, the determination of the state of synchronization by groups is discussed with details for the synchronization and for the synchronization by 3-groups, respectively.     

Robust Tracking and Model Following Control with Zero Tracking Error for Uncertain Dynamical Systems

The problem of robust tracking and model following for a class of linear dynamical systems with time-varying uncertain parameters and disturbances is considered. A class of continuous (nonlinear) state feedback controllers is proposed for robust tracking of dynamical signals. The proposed robust tracking controllers can guarantee that the tracking error decreases asymptotically to zero in the presence of uncertain parameters and disturbances. A procedure for designing such a zero tracking state feedback controller is also introduced. Finally, a numerical example is given to demonstrate the validity of the results.