Refined Asymptotics for the Infinite Heat Equation with Homogeneous Dirichlet Boundary Conditions

The nonnegative viscosity solutions to the infinite heat equation with homogeneous Dirichlet boundary conditions are shown to converge as t → ∞ to a uniquely determined limit after a suitable time rescaling. The proof relies on the half-relaxed limits technique as well as interior positivity estimates and boundary estimates. The expansion of the support is also studied.     

二维波动方程部分Dirichlet边界控制正则性的数值分析

通过数值方法,研究边界充分(逐段)光滑区域上的二维波动方程在部分Dirichlet边界控制下的正则性问题.数值结果表明:在所选条件下,系统是Salamon-Weiss意义下正则的.     

梁振动边界反馈的最优反馈增益的数值解

本用Legendre谱方法估计一端固定,一端加弯矩耗散线性反馈的梁振动的闭环系统使能量最快衰减的最优反馈增益,我们给出了数值产生的图形结果,通过比较发现另一种非耗散的线性反馈在最优反馈增益下比相应的耗散线性反馈有更好的衰减率。     

Timoshenko梁的边界反馈一致镇定

该文研究在Timoshenko梁两端施加边界反馈控制的镇定问题.在某些线性边界反馈作用下,通过分析闭环系统算子的谱,并利用频域方法证明了相应的闭环系统的一致稳定性.     

广义Kdv-Burgers方程的边界能稳性

考虑Kdv—Burgers方程在区间[0，1]上的边界反馈控制，在边界控制律u(0，t)＝ux(1，t)-g1(u(1，t))＝uzx(1，t)-g2(u(1，t))＝0下，证明了该方程是L^2整体指数稳定，H^1整体渐近稳定和H^1半整体指数稳定的．     

Feedback Stabilization for a Scalar Conservation Law with PID Boundary Control

This paper deals with a scalar conservation law in 1-D space dimension, and in particular, the focus is on the stability analysis for such an equation. The problem of feedback stabilization under proportional-integral-derivative (PID for short) boundary control is addressed. In the proportional-integral (PI for short) controller case, by spectral analysis, the authors provide a complete characterization of the set of stabilizing feedback parameters, and determine the corresponding time delay stability interval. Moreover, the stability of the equilibrium is discussed by Lyapunov function techniques, and by this approach the exponential stability when a damping term is added to the classical PI controller scheme is proved. Also, based on Pontryagin results on stability for quasipolynomials, it is shown that the closed-loop system subject to PID control is always unstable.     

波方程的边界镇定性的分析(英文)

利用乘子法研究了带有变系数的波方程的反馈镇定问题．文章中变系数ｑ（ｘ）分为两种情况：（Ｈ_１）ｑ和（Ｈ_２）ｑ．同时还引入了敏感反馈系数和波方程能量的指数衰减域．更为重要的是,本文统一和改善了一些以前已有的结果．     

Sine-Gordon方程的动态边界反馈镇定

考虑一端固定,控制输入在另一端的Sine-Gordon方程的动态边界反馈镇定问题.我们给出闭环系统的适定性,并利用乘子方法得到在动态边界反馈控制下闭环系统的多项式衰减率.     

On a Constrained Approximate Controllability Problem for the Heat Equation: Addendum

In this note, we clarify a technical point of Ref. 1, devoted to studying a constrained approximate controllability for the heat equation.     

具有振动边值的扩散方程Dirichlet问题

本文以上（下）连续函数作为扩散方程u_t=1/2Δu+cu 在Ｄ内的Ｄｉｒｉｃｈｌｅｔ问题边值函数，讨论了振动边值的Ｄｉｒｉｃｈｌｅｔ问题，并用概率方法证明解的存在性、唯一性和稳定性，把古典Ｄｉｒｉｃｈｌｅｔ问题边值条件减弱到最一般情形     

Existence and Exponential Stability for a Euler-Bernoulli Beam Equation with Memory and Boundary Output Feedback Control Term

In this paper, we consider the Euler-Bernoulli beam equation with memory and boundary output feedback control term. We prove the existence of solutions using the Galerkin method and then investigate the exponential stability of solutions by using multiplier technique. This work was supported by grant number KRF-2005-202-C00030 from the Korea Research Foundation. This work was supported by National Institute for Mathematical Sciences.     

A New Boundary Element Method for the Biharmonic Equation with Dirichlet Boundary Conditions

We consider a scalar boundary integral formulation for the biharmonic equation based on the Almansi representation. This formulation was derived by the first author in an earlier paper. Our aim here is to prove the ellipticity of the integral operator and hence establish convergence of and error bounds for Galerkin boundary element methods. The theory applies both in two and three dimensions, but only for star-shaped domains. Numerical results in two dimensions confirm our analysis.     

Morphisms of the Heat Equation

In this article we consider a certain class of mappings between Riemannian manifolds of the type M × R^{+ *}, which preserve two objects associated to the heat equation. First we show that maps which pull back local solutions of the heat equation to local solutions of the heat equation, called heat equation morphisms have to be the product of a homothetic submersion and an affine map of R^{+ *}. Then using the above characterisation, we study maps which pull back the heat kernel to the heat kernel. We call such maps heat kernel morphisms. We show in Theorem 3 that, in the case of compact manifolds, a map : M × M × R^{+ *} N × N × R^{+ *} of the form (x,y,t) = ( (x), (y),h(t)), with surjective, is a heat kernel morphism if and only if is a homothetic covering of N()-sheets and constant dilation such that N() = ^m (m = dim M) and h(t) = ²t. In particular, if is bijective then it must be an isometry and h(t) = t. A similar problem was considered from a different point of view in [6].     

Control and Stabilization Properties for a Singular Heat Equation with an Inverse-Square Potential

The goal of this article is to analyze control properties of parabolic equations with a singular potential ? μ/|x|², where μ is a real number. When μ ≤ (N ? 2)²/4, it was proved in [19 Vancostenoble , J. , Zuazua , E. ( 2008 ). Null controllability for the heat equation with singular inverse-square potentials . J. Funct. Anal. 254 : 1864 – 1902 .[Crossref], [Web of Science ®] , [Google Scholar]] that the equation can be controlled to zero with a distributed control which surrounds the singularity. In the present work, using Carleman estimates, we will prove that this assumption is not necessary, and that we can control the equation from any open subset as for the heat equation. Then we will study the case μ > (N ? 2)²/4, and prove that the situation changes completely: indeed, we will consider a sequence of regularized potentials μ/(|x|² + ?²), and prove that we cannot stabilize the corresponding systems uniformly with respect to ? > 0, due to the presence of explosive modes which concentrate around the singularity.     

Lagrange-SQP Techniques for the Control Constrained Optimal Boundary Control for the Burgers Equation

A Lagrange-Newton-SQP method is analyzed for the optimal control of the Burgers equation. Boundary controls are given, which are restricted by pointwise lower and upper bounds. The convergence of the method is proved in appropriate Banach spaces. This proof is based on a second-order sufficient optimality condition and the theory of Newton methods for generalized equations in Banach spaces. For the numerical realization a primal-dual active set strategy is applied. To illustrate the theoretical investigations, numerical examples are included. Moreover, a globalization technique for the SQP method is tested numerically.     

半线性热方程精确内部能控性

主要采用了构造辅助系统u′ △u=0的方法，同时也利用了强单调算子 的满射性。在条件（H)的修改下，得到在有限时间内的精确可控性，研究半线性热方程系统ey/et=△y-f(x,t,y,倒△y) h(x,t)的精确内部能控性。     

带间断扩散系数热传导方程的高精度数值模拟方法研究

针对热传导方程在间断扩散系数处的数值求解困难,作者已经提出了"孪生逼近"自适应方法.本文以一维问题为例,推广热传导方程解的内蕴连续性到高阶,进而研究构造了更高精度,更经济的"孪生逼近"算法,并用数值结果进行了检验.     

Numerical Solution for the Helmholtz Equation with Mixed Boundary Condition

We consider the numerical solution for the Helmholtz equation in R~2 with mixed boundary conditions.The solvability of this mixed boundary value problem is estab- lished by the boundary integral equation method.Based on the Green formula,we express the solution in terms of the boundary data.The key to the numerical real- ization of this method is the computation of weakly singular integrals.Numerical performances show the validity and feasibility of our method.The numerical schemes proposed in this paper have been applied in the realization of probe method for inverse scattering problems.     

Approximate Controllability for the Semilinear Heat Equation Involving Gradient Terms

This paper deals with the approximate controllability of the semilinear heat equation, when the nonlinear term depends on both the state y and its spatial gradient y and the control acts on any nonempty open subset of the domain. Our proof relies on the fact that the nonlinearity is globally Lipschitz with respect to (y, y). The approximate controllability is viewed as the limit of a sequence of optimal control problems. Another key ingredient is a unique continuation property proved by Fabre (Ref. 1) in the context of linear heat equations. Finally, we prove that approximate controllability can be obtained simultaneously with exact controllability over finite-dimensional subspaces.     

带耗散项的浅水波方程的整体边界稳定性

在[0,1]区间上研究带耗散项的浅水波方程由边界反馈引起的整体指数稳定性.在控制边界条件u(0)=u_x(1)=u_(xx)(0)=δu_(xxx)(1)-εu(1)u_(xx)(1)-(?)(u(1))=0之下,证明了解的H1整体指数稳定性和H4整体渐进稳定性.