半线性热方程的反馈零能控性

考虑具有Lipschitz非线性项,半线性热方程的最优控制问题.我们将运用观测不等式,证明值函数ψ作为相应Hamilton-Jacobi方程的唯一粘性正解是局部Lipschitz连续的.最后,运用动态规划方法,得到系统最优的反馈控制.     

由退化抛物方程支配串联系统的零能控性

该文讨论了由一个半线性退化抛物方程与半线性热方程构成的串联系统的零能控性. 这里控制函数仅施加在一个方程上. 证明的关键是建立适当的能观性不等式.     

含有一个控制的拟线性抛物系统的能控性

该文讨论了一类拟线性抛物系统的全局零能控性和逼近能控性. 文中首先给出并证明了相应的线性系统的Carleman不等式,再由Carleman不等式去证明观测不等式, 最后利用Kakutani不动点定理证明非线性系统的全局零能控性和逼近能控性.     

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。     

带扩散项单种群模型的精确能控性

本文研究了一类单种群生态模型的精确能控性问题.利用线性化系统的Carleman估计和Kakutani不动点定理,证明了该模型稳态解的局部能控性.它反应了外部控制在种群演变过程中的影响作用.     

一类带奇异项的退化Grushin方程的零控问题[英文]

本文研究一类带奇异项的退化Grushin方程的零控问题.不同于Fourier分解的方法,通过选择一类特殊的势函数,我们证明了退化Grushin算子的Carleman估计.然后利用此Carleman估计,给出带奇异项的退化Grushin方程的观测不等式,获得该方程的零控结果.     

带非局部低阶项非线性抛物型方程的时间最优控制

考虑一个带非局部低阶项非线性抛物型方程的时间最优控制问题．首先利用Schauder不动点定理证明了系统的适定性,然后利用Carleman不等式和Kakutani不动点定理证明了容许控制和最优控制的存在性,并且建立了时间最优控制的最大值原理．     

一类边界控制半线性抛物型系统的逼近能控性

本文考虑一类Ｄｉｒｉｃｈｌｅｔ型边界控制半线性抛物型系统，应用半群理论和非线性分析的方法，证明了系统解映射的连续性，给出了系统逼近能控的一个充分条件．对于Ｎｅｕｍａｎｎ型边界控制或混合型边界控制的半线性抛物型系统，可以得到类似的结果．     

一类反应扩散方程组的局部能控性

本文证明了一类反应扩散方程组平衡解的局部精确能控性.     

拟线性双曲型方程(组)的精确能控性

本文为作者在中国科学院数学与系统科学研究院举办的第六届华罗庚数学讲座上的讲稿.§1　引言——从常微分方程谈起考虑如下的线性常微分方程组dXdt=AX+Bu,(1.1)其中,t为自变量(时间),X=(X1,…,XN)为状态变量,u=(u1,…,um)为控制变量,而A及B分别为N×N及N×m常数阵.(1.1)是一个有限维的动力系统.说该系统在时间区间[0,T](T>0)上具有精确能控性,是指对于在t=0时任意给定的初值X0及在t=T时任意给定的终值XT,一定能找到[0,T]上的控制函数u=u(t),使Cauchy问题dXdt=AX+Bu(t),(1.2)t=0:X=X0(1.3)的解X=X(t)精确地满足终端条件t=T:X=X…     

Null Controllability in Unbounded Domains for the Semilinear Heat Equation with Nonlinearities Involving Gradient Terms

We consider the null controllability problem for the semilinear heat equation with nonlinearities involving gradient terms in an unbounded domain of ^N with Dirichlet boundary conditions. The control is assumed to be distributed along a subdomain such that the uncontrolled region \ is bounded. Using Carleman inequalities, we prove first the null controllability of the linearized equation. Then, by a fixed-point method, we obtain the main result for the semilinear case. This result asserts that, when the nonlinearity is ^C1 and globally Lipschitz, the system is null controllable.     

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.     

Exact Local Controllability of a One-Control Reaction-Diffusion System

This work concerns the controllability of some parabolic systems with the control acting in one equation. The exact local controllability of a reaction-diffusion system with one control and the null controllability of a class of linear parabolic systems with one control are obtained. As an application, the existence of a time-optimal control for a reaction-diffusion system with one control is proved.Communicated by D. A. CarlsonThis work was supported by the New Century Excellent Teachers Plan of the Ministry of National Education of China, by the Key Laboratory- Optimal Control and Discrete Mathematics of Hubei Province, and by the National Science Foundation of China.     

极大单调算子的扰动定理及在微分方程上的应用

利用不动点定理,首先证明了在自反Banach空间中极大单调算子扰动的抽象结论,这些结论是对以往一些工作的推广;然后,利用文中的新结论讨论了一类微分方程解的存在性.     

A Family of Enlargements of Maximal Monotone Operators

We introduce a family of enlargements of maximal monotone operators. The Brønsted and Rockafellar -subdifferential operator can be regarded as an enlargement of the subdifferential. The family of enlargements introduced in this paper generalizes the Brønsted and Rockafellar -subdifferential (enlargement) and also generalize the enlargement of an arbitrary maximal monotone operator recently proposed by Burachik, Iusem and Svaiter. We characterize the biggest and the smallest enlargement belonging to this family and discuss some general properties of its members. A subfamily is also studied, namely the subfamily of those enlargements which are also additive. Members of this subfamily are formally closer to the -subdifferential. Existence of maximal elements is proved. In the case of the subdifferential, we prove that the -subdifferential is maximal in this subfamily.     

Banach空间中有限个极大单调算子公共零点的迭代格式

令E为实光滑、一致凸Banach空间,E~*为其对偶空间．令A_i,B_i (?) E×E~*,i= 1,2,…,m,为极大单调算子且(?)(A_i~(-1)0∩B_i~(-1)0)≠φ．引入新的迭代算法,并利用Lyapunov泛函,Q_r算子与广义投影算子等技巧,证明迭代序列弱收敛于极大单调算子A_i,B_i,i= 1,2,…,m的公共零点的结论．     

Exact Controllability of a Semilinear Thermoelastic System with Control Solely in Thermal Equation

A result concerning the exact controllability of a semilinear thermoelastic system, in which the control term occurs solely in the thermal equation, is derived under the influence of rotational inertia and Lipschitz nonlinearity, subject to the clamped/Dirichlet boundary conditions. In the proof, we make use of the result given by Avalos (Differential and Integral Equations, 2000; 13(4–6):613–630), which states that the corresponding linear system is exact controllable.     

Null controllability of a class of Newtonian filtration equations

In this paper, we study the null controllability of a class of Newtonian filtration equations. Using the properties of finite extinction time and finite speed of propagation, we construct a control function in feedback form, such that system is exactly null controllable at any time T>0. If we put a nonnegative constraint on the control function, this system is not exactly null controllable any more, but approximately null controllable for a long time.