Exact Controllability of Semilinear Evolution Systems and Its Application

In this paper, we obtain several abstract results concerning the exact controllability of semilinear evolution systems. First, we prove the null local exact controllability of semilinear first-order systems by means of the contraction mapping principle; in this case, we do not assume any compactness. Next, we derive the global and/or local exact controllability of semilinear second-order systems by means of the Schauder fixed-point theorem; in this case, we assume only the embedding of the related spaces having some compactness, which is reasonable for many concrete problems. Our main result shows that the observability of the dual of the linearized system implies the exact controllability of the original semilinear system. Finally, we apply our abstract results to the exact controllability of the semilinear wave equation.     

非局部半线性积微分系统的可控制性

本文运用半群理论和Schauder不动点定理,在Banach空间研究了一类非局部半线性积微分系统的可控制性.最后,举例说明了所得结果.     

Controllability Properties of Linear Mean-Field Stochastic Systems

We study some controllability properties for linear stochastic systems of mean-field type. First, we give necessary and sufficient criteria for exact terminal-controllability. Second, we characterize the approximate and approximate null-controllability via duality techniques. Using Riccati equations associated to linear quadratic problems in the control of mean-field systems, we provide a (conditional) viability criterion for approximate null-controllability. In the classical diffusion framework, approximate and approximate null-controllability are equivalent. This is no longer the case for mean-field systems. We provide sufficient (algebraic) invariance conditions implying approximate null-controllability. We also present a general class of systems for which our criterion is equivalent to approximate null-controllability property. We also introduce some rank conditions under which approximate and approximate null-controllability are equivalent. Several examples and counter-examples as well as a partial algorithm are provided.     

线性脉冲控制系统的能控性与最优控制

首先讨论了一类线性随机脉冲控制系统的精确能控性质,给出了该类控制系统的脉冲精确能控的等价的代数判据.然后提出了一个确定性的二维线性脉冲控制系统的时间-脉冲强度最优控制问题;利用动态规划原理,给出了脉冲最优控制的反馈形式和值函数的显式表达式;说明了值函数在整个平面上是连续的,在左右两个半平面的内部还是连续可微的.     

Controllability of semilinear systems

The most general results known about controllability of semilinear systems are found in Refs. 1 and 2. The authors of these papers used different approaches, and no connection between them has ever been pointed out. The purpose of this paper is to present a general theorem under which these results easily follow.This paper is taken from a chapter of the author's doctoral thesis written at University of California, Los Angeles, California.     

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。     

Controllability of Impulsive Evolution Inclusions with Nonlocal Conditions

In this paper, we examine controllability problems of evolution inclusions with nonlocal conditions. Using the set-valued and single-valued Mönch fixed-point theorem, we establish some sufficient conditions for the controllability under convex and nonconvex orientor fields respectively. Our approach is different from all previous approaches; we do not assume that the evolution system generates a compact semigroup; so, our method is applicable to a wide class of (impulsive) control systems and evolution inclusions in Banach spaces.     

Controllability of Linear Systems in Non Reflexive Banch Spaces

§ 1.Introduction　 Let Xand Ube Banch spaces,S(t) a C0 semi-group in X,A the infinitesimal gen-erator of S(t) ,and B∈L(U,X) .Consider the systemΣP:x(t) =Ax(t) + Bu(t) ,　 t≥ 0 ,(1 .1 )where u∈ Lploc([0 ,∞ ) ,U) ,1≤p<∞ .　 The problem of controllability of linear systems in Banch spaces has been studied bymany authors.In [1 ] ,the weak and strong reachability and controllability of infinite di-mensional linear systems were investigated.In[2 ,3 ,4 ] ,the controllability of infi…     

On the Existence of Positive Solutions for a Class of Semilinear Elliptic Systems

We study the existence of positive solutions for a class of semilinear ellipcic systems in general domains via the blow up argument and degree theory. The main idea can be used to establish the existence of positive solutions for the Navier problems of polyharmonic semilinear equations in general domains.     

关于一类一阶脉冲微分系统的初值问题

本文的目的是利用单调迭代方法研究一类一阶脉冲微分系统的初值问题的最小最大拟解的存在性及其迭代逼近程序。     

Generalized Quasilinearization Method for a Class of Semilinear Elliptic Systems

In this paper, the method of generalized quasilinearization is extended to a class of semilinear elliptic systems, and the sequences which are the solutions of linear differential equations that converge to the unique solution of the given semilinear elliptic system are obtained.     

一般时间尺度上时滞脉冲系统的双测度稳定性

讨论了一般时间尺度上时滞脉冲系统的双测度稳定性问题.我们引进了一个新的概念即一般时间尺度上时滞脉冲系统的(h_0,h)稳定性.利用Lyapunov函数法和分析法得到了一般时间尺度上时滞脉冲系统解的双测度稳定性判据.最后给出了一个例子以说明本文结论的有效性.     

Controllability of Semilinear Evolution Equations with Nonlocal Conditions

In this paper we examine the controllability problems of certain evolution equations with nonlocal conditions. Using the Schaefer fixed-point theorem, we obtain sufficient conditions for controllability and we give an application.     

奇异系统的某些性质及其与奇异脉冲系统的稳定性等价

本文研究了与特征值相关的奇异系统解的构造及其性态，并讨论了奇异脉冲系统与奇异系统的稳定性等价。     

Banach空间中非线性脉冲Volterra型积分方程组的可解性

在较弱的条件下，利用锥理论和单调迭代方法首先建立了Banach空间中一类非线性算子方程组最小最大解的存在性定理；然后作为应用，利用一个新的比较结果，得到了Banach空间中非线性脉冲Volterra型积分方程组的整体解，改进了最近的许多结果．     

线性控制系统对小时滞的鲁棒能控性

对于在Banach空间中线性控制系统对时滞的鲁棒能控性．给出一个充分条件,在这个条件下,证明对一个能控的线性系统施加某个小的时滞扰动之后,其能控性保持不变．     

Controllability analysis of linear time-varying systems with multiple time delays and impulsive effects

In this paper, the issue of controllability for linear time-varying systems with multiple time delays in the control and impulsive effects is addressed. The solution of such systems based on the variation of parameters is derived. Several sufficient and necessary algebraic conditions for two kinds of controllability, i.e., controllability to the origin and controllability, are derived. The relation among these conditions are established. A numerical example is provided to illustrate the effectiveness of the proposed methods.     

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.     

Controllability of nonlinear delay systems

A necessary and sufficient condition for complete controllability of differential systems with delay is proved, which extends analogous results for usual differential systems.     

On the Geometry of Inner Approximations to the Set of Null Controllability for Constrained Linear Systems

The problem of obtaining inner approximations to the set of null controllability for linear systems is considered, and the geometry of certain sets arising in the approximation of using linear feedback is explored. A two-dimensional example is worked in detail in order to delineate the limitations to the use of linear feedback.