退化双曲方程的能控性和Ginzburg-Landau方程的不灵敏控制

发布时间：2018-07-16 17:34

【摘要】：本文主要研究了一维退化双曲方程的能控性和Ginzburg-Landau方程的不灵敏控制问题.对退化双曲方程.根据控制所施加的位置不同.我们分别研究了其边界能控性和内部能控性:而对于某些不能做到零能控的退化双曲方程.我们研究其较弱的能控性质.包括区域能控性和延迟区域能控性.Ginzburg-Landau方程可以描述非线性波的许多超导现象并且在振幅方程理论中起到重要作用.我们主要研究非线性Ginzburg-Landau方程不灵敏控制的存在性.本论文的主要内容分为以下四部分.在本文的第2章中,我们致力于研究一类具有齐次Dirichlet边界条件和内部控制的非线性复Ginzburg-Landau方程不灵敏控制的存在性.当方程中的非线性项在无穷远处满足适当的超线性增长条件时.我们证明了相应半线性Ginzburg-Landau方程不灵敏控制的存在性.同时.当方程中非线性项仅是光滑函数不加任何增长条件时,我们得到了不灵敏控制的局部性结果.按通常的方法,我们将不灵敏控制问题转化为一个线性和半线性Ginzburg-Landau 方程耦合而成的方程组在单个控制下的能控性问题,关键是建立线性耦合 Ginzburg-Landau方程组在单个观测下的一个能观不等式.在本文的第3章中.我们致力于研究一维线性退化双曲方程的边界零能控问题.因退化双曲方程仍具有时间可逆性,所以其零能控与精确能控等价.首先,我们讨论了线性退化双曲方程的适定性.然后.给出了当控制施加在非退化边界时某些退化双曲方程的零能控性.不同于已知的控制施加在退化边界的能控性结果,在这种情况下状态空间中的任意初值都是零能控的.同时,我们给出了能控性时间的精确表达式.另外.对某些其他的退化双曲方程我们给出了不零能控的反例.在本文的第4章中,我们致力于研究一维半线性退化双曲方程的内部零能控问题.应用Hilbert唯一性方法,我们需建立线性退化双曲方程的一个能观性估计.由特征线法我们先证明退化双曲方程的唯一延拓性,再由唯一延拓性结合乘子法证明能观不等式.关键在于乘子的构造.在本文的第5章中,我们致力于研究一维线性退化双曲方程施加内部控制时的延迟区域零能控问题.不同于非退化双曲方程,对某些退化双曲方程经典的零能控结果不成立.因此,引入了延迟区域零能控性,它意味着找一个控制使得退化双曲方程的相应状态在空间区域的某个子集里和一段时间内恒为零.为此.我们先建立退化双曲方程的区域零能控性.此问题也可转化为线性退化双曲方程一个适当的能观性问题.关键是构造合适的乘子来证明此能观不等式.
[Abstract]:In this paper, the controllability of one-dimensional degenerate hyperbolic equation and insensitive control of Ginzburg-Landau equation are studied. For degenerate hyperbolic equations. Different positions are applied according to the control. We study the boundary controllability and internal controllability respectively for some degenerate hyperbolic equations which can not be controlled from zero. We study its weaker controllability. The region controllability and delay region controllability. Ginzburg-Landau equation can describe many superconducting phenomena of nonlinear waves and play an important role in amplitude equation theory. We study the existence of insensitive control for nonlinear Ginzburg-Landau equation. The main content of this paper is divided into the following four parts. In chapter 2, we study the existence of insensitive control for a class of nonlinear complex Ginzburg-Landau equations with homogeneous Dirichlet boundary conditions and internal control. When the nonlinear term in the equation satisfies the appropriate superlinear growth condition at infinity. We prove the existence of insensitive control for the corresponding semilinear Ginzburg-Landau equation. meanwhile When the nonlinear term in the equation is only a smooth function without any growth condition, we obtain the local results of insensitive control. According to the usual method, we transform the insensitive control problem into the controllability problem of a linear and semi-linear Ginzburg-Landau equation coupled with a single control system. The key is to establish an observable inequality for linearly coupled Ginzburg-Landau equations under a single observation. In chapter 3 of this paper. We study the boundary zero controllability problem of one dimensional linear degenerate hyperbolic equation. Since the degenerate hyperbolic equation still has time reversibility, its zero controllability is equivalent to exact controllability. First, we discuss the fitness of the linear degenerate hyperbolic equation. And then. The zero controllability of some degenerate hyperbolic equations when the control is applied to the nondegenerate boundary is given. In this case, any initial value in the state space is zero controllable, which is different from the known controllability of the control applied on the degenerate boundary. At the same time, we give the exact expression of controllability time. In addition. For some other degenerate hyperbolic equations, we give a counterexample of nonzero controllability. In chapter 4, we study the internal zero controllability of one dimensional semilinear degenerate hyperbolic equations. By using Hilbert uniqueness method, we need to establish an observability estimate for linear degenerate hyperbolic equations. We first prove the unique continuation of the degenerate hyperbolic equation by the characteristic line method, and then prove the observable inequality by the unique continuation method combined with the multiplier method. The key lies in the construction of multipliers. In the fifth chapter of this paper, we focus on the problem of zero controllability in the delay domain of one dimensional linear degenerate hyperbolic equation with internal control. Different from the nondegenerate hyperbolic equations, the zero controllability results of some degenerate hyperbolic equations do not hold true. Therefore, the zero controllability of the delay region is introduced, which means finding a control such that the corresponding state of the degenerate hyperbolic equation is constant to zero in a subset of the space region and for a period of time. To this end. We first establish the domain zero controllability of degenerate hyperbolic equations. This problem can also be transformed into an appropriate observability problem for linear degenerate hyperbolic equations. The key is to construct appropriate multipliers to prove this observable inequality.
【学位授予单位】：东北师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O231

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