几类非线性波方程的能控性

发布时间：2018-04-23 19:09

本文选题：精确能控性 + 同时能控性　；参考：《山西大学》2017年硕士论文

【摘要】：能控性是分布参数控制理论的一个重要问题。本文研究波方程系统的能控性问题.在第一章中,我们简要地介绍了分布参数控制理论的研究进展和本文的主要内容。在第二章中,我们研究由若干个具有不同传播速度的线性阻尼波解耦形成的方程组的同时精确能控性,证明可以用同一控制将所有波系统同时精确控制(这一性质被称为同时精确能控性).主要方法是,采用经典的Hilbert唯一方法,建立原系统的能控性与其对偶系统的能观测性的等价关系,通过证明能观性不等式,得到原系统的同时精确能控性.在第三章中,我们考虑下列一维非线性波方程yu-yxx+g(y)yt=hlw的内部精确能控性,其中,g(·)是非负连续函数,g(Y)Yt称为退化阻尼项,h是施加在方程内部的控制函数.假设g(·)满足(?)γ(γ为充分小的正数)时,我们应用线性化原理和对线性化阻尼波方程的显式能观性不等式估计,证明在齐次狄利克雷边界条件下退化阻尼波系统是整体精确能控的;假设g(s)=|s|,利用Banach不动点定理,我们得到退化阻尼波系统的局部精确能控性.在第四章中,我们讨论了半线性波方程的两个脱敏控制问题.假设非线性项f满足(?)=(?)=0,证明了纽曼边界观测量对于初始扰动可以实现脱敏控制;假设非线性项f是整体Lipschitz连续性的,我们证明内部观测量对于初始扰动可以实现近似脱敏控制.
[Abstract]:Controllability is an important problem in distributed parameter control theory. In this paper, we study the controllability of wave equation systems. In the first chapter, we briefly introduce the research progress of distributed parameter control theory and the main contents of this paper. In the second chapter, we study the simultaneous exact controllability of several linear damped wave decoupling equations with different propagation velocities. It is proved that all wave systems can be controlled precisely at the same time by the same control (this property is called simultaneous precise controllability). The main method is to establish the equivalent relation between the controllability of the original system and the observability of the dual system by using the classical Hilbert unique method. By proving the observability inequality, the simultaneous exact controllability of the original system is obtained. In chapter 3, we consider the following internal exact controllability of yu-yxx g(y)yt=hlw, where g () is a nonnegative continuous function called degenerate damping term h is a control function applied within the equation. Assuming that g () satisfies (纬) 纬 (纬 is a sufficiently small positive number), we apply the linearization principle and estimate the explicit observability inequality for the linearized damped wave equation, and prove that the degenerate damped wave system is globally accurately controllable under the homogeneous Delikley boundary condition. By using the Banach fixed point theorem, we obtain the locally exact controllability of degenerate damped wave systems. In chapter 4, we discuss two desensitization control problems for semilinear wave equations. Assuming that the nonlinear term f is satisfied, it is proved that the Newman boundary observation can realize desensitization control for the initial disturbance, and assuming that the nonlinear term f is global Lipschitz continuity, we prove that the internal observation can realize the approximate desensitization control for the initial disturbance.
【学位授予单位】：山西大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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