Kuramoto-Sivashinsky方程的能控性

发布时间：2018-06-02 17:53

本文选题：Kuramoto-Sivashinsky方程 + 随机Kuramoto-Sivashinsky方程　；参考：《吉林大学》2015年博士论文

【摘要】：Kuramoto-Sivashinsky (KS)方程是Kuramoto等在研究反应扩散系统相湍流和Sivashinsky在研究火焰燃烧传播模型中独立提出的,它反映了气体的扩散和热传导对于平面火焰前缘稳定性的综合影响,同时这个非线性偏微分方程描述了在各种物理和化学系统中早期的不稳定性.这个模型也来源于粘性液体倒在倾斜的平面受外加电场作用的建模中.KS方程已经成为无穷维动力系统的一个典型例子.对比确定的KS方程,随机KS方程也吸引了许多学者的关注.随机KS方程源于溅射侵蚀面动态的粗化研究中,原则上,任何受时间噪声影响的可以用确定KS方程描述的物理模型都可以用随机KS方程描述.目前,控制理论是一个研究较多的科学领域.控制理论出现在大部分现代应用中,同时也被称为工业革命的重大发现.另一方面,由于众多数学思想和数学方法的融合,控制理论已经成为一个新的重要的数学分支.近四十年来,偏微分方程的控制理论迅速发展,并由于实际问题的驱动,今天已经吸引了许多学者的关注.但据我们所知,关于KS方程能控性的结果相对较少.本文致力于研究KS方程的能控性.在第一部分中,我们研究了具有内部控制的KS方程的轨线精确能控性.我们首先对四阶抛物算子建立了一个单参数形式的新的整体Carleman估计.这个整体Carleman估计与已有的整体Carleman估计的主要区别是估计中的权函数只依赖于一个参数.为了得到这个估计,我们重新构造了一个权函数.然后,我们研究KS方程线性化系统的零能控性问题.应用已得到的整体Carleman估计和能量估计,我们获得了KS方程线性化系统对偶系统的能观性不等式,基于此能观性不等式,利用经典的对偶讨论,我们建立了KS方程线性化系统的零能控性.结合线性化系统的零能控性结果和Kakutani不动点定理可以得到KS方程的局部轨线能控性.在第二部分中,我们研究了正倒向线性随机KS方程的零能控性.首先,我们利用Galerkin方法建立了正倒向线性随机KS方程的适定性.然后我们建立了正倒向随机四阶抛物方程的整体Carleman估计.与确定情形本质的不同是随机情形建立这个估计有噪声项的影响,所以确定情形建立整体Carleman估计的方法是不可以运用到随机情形的.为此我们采用了一种新的方法,这种方法已经被应用到建立正倒向线性随机热方程的整体Carleman估计中.由这个整体Carleman估计和倒正向线性随机KS方程的能量估计,我们可以建立倒正向线性随机KS方程的能观性不等式.由此能观性不等式和Hahn-Banach定理,可以建立正倒向线性随机KS方程的零能控性.
[Abstract]:Kuramoto-Sivashinsky KS) equation is proposed by Kuramoto et al in the study of phase turbulence in reaction diffusion system and Sivashinsky in the study of flame combustion propagation model. It reflects the comprehensive influence of gas diffusion and heat conduction on the stability of plane flame front. At the same time, this nonlinear partial differential equation describes the early instability in various physical and chemical systems. This model is also derived from the modeling of viscous fluid inverted on an inclined plane subjected to an applied electric field. KS equation has become a typical example of infinite dimensional dynamic system. Compared with the determined KS equation, the stochastic KS equation has attracted the attention of many scholars. The stochastic KS equation originates from the coarsening study of sputtering erosion surface dynamics. In principle, any physical model which can be described by deterministic KS equation can be described by stochastic KS equation. At present, control theory is a scientific research field. Control theory appears in most modern applications and is also known as an important discovery of the Industrial Revolution. On the other hand, control theory has become a new important branch of mathematics because of the fusion of many mathematical ideas and methods. In the past 40 years, the control theory of partial differential equations has developed rapidly, and has attracted the attention of many scholars because of the driving of practical problems. But as far as we know, there are few results about controllability of KS equation. The controllability of KS equations is studied in this paper. In the first part, we study the orbit exact controllability of KS equation with internal control. We first establish a new global Carleman estimator in the form of a single parameter for a fourth order parabolic operator. The main difference between the global Carleman estimator and the existing global Carleman estimator is that the weight function in the estimator is dependent on only one parameter. In order to obtain this estimate, we reconstruct a weight function. Then, we study the zero controllability of the linearized system of KS equation. By using the global Carleman estimate and energy estimate, we obtain the observability inequality for the dual system of the linearized system of KS equation. Based on this observability inequality, we use the classical duality discussion. We establish the zero controllability of the linearized system of KS equation. The local orbit controllability of KS equation can be obtained by combining the zero controllability result of linearized system and Kakutani fixed point theorem. In the second part, we study the zero controllability of forward linear stochastic KS equations. First of all, we use the Galerkin method to establish the fitness of forward linear stochastic KS equation. Then we establish the global Carleman estimate for the forward backward stochastic fourth order parabolic equation. The essential difference from the deterministic case is that the establishment of this estimate in a random case has the effect of noise terms, so the method of establishing a global Carleman estimate in a given case can not be applied to a random case. For this purpose, we adopt a new method, which has been applied to the global Carleman estimation of forward linear stochastic heat equations. Based on the global Carleman estimate and the energy estimation of the backward linear stochastic KS equation, we can establish an observability inequality for the backward linear stochastic KS equation. Based on the observability inequality and Hahn-Banach theorem, the zero controllability of forward linear stochastic KS equations can be established.
【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O175

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