一类反应扩散方程组的零能控性

发布时间：2018-06-24 13:47

本文选题：反应-扩散方程组 + 零能控性　；参考：《哈尔滨工业大学》2015年硕士论文

【摘要】：自然界中的很多现象都可以用反应扩散方程组来描述,这些方程组的能控性研究也显得越来越重要,本文研究了一类含梯度项的非线性反应扩散方程组在内部一个任意小的区域上施加控制函数时的零能控性。本文首先考虑该非线性方程组所对应的线性方程组,给出了这类线性方程组解的存在性、唯一性和正则性。在此基础上构造相应的泛函,利用该泛函的极小元来给出控制函数,而泛函极小元的存在性依赖于该线性方程组的对偶方程组的能观性估计。在证明能观不等式时,由于所考虑的方程组中含有梯度项,经典的Carleman不等式不再适用,本文对此不等式进行改进。最终,证明了该线性系统在内部一个任意小的区域上施加两个控制函数时是零能控的。进一步,减少控制函数的个数是控制理论的一个重要研究部分。可以发现,控制函数个数的减少可以通过改进Carleman不等式来实现,为此本文对该线性方程组施加一些限制条件,在方程的Carleman不等式的基础上,给出了方程组的Carleman不等式,从而证明在内部一个任意小的区域上只施加一个控制函数时该系统也是零能控的。另一方面,“虚拟函数法”也是一种减少控制函数个数的有效方法,本文在施加两个控制时方程组的零能控性结论的基础上,利用“虚拟函数法”,亦证明了当该方程组满足一定条件时,仅施加单一控制函数,该线性方程组也是零能控的。最后,将原来的非线性方程组线性化,结合Schauder不动点定理,证明了上述有关线性方程组的零能控性结论对于相应的非线性方程组也是成立的,从而证明了本文的主要结论。
[Abstract]:Many phenomena in nature can be described by reaction-diffusion equations, and the controllability of these equations is becoming more and more important. In this paper, we study the zero controllability of a class of nonlinear reaction-diffusion equations with gradient terms when a control function is applied to an arbitrary small region within the system. In this paper, we first consider the system of linear equations corresponding to the system of nonlinear equations, and give the existence, uniqueness and regularity of the solutions of the system of linear equations. On this basis, the corresponding functional is constructed and the control function is given by using the minimal element of the functional. The existence of the minimal element of the functional depends on the estimation of the observability of the dual equations of the system of linear equations. In order to prove the observable inequality, the classical Carleman inequality is no longer applicable because there are gradient terms in the system of equations under consideration, so the inequality is improved in this paper. Finally, it is proved that the linear system is zero controllable when two control functions are applied on an arbitrary small region within the system. Further, reducing the number of control functions is an important part of control theory. It can be found that the reduction of the number of control functions can be achieved by the improved Carleman inequality. In this paper, some restrictions are imposed on the system of linear equations, and on the basis of the Carleman inequality of the equation, the Carleman inequality of the equations is given. It is proved that the system is also zero controllable when only one control function is applied to an arbitrary small region. On the other hand, "virtual function method" is also an effective method to reduce the number of control functions. It is also proved that when the system of equations satisfies certain conditions, only a single control function is applied, and the system of linear equations is also zero controllable. Finally, by linearizing the original nonlinear equations and combining with Schauder fixed point theorem, it is proved that the zero controllability of the system of linear equations mentioned above is also true for the corresponding nonlinear equations, and the main conclusions in this paper are proved.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175

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