一类高维区域上反应扩散方程的边界控制

发布时间：2018-10-30 08:29

【摘要】：当系统所在区间不规则时,将变量简化为一维的方法便不再适用.因此,对于此情况需要建立一个新的模型以解决其边界控制问题.本文主要研究了曲边矩形和曲顶柱体上反应扩散方程的边界控制.首先本文给出了二维曲边矩形上的反应扩散方程的边界控制,随后将此结论推广至三维,以达到本文的目的.在方法上,本文通过应用Krstic等人提出的偏微分方程反步控制法,使得给定的系统转换至指数稳定的目标系统,从而达到控制目标.随后,利用目标系统的稳定性和Volterra积分变换及其逆变换的有界性,证明闭环系统的指数稳定.本文的难点在于闭环系统的稳定性的证明.通过利用预备知识中提及的公式,证明了闭环系统是指数稳定的.最后,本文利用MATLAB软件仿真出相关结果.仿真结果表明理论推导与实际一致.
[Abstract]:When the system is in an irregular interval, the method of simplifying variables to one dimension is no longer applicable. Therefore, a new model is needed to solve the boundary control problem. In this paper, the boundary control of the reaction diffusion equation on curved rectangular and curved cylinder is studied. In this paper, the boundary control of the reaction-diffusion equation on a two-dimensional curved rectangle is given, and then the conclusion is extended to three dimensions to achieve the purpose of this paper. In this paper, by using the partial differential equation backstepping control method proposed by Krstic et al., the given system is converted to an exponential stable target system, and the control goal is achieved. Then, by using the stability of the target system and the boundedness of the Volterra integral transformation and its inverse transformation, the exponential stability of the closed-loop system is proved. The difficulty of this paper lies in the proof of the stability of the closed loop system. By using the formula mentioned in the preparatory knowledge, it is proved that the closed-loop system is exponentially stable. Finally, the related results are simulated by MATLAB software. The simulation results show that the theoretical derivation is consistent with the practice.
【学位授予单位】：西南大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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