一类反应扩散方程的边界控制

【作者】 司元超；

【导师】 谢成康；

【作者基本信息】 西南大学 ， 运筹学与控制论， 2015， 硕士

【摘要】 考虑了一类参数与时间和空间相关的反应扩散系统的正定问题.该系统既包含了反应、对流、扩散等项,又含有关于空间的积分项,其中纽曼边界条件是根据热传导中的傅里叶定律所得.该系统可以表征燃烧过程等化学变化,其中积分项表明该系统关于空问具有记忆性.根据偏微分方程的反步控制法,通过选择核函数,建立一个可逆的Volterra型积分变换,将原系统变换到一个预先选择的稳定的目标系统.根据所需满足的目标系统的边界条件,得到设计所要的控制输入.所选的核方程难以求出显示解.为此,需要证明核方程解的存在性.利用偏微分方程特征线,将核方程转换为一个积分方程,再利用逐次逼近法和数学归纳法,证明了核方程在假设条件下解的存在性.根据Volterra型积分变换及其逆变换的有界性,利用Lyapunov函数,证明了闭环系统是指数稳定的.利用数值分析等知识,论文给出了闭环系统和核方程所对应的离散方程,通过编程实现了对闭环系统的仿真,绘制出了其相关图像.仿真的结果表明与理论推导一致.与已有的成果相比,论文考虑了一类更广、更复杂的反应扩散系统.更多还原

【Abstract】 Boundary control for a class of partial integro-differential systems with space and time dependent coefficients is considered. This system not only consists of reaction, ad-vection, diffusion terms, but also the integral item about space. The Neumann boundary condition comes from the Fourier Law in heat conduction. This system models the phys-ical phenomenons like burning process with a chemical reaction, the integral term means that the system has memory function with spatial variable.According to the partial differential equation(PDE) backstepping method, a Volterra-type integral transformation is established that convert the system into a target system, which is exponentially stable, thought selecting a kernel function. A control law is derived from the boundary condition of the target system that transformation satisfied. There exits a mathematical difficulty to solve the kernel equations analytically. Alter-nately, it is to show the existence of the kernel function. Kernel equation is converted into an integral equation by using the characteristic line of PDE. Based on the method of successive approximation and mathematical induction, the existence of kernel equation is proved in the condition of an assumption. Exponential stability of the closed-loop system is achieved via the transformation and it’s inverse. Simulation equations of the closed-loop and the kernel equation are presented, and the simulation results are shown thought figures.Comparing with existed results, this paper considers a more complex reaction diffu-sion system which extends the existed results to more general situation.更多还原