K-S方程的精确边界控制

Exact boundary control of Kuramoto-Sivashinsky equation

运用Fourier基函数的展开以及Fourier变换的方法研究带有周期边界条件的Kuramoto-Sivashinsky方程在有限时间区间[0,T]上的精确控制.首先研究线性化K-S方程的精确控制,运用Reimann-Lebesgue收敛定理以及R iese基函数的性质证明了在给定的时间T>0,对于两个任意给定的函数u0(x),u1(x)属于一定的Sobolev空间,总能找到一个控制函数使得线性化K-S方程有一个存在于某一合适的空间的解u(x,t)使其满足u(x,0)=u0(x),u(x,T)=u1(x).然后结合线性化K-S方程的精确控制,再通过定义Fredholm算子并应用此算子的一些理论可以找到K-S方程的控制函数,使其达到精确控制.

By the expansions of the Fourier basis functions and the propositions of Fourier transformations, the exact boundary control problem of the Kuramoto-Sivashinsky Equation with periodic boundary conditions in the limited time interval [0, T] is studied. Firstly, the exact control of the linearized K- S Equation is considered. By Reimann-Lebesgue theorem and the propositions of Riesz basis functions, it proves that for given time T 〉 0, for any two functions uo （x）, u1 （x） given in a suitable Sobolev space, one can always find a control function so that the linearized K -S Equation has a solution satisfying the initial state u（x ,0） = u0 （x） and the final state u（x, T） = u, （x）. Then by defining a Fredholm operator and utilizing its theories, the control function of K - S Equation can be found and exactly controlled.