论文标题
利用总质量来确定扩散问题的对称边界控制中的开关点
The utilization of total mass to determine the switching points in the symmetric boundary control of a diffusion problem
论文作者
论文摘要
作者研究问题$ u_t = u_ {xx}，\ 0 <x <1，\ t> 0; \ u（x，0）= 0，$和$ u（0，t）= u（1，t）=ψ（t），$ where $ψ（t）= u_0 $ for $ t_ {2k {2k} <t <t <t <t <t <t <t <t <t <t <t <t <t <t <t <t <t <t <t <t <t <t <t <t <t <t <t_ {2k+1} $ and $ψ（t）= 0 k = 0,1,2，\ ldots $带$ t_0 = 0 $，序列$ t_ {k} $由等式$ \ int_0^1 u（x，x，t_k）dx = m，$ k = 1,3,3,5，\ dots，$ k = 1,3,5，\ dots，＆\ dots and $ \ int_0^1 u（x，x，x，x，x，$ k = 2，$ k = 2，其中$ 0 <m <m <u_0 $。请注意，开关点$ t_k，\ quad k = 1,2,3，\ ldots $是未知的。证明了存在和独特性。获得$ t_k $和$ t_ {k+1} -t_k $的理论估计值，并提供了估计值的数值验证。
The authors study the problem $u_t=u_{xx},\ 0<x<1,\ t>0; \ u(x,0)=0,$ and $u(0,t)=u(1,t)=ψ(t),$ where $ψ(t)=u_0$ for $t_{2k} < t<t_{2k+1}$ and $ψ(t)=0$ for $t_{2k+1} <t<t_{2k+2},\ k=0,1,2,\ldots$ with $t_0=0$ and the sequence $t_{k}$ is determined by the equations $\int_0^1 u(x,t_k)dx = M,$ for $k=1,3,5,\dots,$ and $\int_0^1 u(x,t_k)dx = m,$ for $k=2,4,6,\dots$ and where $0<m<M<u_0$. Note that the switching points $t_k,\quad k=1,2,3,\ldots$ are unknown. Existence and uniqueness are demonstrated. Theoretical estimates of the $t_k$ and $t_{k+1}-t_k$ are obtained and numerical verifications of the estimates are presented.
