论文标题
使用总质量的Neumann边界控制的扩散问题
A Diffusion Problem with Neumann Boundary Control Utilizing Total Mass
论文作者
论文摘要
作者研究扩散问题$ u_t = u_ {xx}，\ 0 <x <1，\ t> 0; \ u（x，0）= 0，$和$ -U_X（0，t）= u_x（1，t）= ϕ（t），$，其中$ ϕ（t）$是一个控制函数，可确保总质量$ \ int_0^1 u（x，x，t_k）dx $在两个偏置值之间停留。从数学上讲，$ ϕ（t）= 1 $ for $ t_ {2k} <t <t <t <t <t <t <t <t <t <t <t <t <t_ {2k+1} $和$ ϕ（t）= - 1 $ for $ t_ {2k+1} <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 <t <t <t <t <t <t <t <t <t <t <t <t <t_ {2k+2} $ \ int_0^1 u（x，t_k）dx = m，$ k = 1,3,5，\ dots，$和$ \ int_0^1 u（x，x，t_k）dx = m，$ k = 2,4,6，\ dots $，其中$ 0 <m <m <m <m <m <m <u_0 $。请注意，切换时间$ t_k $是未知数。确定开关时间$ t_k $及其差异$ t_ {k+1} -t_k $。提出了数值验证示例。
The author studies the diffusion problem $u_t=u_{xx},\ 0<x<1,\ t>0; \ u(x,0)=0,$ and $-u_x(0,t)=u_x(1,t)=ϕ(t),$ where $ϕ(t)$ is a control function that ensures that the total mass $\int_0^1 u(x,t_k)dx$ stays between two predetermined values. Mathematically, $ϕ(t)=1$ for $t_{2k} < t<t_{2k+1}$ and $ϕ(t)=-1$ 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 times $t_k$ are unknowns. Determination of switching times $t_k$ and their differences $t_{k+1}-t_k$ are obtained. Numerical verifying examples are presented.
