A Diffusion Problem with Neumann Boundary Control Utilizing Total Mass

Abstract

The author studies the diffusion problem $u_t=u_{xx},0<x<1,t>0;u(x,0)=0,$ and $-u_x(0,t)=u_x(1,t)=\varphi (t),$ where $\varphi (t)$ is a control function that ensures that the total mass ${\int }_0^{1}u(x,t_k)dx$ stays between two predetermined values. Mathematically, $\varphi (t)=1$ for $t_{2k}<t<t_{2k+1}$ and $\varphi (t)=-1$ for $t_{2k+1}<t<t_{2k+2},k=0,1,2,\dots$ 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.

Keywords: Diffusion Problem, Neumann Boundary Conditions, Total Mass Control, Finite Difference