A Diffusion Problem with Neumann Boundary Control Utilizing Total Mass

TL;DR

This paper investigates a diffusion control problem with Neumann boundary conditions, where the control function switches periodically to keep the total mass within specified bounds, and determines the switching times through analysis and numerical examples.

Contribution

It introduces a method to determine switching times for boundary control in a diffusion process to maintain total mass within set limits, including analytical and numerical results.

Findings

Explicit formulas for switching times and their differences.

Numerical examples verifying the control strategy.

Demonstration of control effectiveness in maintaining mass bounds.

Abstract

The author studies the diffusion problem $u_t=u_{xx},\ 0<x<1,\ t>0; \ u(x,0)=0,$ and $-u_x(0,t)=u_x(1,t)=\phi(t),$ where $\phi(t)$ is a control function that ensures that the total mass $\int_0^1 u(x,t_k)dx$ stays between two predetermined values. Mathematically, $\phi(t)=1$ for $t_{2k} < t<t_{2k+1}$ and $\phi(t)=-1$ 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.