The utilization of total mass to determine the switching points in the symmetric boundary control of a diffusion problem

The authors study the problem $u_t=u_{xx},0<x<1,t>0;u(x,0)=0,$ and $u(0,t)=u(1,t)=\psi (t),$ where $\psi (t)=u_0$ for $t_{2k}<t<t_{2k+1}$ and $\psi (t)=0$ 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 points $t_k,k=1,2,3,\dots$ 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.