Electronic Journal of Differential Equations (Apr 2009)
A boundary control problem with a nonlinear reaction term
Abstract
The authors study the problem $u_t=u_{xx}-au$, $0<x<1$, $t>0$; $u(x,0)=0$, and $-u_x(0,t)=u_x(1,t)=phi(t)$, where $a=a(x,t,u)$, and $phi(t)=1$ for $t_{2k} < t<t_{2k+1}$ and $phi(t)=0$ 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 $k=2,4,6,dots$, where $0<m<M$. Note that the switching points $t_k$, are unknown. A maximum principal argument has been used to prove that the solution is positive under certain conditions. 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.
