A lower bound of the norm of the control operator for the heat equation (Q1067202)

Consider the one-dimensional heat equation \(u_ t=u_{xx}\), \(0<x<\pi\), \(0<t<T\), \(u(x,0)=u_ 0(t)\in L_ 2[0,\pi]\), \(u(0,t)=0\), \(u(\pi,t)=f(t)\) on [0,T]. It is desired to choose the control f in \(L_ 2[0,T]\) so that \(u(x,T)=0\), \(0\leq x\leq \pi\) and which has minimum norm among all control functions which satisfy these conditions. The author defines a control operator \(C_ T: L_ 2[0,\pi]\to L_ 2[0,T]\) by \(C_ Tu_ 0=f\), and studies its norm. It is of interest to determine the behaviour of the norm of this operator as a function of T, specially as \(T\to 0+.\) A lower bound of the form K exp(b/T) is obtained.