On the lack of null-controllability of the heat equation on the half-line (Q2701673)

It is considered the following boundary control problem NEWLINE\[NEWLINE\begin{aligned} u_t(x,t) &- u_{xx}(x, t)=0,\quad x>0,\quad 0< t< T,\\ u(0,t) &= v(t),\quad 0< t< T,\\ u(x,0) &= u_0(x)\end{aligned}NEWLINE\]NEWLINE with \(v\in L^2(0,T)\). The null-controllability is considered and the main difficulty comes from the unboundedness of the domain for \(x\). The problem is solved introducing similarity variables NEWLINE\[NEWLINEy= x/\sqrt{t+ 1},\quad s= \log(t+1)NEWLINE\]NEWLINE and the associated system. An analysis of the control problem is performed by reducing it to a problem of moments which is critical in the sense that some usual convergence condition does not hold. Unlike the case of a finite \(x\)-domain, the controllable data have exponentially increasing Fourier coefficients.