  An optimal boundary control problem for the one-dimensional heat equation is
considered. The objective functional includes a standard quadratic terminal
observation, a Tikhonov regularization term with regularization parameter
$\nu$, and the $L^1$-norm of the control that accounts for sparsity. The
switching structure of the optimal control is discussed for $\nu \ge 0$. Under
natural assumptions, it is shown that the set of switching points of the
optimal control is countable with the final time as only possible accumulation
point. The convergence of switching points is investigated for $\nu \searrow
0$.
