In the case of the wave equation, defined on a sufficiently smooth bounded domain of arbitrary dimension, and subject to Dirichlet boundary control, the operator from boundary to boundary is bounded in the -sense. The proof combines hyperbolic differential energy methods with a microlocal elliptic component.
"The operator $B^*L$ for the wave equation with Dirichlet control." Abstr. Appl. Anal. 2004 (7) 625 - 634, 29 June 2004. https://doi.org/10.1155/S1085337504404011