The operator B∗L for the wave equation with Dirichlet control

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 B∗L from boundary to boundary is bounded in the L2-sense. The proof combines hyperbolic dif-ferential energy methods with a microlocal elliptic component. 1. Corrigendum and addendum to [10, Section 5.2] In this paper, we primarily make reference to [10, Section 5.2, pages 1117–1120]. At the end, in Section 3 below, we will also examine its impact on [10, Section 7.1], which is a direct consequence of [10, Section 5.2]. Section 5.2 of [10] deals with the regularity of the map g → B∗Lg, where v = Lg is the solution of the two-dimensional wave equation [10, equation (5.2.2)] in the half-space, with zero initial conditions and Dirichlet boundary control g. (See problem (1.9) below for the general case on a bounded domain in Rn, n ≥ 2.) The claim made in [10, Section 5.2] that B∗L /∈(L2(0,T;U)) is incorrect, due to a spurious appearance of the symbol “Re ” (real part) in [10, equation (5.2.18)]—and, con-sequently, in [10, equation (5.2.22)]—while in view of the correct [10, equation (5.2.10)], the symbol “Re ” should have been omitted. Luckily, the same analysis given in [10, Section 5.2], once the spurious symbol “Re ” is omitted from [10, equation (5.2.18)] (as it should be), provides, in fact, a direct proof of the positive result that g − → B∗Lg is continuous on L2(Σ), Σ = (0,T)×Γ. (1.1) This result in (1.1)—at this step, with the correction noted above, valid just for the two-dimensional wave equation on the half-space—can, in fact, be generalized to hold true for the wave equation with Dirichlet boundary control g defined on any bounded, sufficiently smooth domain inRn, n ≥ 2—see Theorem 1.1 below. The positive conclusion in the case n = 1 was already noted in [10, Section 4.7 and equation (5.1.19)]. Thus, this note serves a two-fold purpose: (i) on the one hand, it provides a corrigendum to the counter-exampl