THE OPERATOR B∗L FOR THE WAVE EQUATION WITH DIRICHLET CONTROL

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, consequently, 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

626 The operator B * L for the wave equation with Dirichlet control in the two-dimensional half space of [10, Section 5.2]; (ii) on the other hand, it provides its replacement in the addendum-the positive statement of Theorem 1.1 below.
(ii) As a consequence of (i), in equation (5.2.22), page 1120, suppress the symbol "Re," so that the corrected equation becomes, for (σ,τ (1.2) (iii) As a consequence of (ii), in equation (5.2.23), page 1120, suppress the symbol "Re," so that the corrected equation becomes by (1.2), with 3) The very same argument with "Re" omitted, as it should be, instead of a negative result, gives the positive result in (1.1) in the half-space; in fact, for any n ≥ 2. We will see this below.
Positive result on a half-space, n ≥ 2. The proof is essentially contained in [10, Section 5.2], modulo the corrections as stated above.We consider the half-space wave equation problem in [10, equation (5.2.2)].Let u ∈ L 2 (0,∞;L 2 (Γ)).Then, the corresponding version of [10, equation (5.2.10), page 1119] is (1.7) I. Lasiecka and R. Triggiani 627 Then, (1.4) and (1.7) yield the desired conclusion: and thus (1.1) holds true for the wave equation on the n-dimensional half-space n ≥ 2. The argument above is very transparent and shows exactly what is going on in order to gain the additional derivative on the boundary in the present case.
Addendum.We now state the general positive result.
For future reference in the proof of Section 2, we recall from [10, equations (5.1.3),(5.1.10),(5.1.13)]that (1.11) (1.12) Remark 1.2.The above Theorem 1.1 was first stated in [1] (see estimate (2.7), page 121).We believe that the proof that we will give below in Section 2 is essentially self-contained and much simpler than the sketch given in [1].The idea pursued in [1] is based on a full microlocal analysis of the fourth-order operator ∆(D 2 t − ∆) (where the extra ∆ is used to eliminate Dg from the z-dynamics z tt = ∆z + Dg t , see [10, equation (5.1.11b)],as ∆Dg t ≡ 0).The subsequent microlocal analysis of [1] considers, as usual [8], three regions: the hyperbolic region, the elliptic region, and the "glancing rays" region.The latter is the most demanding, and it is unfortunate that no details are provided in [1] for the analysis in the glancing region, except for reference to the author's Ph.D. thesis.
By contrast, our proof in Section 2 below invokes, for the most critical part, the sharp regularity of the wave equation from [5]-which is obtained via differential, rather than pseudodifferential/microlocal analysis methods.In addition, standard elliptic (interior and) trace regularity of the Dirichlet map D is used.Thus, by simply invoking these results in (1.12) above for z t , we obtain-by purely differential methods-the critical result on ∂z t /∂ν of Step 1,(2.3).This then provides automatically the desired regularity of ∂z/∂ν microlocally outside the elliptic sector of the D'Alambertian = D 2 t − ∆, where the time variable dominates the tangential space variable in the Fourier space, see (2.11) below.
Thus, the rest of the proof follows from pseudodifferential operator (PDO) elliptic regularity of the localized problem.
Step 2. It remains to show that the L 2 regularity of ∂z/∂ν holds also in the elliptic sector.This is done by standard arguments using localization of the PDO symbols.We use standard partition of unity procedure and local change of coordinates by which Ω and Γ can be identified (locally x + r(x, y)D 2 y + lot, where lot (which result from commutators) are first-order differential operators and r(x, y)D 2 y stands for the secondorder tangential (in the y variable) strongly elliptic operator.Since solutions v satisfy zero initial data, we can also extend v(t) by zero for t < 0. For t > T we multiply the solution by a smooth cutoff function φ(t) = 0, t ≥ (3/2)T, φ(t) = 1, t ≤ T. Thus, in order to obtain the desired solution, it amounts to consider the following problem: where ∆ 0 = D 2 x + r(x, y)D 2 y is the principal part of ∆ and v is the original solution v = Lg of problem (1.9).Below, we will write w = u + y, where u, y satisfy (2.5) and (2.6), respectively.As a consequence, we will obtain (2.4b) I. Lasiecka and R. Triggiani 629 Below we will denote by u the solution of the counterpart regularity statement of (2.1) for v in Ω.Likewise, we introduce the following nonhomogenous problem: where f = lot(v) results from the presence of the lower-order terms applied to the original variable v in (2.4), that is, in (1.9).Thus, recalling that v ∈ C([0,T];L 2 (Ω)) by (2.1), we obtain By the principle of superposition, we have w = u + y, as announced above.
Step 3. In this step, we handle the y-problem (2.6).We first recall from (1.10) that our original objective is showing that D * v t ∈ L 2 (Σ) continuously in g ∈ L 2 (Σ).Moreover, we recall that v in Ω is transferred into w = u + y, on the half-space Ω (locally).Thus, by (2.6), (2.7), what suffices to show for y is the following regularity property: whereby D * y t is ultimately continuous in g ∈ L 2 (Σ).However, the above property (2.8) is known from [5, Theorem 3.11, page 182] and has been used in the past several times.In fact, set A = −∆ 0 , with Ᏸ(A) = H 2 ( Ω) ∩ H 1 0 ( Ω) and rewrite (2.6) abstractly as y tt = −Ay + f .Apply A −1 throughout and set , again by (2.7).Thus, Ψ solves the problem (2.9) We further have that A −1 y t ∈ C([0,T];H 1 0 ( Ω)), again by (2.7).Finally we recall that D * AA −1 y t = −(∂/∂ν)Ψ t (see [9], [10, equation (5.1.9)]).One can simply quote [5, Theorem 3.11, page 182] or [9, equation (10.5.5.11), page 952] to obtain the desired regularity (2.8): where henceforth we take for Q an extended cylinder based on Ω × [−T,2T].Indeed, this last inclusion follows from [ᐄ, ] ∈ S 1 ( Q) and the priori regularity (2.5b) for u imply- Furthermore, still by (2.5b) and the fact that suppu ∈ [0,(3/2)T], we have, by the pseudolocal property of pseudodifferential operators, that (ᐄu)(2T) ∈ C ∞ ( Ω), (ᐄu)(−T) ∈ C ∞ ( Ω).We conclude that ᐄu| ∂ Q ∈ L 2 (∂ Q), a boundary condition to be associated to (2.12).Since ᐄ is a pseudodifferential elliptic operator, classical elliptic theory, applied to where the first containment on the right-hand side of (2.13) is due to the boundary term, and the second to the interior term.Next, we return to the elliptic problem ∆z = −v t in Q, z| Σ = 0 from (1.11), with a priori regularity noted in (1.11).The counterpart of the above elliptic problem in the half-space Q (locally) is ∆z = −u t in Q, z|Σ = 0 (we retain the symbol z in Q), as we are identifying w with u in the present Step 4 (due to the results of Step 3).Applying ᐄ throughout yields Hence, by the a priori regularity in (2.5b) for u and in (1.11) for z, we conclude I. Lasiecka and R. Triggiani 631 Moreover, by virtue of (2.13), (d/dt)ᐄu ∈ H (0,−1/2) ( Q) where we have used the anisotropic Hörmander's spaces [3, Volume III, page 477], H (m,s) ( Q), where m is the order in the normal direction to the plane x = 0 (which plays a distinguished role) and (m + s) is the order in the tangential direction in t and y.Via (2.15), we are thus led to solving the problem (2.16) By elliptic regularity (note that ∆ᐄ is elliptic in Q), we obtain again (2.17) Combining (2.17) and (2.11) yields the final conclusion and Theorem 1.1 is proved.that is, Then the map g → (∂/∂ν)v t is continuous on L 2 (Σ).Proposition 4.1.In addition to the standing hypotheses (i) and (ii) above, assume that (a) A is skew adjoint: A * = −A, so that e A * t = e −At , t ∈ R, and (b) Then, in fact,   Finally, recalling L T in (4.1) and its adjoint L * T [9], we rewrite (4.8) in the following attractive form: (from which (4.5) follows, by taking the L 2 (0,T;U)-inner product with u).Equation (4.9) shows the implication (4.5)⇒(4.3).

3 .Theorem 3 . 1 .
Impact on [10, Section 7.1] Theorem 1.1 and the decomposition argument in [10, Section 7.1, page 1129] allow one to deduce the analogous positive result valid for the Kirchhoff plate with moment controls.Indeed, with reference to the model in [10, equations (7.1.1)],we have the following theorem.Let Ω be as in Theorem 1.1, and let v be a solution to [10, equations (7.1.1)],
Theorem 1.1.Let Ω be a sufficiently smooth bounded domain in R n , n ≥ 2. Consider the v-problem in [10, equation (5.1.1),page 1114], that is, Having accounted for the lot(v) in Step 3-which are responsible for the yproblem-we may in this step set y ≡ 0 and thus identify w with u : w ≡ u.Thus it remains to consider problem (2.5) in u, involving only the principal part of the D'Alambertian.Let ᐄ ∈ S 0 ( Q) denote the PDO operator ᐄ(x, y,t) with smooth symbol of localization χ(x, y,t,σ,η) supported in the elliptic sector of ≡ D 2 .10) 630 The operator B * L for the wave equation with Dirichlet control Step 4.