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Abstract and Applied Analysis L2(Σ)regularity of the boundary to boundary operator B∗L for hyperbolic and Petrowski PDEs
L2(Σ)regularity of the boundary to boundary operator B∗L for hyperbolic and Petrowski PDEs
Lasiecka, I., Triggiani, R.Quanto Você gostou deste livro?
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Volume:
2003
Ano:
2003
Idioma:
english
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Abstract and Applied Analysis
DOI:
10.1155/s1085337503305032
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L2 (Σ)REGULARITY OF THE BOUNDARY TO BOUNDARY OPERATOR B∗ L FOR HYPERBOLIC AND PETROWSKI PDEs I. LASIECKA AND R. TRIGGIANI Received 20 February 2003 This paper takes up and thoroughly analyzes a technical mathematical issue in PDE theory, while—as a bypass product—making a larger case. The technical issue is the L2 (Σ)regularity of the boundary → boundary operator B∗ L for (multidimensional) hyperbolic and Petrowskitype mixed PDEs problems, where L is the boundary input → interior solution operator and B is the control operator from the boundary. Both positive and negative classes of distinctive PDE illustrations are exhibited and proved. The larger case to be made is that hard analysis PDE energy methods are the tools of the trade—not soft analysis methods. This holds true not only to analyze B∗ L but also to establish three interrelated cardinal results: optimal PDE regularity, exact controllability, and uniform stabilization. Thus, the paper takes a critical view on a spate of “abstract” results in “infinitedimensional systems theory,” generated by unnecessarily complicated and highly limited “soft” methods, with no apparent awareness of the high degree of restriction of the abstract assumptions made—far from necessary—as well as on how to verify them in the case of multidimensional dynamical systems such as PDEs. 1. A historical overview: hard analysis beats soft analysis on regularity, exact controllability, and uniform stabilization of hyperbolic and Petrowskitype PDEs under boundary control At first, naturally, PDEs boundary control theory for evolution equations tackled the most established PDE classes—parabolic PDEs—whose Hilbert space theory for mixed problems was already available in a form close to an optimal book form [51, 56, 57, 58] since the early 1970s. Next, in the early 1980s, when the study of boundary control problems for (linear) PDEs began to address hyperbolic and Petrowskitype systems on a multidimensional bounded domain [10, 26] (see [5, 6, 35, 44, 45] for overview), Copyri; ght © 2003 Hindawi Publishing Corporation Abstract and Applied Analysis 2003:19 (2003) 1061–1139 2000 Mathematics Subject Classification: 35Lxx, 35Qxx, 93xx URL: http://dx.doi.org/10.1155/S1085337503305032 1062 Regularity of B∗ L it faced at the outset an altogether new and fundamental obstacle, which was bound to hamper any progress. Namely, an optimal, or even sharp, theory on the preliminary, foundational questions of wellposedness and global regularity (both in the interior and on the boundary, for the relevant solution traces) was generally missing in the PDEs literature of mixed (initial and boundary value) problems for hyperbolic and Petrowskitype systems [51]. Available results were often explicitly recognized as definitely nonoptimal [57, page 141]. Hard analysis energy methods. A happy and quite challenging exception was the optimal—both interior and boundary—regularity theory for mixed, nonsymmetric, noncharacteristic firstorder hyperbolic systems culminated through repeated eﬀorts in the early 1970s [16, 63, 64]. Its final, full success required eventually the use of pseudodiﬀerential energy methods (Kreiss’ symmetrizer). Apart from this isolated case, mathematical knowledge of global optimal regularity theory of hyperbolic and Petrowskitype mixed problems was scarce, save for some trivial onedimensional cases. Thus, in this gloomy scenario, one may say that optimal control theory [10, 26, 51] provided a forceful impetus in seeking to attain an optimal global regularity theory for these classes of mixed PDEs problems. To this end, PDEs (hard analysis) energy methods—both in diﬀerential and pseudodiﬀerential form—were introduced and brought to bear on these problems. The case of secondorder hyperbolic equations under Dirichlet boundary control was tackled first. The resulting theory that emerged turns out to be optimal and does not depend on the space dimension [22, 24, 25, 43, 52]. It was best achieved by the use of energy methods in a diﬀerential form. The case of secondorder hyperbolic equations, this time under Neumann boundary control, proved far more recalcitrant and challenging (in space dimension strictly greater than one) and was conducted in a few phases. The additional degree of diﬃculties for this mixed PDE class stems from the fact that the Lopatinski condition is not satisfied for it. Unlike the Dirichlet’s, the Neumann boundary control case requires pseudodiﬀerential analysis. Final results depend on the geometry [32, 34, 38, 43, 69]. Naturally, in investigative eﬀorts which moved either in a parallel or in a serial mode, the conceptual and computational “tricks” that had proved successful in obtaining an optimal, or sharp, regularity theory for secondorder hyperbolic equations were exported, with suitable variations and adaptations, to certain Petrowskitype systems. The lessons learned with secondorder equations served as a guide and a benchmark study for these other classes. To be sure, not all cases have been, to date, completely resolved. The problem of optimal regularity of some Petrowski systems with “high” boundary operators is not yet fully solved. However, a large body of optimal regularity theory has by now emerged, dealing with systems such as Schrödinger equations, platelike equations of both hyperbolic (Kirchhoﬀ model) and nonhyperbolic types (EulerBernoulli model), and so forth. Subsequently, additional more complicated dynamics followed such as system of elasticity, Maxwell equations, dynamic shell equations, and so forth. I. Lasiecka and R. Triggiani 1063 Shared by all these endeavors, there is one common loud message that hard analysis energy methods have been responsible for the resulting successes. A rather broad account of these issues under one cover may be found in [35, 43, 45, 53]. Abstract models of PDEs mixed problems. Simultaneously, and in parallel fashion, the aforementioned investigative eﬀorts since the mid 1970s also produced “abstract models” for mixed PDE problems subject to control either acting on the boundary of, or else as a point control within, a multidimensional bounded domain, see [2, 82, 83] for parabolic problems and [24, 25, 73] for hyperbolic problems. Though, in particular, operators arising in the abstract model depend on both the specific class of PDEs and its specific homogeneous and nonhomogeneous boundary conditions, one cardinal point reached in this line of investigation was the following discovery: most of them—but by no means all of them [9, 23, 78]—are encompassed and captured by the abstract model ẏ = Ay + Bu in Ᏸ A∗ , y(0) = y0 ∈ Y, (1.1) where U and Y are, respectively, control and state Hilbert spaces, and where (i) the operator A : Y ⊃ Ᏸ(A) → Y is the infinitesimal generator of a strongly continuous (s.c.) semigroup eAt on Y , t ≥ 0; (ii) B is an “unbounded” operator U → Y satisfying B ∈ ᏸ(U;[Ᏸ(A∗ )] ) or, equivalently, A−1 B ∈ ᏸ(U;Y ). Above, as well as in (1.1), [Ᏸ(A∗ )] denotes the dual space with respect to the pivot space Y of the domain Ᏸ(A∗ ) of the Y adjoint A∗ of A. Without loss of generality, we take A−1 ∈ ᏸ(Y ). Many examples of these abstract models are given under one cover in [5, 6, 35], [44, 45]; they include the case of firstorder hyperbolic systems quoted before, where again the need for an abstract model came from boundary PDE control theory and was not available in the purely PDE theory per se. See Section 4.1. Accordingly, having accomplished a first abstract unification of many dynamical PDEs mixed problems, it was natural to attempt to extract—wherever possible—additional, more indepth, common “abstract properties,” shared by suﬃciently many classes of PDE mixed problems. For the purpose of this paper, we will focus on three “abstract properties”: (optimal) regularity, exact controllability, and uniform stabilization. Regularity. The variation of parameter formula for (1.1) is (Lu)(t) = t 0 y(t) = eAt y0 + (Lu)(t), eA(t−τ) Bu(τ)dτ, LT u = (Lu)(T) = T 0 (1.2a) eA(T −t) Bu(t)dt. (1.2b) Per se, the abstract diﬀerential equation (1.1) is not the critical object of investigation. It is good to have it inasmuch as it yields (1.2). The key element that 1064 Regularity of B∗ L defines the crucial feature of a particular PDE mixed problem is, however, the regularity of the operators L and LT . This is what was referred to above as “interior regularity”: the control u acts on the boundary, while Lu is the corresponding solution acting in the interior. Accordingly, this pursued line of investigation brought about a second, abstract realization [24, 25, 26, 43] that of determining the “best” function space Y for each class of mixed hyperbolic and Petrowskitype problems such that the following interior regularity property holds true: L : continuous L2 (0,T;U) −→ C [0,T];Y , (1.3) for one, hence for all positive, finite T. Presently, such space Y is explicitly identified in most (but by no means all) of the mixed PDE problems of hyperbolic or Petrowski type. (The case Y = [Ᏸ(A∗ )] is always true in the present setting, and not much informative, save for oﬀering a backup result for (1.1).) An equivalent (dual) formulation is given in (1.4), see [10, 25, 26]. Hard beats soft on regularity. It is hard analysis that delivers the softexpressed interior regularity result (1.3). For the mixed PDEs classes under consideration, achieving the regularity property (1.3) with the “best” function space Y is, as amply stressed above, not an accomplishment of soft analysis methods (say, semigroup theory or cosine operator theory, which instead gives the lousy result of (1.3) with Y = [Ᏸ(A∗ )] , and, in fact, something “better” such as [Ᏸ(A∗α )] for some 0 < α < 1 depending on the equation and the boundary conditions [24, 56, 57, 58], but far from optimal). On the contrary, it is the accomplishment of hard analysis PDE energy methods, tuned to the specific combination of PDE and boundary control, which first produces, for each such individual combination, a PDE estimate for the corresponding dual PDE problem. The precursor was the multidimensional wave equation with Dirichlet control [22, 24, 25]. All such a priori estimates thus obtained on an individual basis admit the following “abstract version”: ∗ L∗T ≡ B ∗ eA t : continuous Y −→ L2 (0,T;U), (1.4) where LT is defined by (1.2b) [22, 24, 25]. In PDE mixed problems, property (1.4) is a (sharp) “trace regularity property” of the boundary homogeneous problem, which is dual to the corresponding map LT in (1.2b): from the L2 (0,T;U)boundary control to the PDE solution at time T, see many examples in [35, 44, 45]. Indeed, such PDE estimate is both nontrivial and unexpected, and typically yields a finite gain (often 1/2) in the space regularity of the solution trace, which does not follow even by a formal application of trace theory to the optimal interior regularity of the PDE solution. Some PDE circles have come to call it “hidden regularity,” and with good reasons. It was first discovered in the case of the wave equation with Dirichlet control [25]. I. Lasiecka and R. Triggiani 1065 Only after the fact, if one so wishes, soft methods can be brought into the analysis to show that, in fact, the abstract trace regularity (1.4) is equivalent to the interior regularity property (1.3) [10, 25, 26]. (Needless to say, this can actually be done also on a casebycase basis for each PDE class.) Thus, one key message is clear: that for all such questions of regularity of mixed PDE problems, the slogan “hard beats soft” holds definitely true. It is hard analysis PDE energy methods (diﬀerential or pseudodiﬀerential) that produce the key—and unexpected—a priori estimates which shine within (1.4). Soft analysis then takes advantage of these single a priori estimates into a common abstract formulation only afterwards, for the purpose of unification; for instance, in carrying out the study of optimal control theory with quadratic cost, and so forth. This is the spirit of abstract, unifying treatments of optimal control problems for PDE subject to boundary (and point) control that can be found in books such as [5, 6, 35, 45]. As mentioned above, the regularity (1.4) is equivalent to the regularity (1.3) by a duality argument [10, 25, 26]. Surjectivity of LT or exact controllability. In a similar vein, we can describe the second abstract dynamic property of model (1.1) or (1.2); namely, the property that the inputsolution operator LT , defined in (1.2b), satisfies LT be surjective : L2 (0,T;U) −→ onto Y1 , (1.5) where Y1 ⊂ Y . In the most desirable case, Y1 is the same space Y as in (1.3). In fact, this is often the case with hyperbolic and Petrowskitype systems, but is by no means always true (e.g., secondorder hyperbolic equations with Neumann control, EulerBernoulli plate equations with control in “high” boundary conditions). For time reversible dynamics such as the hyperbolic and Petrowskitype systems under consideration, the functional analytic property (1.5) is relabelled “exact controllability in Y1 at t = T” in the PDE control theory literature. By a standard functional analysis result [70, page 237], property (1.5) is equivalent by duality to the following socalled “abstract continuous observability” estimate: ∗ L z ≥ cT z T T or 0 ∗ A∗ t 2 B e x dt ≥ cT x U 2 Y1 ∀x ∈ Y1 , (1.6) perhaps only for T suﬃciently large in hyperbolic problems with finite speed of propagation, which we recognize as being the inverse inequality of (1.4), at least when Y1 = Y and T is large. So far, so good: the abstract condition (1.6) shines for its unifying value (and for the utter simplicity by which it is obtained—just a duality step). But the crux of the matter begins now: how does one establish the validity of characterization (1.6) for exact controllability in the appropriate function spaces U and Y1 —in particular, if we can take Y1 = Y —for the classes of multidimensional hyperbolic 1066 Regularity of B∗ L and Petrowskitype PDE with boundary control? The answer is the same as in the case of regularity of the operator L discussed before, except even more emphatically: again, for each single class, one establishes by appropriate PDE energy (hard analysis) methods the a priori concrete versions of the continuous observability inequality of which (1.6) is an abstract unifying reformulation. Thus, we can extract a second lesson, this time for the exact controllability problem. It is “hard beats soft on exact controllability,” an extension of the same slogan, now duplicated from global regularity to exact controllability as well. It is hard PDE analysis that permits one to obtain inversetype inequalities such as (1.4), bounding the initial energy of the corresponding boundary homogeneous problem by the appropriate boundary trace. Uniform stabilization. One may repeat the same set of considerations, in the same spirit, when it comes to establishing uniform stabilization of an originally conservative hyperbolic or Petrowskitype system, by means of a suitable boundary dissipation. The abstract characterization is an inversetype inequality such as (1.6), except that it refers now to the boundary dissipative mixed PDE problem, not the boundary homogeneous conservative PDE problem. The particular abstract inequality will be given in (2.12) in the context under discussion. However, the common lesson is duplicated once more. It is again the slogan “hard beats soft,” this third time applied to the uniform stabilization problem. Indeed, this conclusion is even more acute in this case than in the preceding two cases, as, typically, establishing the uniform stabilization inequality for the class of hyperbolic or Petrowskitype PDEs under discussion is more challenging, sometimes by much than obtaining the corresponding specialization of the continuous observability inequality (1.6). Enter “infinitedimensional systems theory”. To repeat ad nauseam, the distinctive thrust described above in connection with the problems of regularity, exact controllability, and uniform stabilization of hyperbolic and Petrowskitype mixed PDE problems is: one proves the concrete required estimates in each of the three issues by hard PDE analysis in the energy method, and only afterwards extracts and delivers the corresponding abstract version for unification purposes. One unfortunate consequence of all this is that a wanderer coming from outside may choose to see only the clean, shining abstract version, not the “dirty” technical hard analysis that went into proving it in the first place. Thus, such a traveller may be tempted to move around only within the abstract level, in the comfort of some standard semigroup setting, and be induced to prove “significant” results without descending into the arena of hard analysis. Indeed, in this way, while holding the neck above the Hilbert or Banach space clouds, one can show some results. The key is: under what assumptions? Consistently with the care to remain in lofty land, the assumptions will be “abstract,” of course, meaning now “soft.” And here is the key of this whole matter, the moral of the present introductory section. I. Lasiecka and R. Triggiani 1067 (i) Are the “abstract” soft assumptions introduced by an alternative, indirect approach ever true, hopefully at least for some nontrivial classes of multidimensional PDEs? How does one verify them? How does the eﬀort to verify the assumptions of these indirect routes compare with the more gratifying eﬀort of establishing directly the relevant, a priori characterizing inequality, as already available in the literature of the past 20 years? (ii) In case a hypothesis of the indirect route is indeed true at least for some classes of relevant PDEs, is it too strong for the final goal that is claimed? That is, how far is it from being necessary? (iii) If the proposed “new” route avoids the direct proof of the past literature to establish the desired result, by going around the circle instead of moving straight along the relevant diameter, is there anything gained in a detour oﬀered as an alternative approach? Infinitedimensional systems theory oﬀers many illustrations where the answer to the basic questions above is, overall and cumulatively, negative. A most recent case in point is displayed by [12]. It oﬀers an eloquent opportunity to analyze and discuss the conceptual thrust of the present paper, which is multifold. It includes, deliberately, a tutorial component for the purpose of enlightening and guiding those who are lured to the field, coming from (the smooth avenue of) Banach spaces, happily unaware of, and recalcitrant to learn, PDE techniques (save for the eigenfunctions or at most standard Riesz basis, methods of onedimensional domains, when applicable). How many times is the word “semigroup” or the combination “Riesz basis” ever used in Hörmander’s volumes? Yet, the object of those volumes, a thorough description of dynamical properties of linear PDEs, though scarce on global properties of mixed PDE problems, should represent a preliminary setting for the most important and relevant classes of “infinitedimensional systems theory.” 2. A first analysis of the stabilization problem via B∗ L in light of the content of Section 1 The recent paper [12] furnishes clear support for the analysis set forth in Section 1 of the present paper. To begin, we point out some information for readers less acquainted with the topic and the literature. (a) [12, Theorem 1, page 47] has been known in a much stronger nonlinear and multivalued version, see [19]. Moreover, a rather comprehensive treatment of this and other related problems, including references and numerous applications can be found in [21, Chapter 1]. For the linear model (which is the case considered in [12]) stronger results are given in the monograph [45, Theorem 7.6.2.2, page 665]. The fact that “admissibility” of the control operator has nothing to do with the issue of generation (which seems surprising compared to [12]) has been known at least from these references. (b) [12, Theorem 2, page 50] is well known as the socalled Russell’s principle “controllability via stabilizability” for time reversible dynamics, put forward by 1068 Regularity of B∗ L Russell also for infinitedimensional systems [65, 66]. It has since been openly invoked in the literature of boundary control for PDE many times, including the first case of a boundary controllability result of the wave equation with Neumann control, in the energy space H 1 (Ω) × L2 (Ω), obtained in [7]. By the way, in the spirit of the content of Section 1, this “principle” turned out to be a not so sound strategy as it traded the generally easier exact controllability problem with the generally harder uniform stabilization result. (c) The statement reported in [12, page 46, 3rd paragraph] about the lack of exact controllability on any [0,T] in the case of a bounded finitedimensional control operator B has likewise been known, and in a much stronger version since the University of Minnesota, 1973 Ph.D. thesis by the second author, where the relevant topic was published in [71, 72], and has been reported widely also in a book form. Indeed, various more demanding extensions motivated by boundary control of PDE have been later provided, in [75, 77]; see also the lack of uniform stabilization in [75, 76]. In light of Section 1 of the present paper, we intend to concentrate on [12, Theorem 3, page 53], which, apparently, is also announced in [1, Proposition 3.3]. This result deals with the relationship between exact controllability and stabilization. First, we give some background. This is the setting of [19] and [45, Chapter 7, page 663]. A secondorder equation setting. Let H, U be Hilbert spaces and (h1) let Ꮽ : H ⊃ Ᏸ(Ꮽ) → H be a positive selfadjoint operator; (h2) Ꮾ ∈ ᏸ(U;[Ᏸ(Ꮽ1/2 )] ); equivalently, Ꮽ−1/2 Ꮾ ∈ ᏸ(U;H). We consider the openloop control system vtt + Ꮽv = Ꮾu, v(0) = v0 , vt (0) = v1 , (2.1) as well as the corresponding closedloop, dissipative feedback system wtt + Ꮽw + ᏮᏮ∗ wt = 0, w(0) = w0 , wt (0) = w1 . (2.2) We rewrite (2.1) and (2.2) as firstorder systems of the form (1.1) in the space Y = Ᏸ(Ꮽ1/2 ) × H: v(t) d v(t) + Bu, =A vt (t) dt vt (t) 0 I , A= −Ꮽ 0 w(t) d w(t) , = AF wt (t) dt wt (t) 0 I AF = = A − BB ∗ , −Ꮽ −ᏮᏮ∗ (2.3) B= 0 , Ꮾ (2.4) with obvious domains. The operator AF is maximal dissipative and thus the generator of a s.c. contraction semigroup eAF t , t ≥ 0, on Y [45, Proposition 7.6.2.1, page 664]. I. Lasiecka and R. Triggiani 1069 Setting y(t) = [w(t),wt (t)], y0 = [w0 ,w1 ], we have that the variation of parameter system for the wproblem is w(t) = y(t) = eAF t y0 = eAt y0 − wt (t) t 0 eA(t−τ) BB ∗ eAF τ y0 dτ = eAt y0 − L B ∗ eAF · y0 (t), (2.5a) (2.5b) recalling the operator L defined in (1.2b). A firstorder equation setting. We now consider a firstorder model with skewadjoint generator. Let Y and U be two Hilbert spaces. The basic setting is now as follows: (a1) A = −A∗ is a skewadjoint operator Y ⊃ Ᏸ(A) → Y , so that A = iS, where S is a selfadjoint operator on Y , which (essentially without loss of generality) we take positive definite (as in the case of the Schrödinger equation of Section 4.2 below). Accordingly, the fractional powers of S, A, and A∗ are well defined; (a2) B is a linear operator U → [Ᏸ(A∗1/2 )] , duality with respect to Y as a pivot space; equivalently, Q ≡ A−1/2 B ∈ ᏸ(U;Y ) and B∗ A∗−1/2 ∈ ᏸ(Y ;U). Under assumptions (a1) and (a2), we consider the operator AF : Y ⊃ Ᏸ(AF ) → Y defined by AF x = A − BB ∗ x, x ∈ Ᏸ AF = x ∈ Y : A − BB ∗ x ∈ Y . (2.6) Proposition 2.1. Under assumptions (a1) and (a2) above, and, with reference to (2.6), (i) the domain of the operator AF is Ᏸ AF = A−1/2 I − iQQ∗ −1/2 A−F 1 = A −1 A−1/2 Y ⊂ Ᏸ A1/2 ⊂ Ᏸ B ∗ , I − iQQ ∗ −1 A−1/2 ∈ ᏸ(Y ); (2.7a) (2.7b) (ii) the operator AF is dissipative, in fact, maximal dissipative, and hence the generator of a s.c. contraction semigroup eAF t on Y , t ≥ 0; (similarly, the Y 1 adjoint A∗F is the generator of a s.c. contraction semigroup on Y , with A∗− F given by the same expression (2.7b) with “+” sign rather than “−” sign for the operator in the middle); (iii) hence, the abstract firstorder, closedloop equation ẏ = A − BB ∗ y, y(0) = y0 ∈ Y, (2.8a) 1070 Regularity of B∗ L (obtained from the openloop equation η̇ = Aη + Bu (2.8b) with feedback u = −B∗ y) admits the unique solution eAF t y0 , t ≥ 0. Proof. (i) Let x ∈ Ᏸ(AF ). Then we can write AF x = A − BB ∗ x = A1/2 I − A−1/2 B B ∗ A−1/2 A1/2 x = A1/2 I − iQQ∗ A1/2 x (2.9) = f ∈ Y, with Q ≡ A−1/2 B ∈ ᏸ(U;Y ) by assumption, and Q∗ ≡ B ∗ A∗−1/2 ∈ ᏸ(Y ;U) its dual or conjugate. Here, we have used (a.1): A∗ = −A so that A∗1/2 = iA1/2 , hence A∗−1/2 = −iA−1/2 , finally B∗ A−1/2 = iB ∗ A∗−1/2 = iQ∗ . It is clear that the operator [I − iQQ∗ ], where QQ∗ ∈ ᏸ(Y ) is nonnegative, selfadjoint on Y , is boundedly invertible on Y . Thus, (2.9) yields x = A−F 1 f = A−1/2 I − iQQ∗ −1 A−1/2 f ∈ Ᏸ AF , f ∈ Y, (2.10) and (2.7a) and (2.7b) are proved. Then, the identity in (2.7a) plainly shows that Ᏸ(AF ) ⊂ Ᏸ(A1/2 ), while Ᏸ(A1/2 ) ⊂ Ᏸ(B∗ ) by assumption (a.2). Part (i) is proved. (ii) We next show that AF is dissipative. Let x ∈ Ᏸ(AF ). Thus, x ∈ Ᏸ(A1/2 ) = Ᏸ(A∗1/2 ) ⊂ Ᏸ(B∗ ) by part (i). Hence, we can write, if (·, ·) is the Y inner product, then Re AF x,x = Re A − BB ∗ x,x 2 = Re(x,x) − B ∗ x 2 ≤ −B ∗ x ≤ 0 ∀x ∈ Ᏸ AF , (2.11) since Re(Ax,x) = Re{−i A1/2 x 2 } = 0, where each term in (2.11) is well defined. Thus, AF is dissipative. Finally, since A−F 1 ∈ ᏸ(Y ) by part (i), then (λ0 − AF )−1 ∈ ᏸ(Y ) as well for a suitable small λ0 > 0, and then the range condition range(λ0 − AF ) = Y is satisfied, so that AF is maximal dissipative. By the LumerPhillips theorem [62, page 14], AF is the generator of a s.c. contraction semigroup on Y . The same argu ment shows that A∗F is maximal dissipative. Remark 2.2. One can, of course, extend the range of Proposition 2.1 by adding to A a suitable perturbation P: either P ∈ ᏸ(Y ) or else P relatively bounded dissipative perturbations as in known results [62, Corollary 3.3, Theorem 3.4, pages 82–83] for instance, and still obtain that [(A + P) − BB∗ ] is the generator of a s.c. semigroup (of contractions in the last two cases). I. Lasiecka and R. Triggiani 1071 An extension of the key question in [12]. The question which follows was raised in [12, Theorem 3] only in connection with the secondorder system (2.1), (2.2), subject to the assumptions (h1), (h2), that precede (2.1). However, in view of Proposition 2.1, we may likewise extend the same question to the firstorder systems (2.8a) and (2.8b) subject to the assumptions (a.1), (a.2) that precede Proposition 2.1. For both problems, we have A∗ = −A, the skewadjoint property of the free dynamics generator. In [12], the following question has been asked with reference to system (2.1), (2.2): is it true that exact controllability of (2.1) on the state space Y = Ᏸ(Ꮽ1/2 ) × H by means of L2 (0,T;U)controls is equivalent to uniform stabilization of (2.2) on the same space Y ? Here we will extend this question also in reference to systems (2.8a) and (2.8b) in order to include, for instance, also the Schrödinger equation case of Section 4.2. Henceforth, {A,B,AF ,Y,U } refers either to (2.5) or to (2.8) indiﬀerently. Quantitatively, we may reformulate the above question as follows: is the continuous observability inequality (1.6) (which characterizes exact controllability of (1.1) with A and B as in (2.4) or as in (2.6)) equivalent to the inequality T 0 ∗ A t 2 B e F x dt ≥ cT eAF T x2 U Y ∀x ∈ Y, (2.12) which characterizes the uniform stability of the wproblem (2.2) or the yproblem (2.8a)? In our case, A is skewadjoint A∗ = −A. Thus, exact controllability of {A,B } (that is of (2.1) or (2.8a)) over [0,T] is equivalent to exact controllability of {A∗ ,B } over [0,T]. In other words, in our case, inequality (1.6) is equivalent to T 0 ∗ At 2 B e x dt ≥ cT x U 2 Y ∀x ∈ Y. (2.13) Thus, the present question is rephrased now as follows: is inequality (2.12) equivalent to inequality (2.13)? In one direction, the implication, uniform stabilization of (2.1) or (2.8b) (i.e., (2.12))→ exact controllability of (2.1) or (2.8b) (i.e., (2.13)) was shown by Russell [65, 66] some 30 years ago by virtue of a clean soft argument. This result is what paper [12] labels Theorem 2. The proof in [12] is exactly the same as the original wellknown proof of Russell [65]. In the opposite direction, we have the following corollary. Claim 2.3. With reference to the secondorder equations (2.1), (2.2) (resp., the firstorder equations (2.8a) and (2.8b)), assume the preceding assumptions (h1), (h2) (resp., (a1), (a2)). Then, the implication, exact controllability of (2.1) or (2.8b) (i.e., (2.13)) ⇒ uniform stabilization of (2.2) or of (2.8a) (i.e., (2.12)) holds true if one adds the assumption that the operator B ∗ L : continuous L2 (0,T;U) −→ L2 (0,T;U). (2.14) 1072 Regularity of B∗ L This result, which is almost trivial (see a standard short proof in Section 3), is stronger than what paper [12] labels Theorem 3, see Remark 2.4, even in connection with the secondorder equations (2.1), (2.2) considered in [12]. Remark 2.4. We remark that if B is, in particular, a bounded operator, B ∈ ᏸ(U;Y ), then (condition (1.3) and) condition (2.14) is, a fortiori, satisfied. Thus, in this case, exact controllability of (2.1) or (2.8b) implies (and is implied by [65, 66]) uniform stabilization. We recover (with the simple proof of Section 3) a 30yearsold wellknown result of [67] (based on the same finitedimensional proof of [59]). Yet, there are still contemporary papers (say on a simply supported plate with internal velocity damping) on this topic!. Remark 2.5. Actually [12, Theorem 3] assumes, instead of (2.14) for B∗ L, a property which amounts to a “frequency domain” reformulation of property (2.12); the latter is less direct, less enlightening than the former and at any rate unnecessary. Moreover, [12, Theorem 3] assumes, in addition, the regularity property (1.3) for L or its dual equivalent version (1.4), which the subsequent [12, Remark 3] states that it may be dispensed with, as learned via the review process, but with no proof being presented. In the appendix, we provide a proof that (2.14) for B ∗ L implies (1.4) or (1.3) for L; this is, in fact, a simple implication. Apparently, [12, Theorem 3] was also announced in [1, Proposition 3.3]. At any rate, the statement of Claim 2.3 is also known to specialized PDE circles, and we will provide several references below, where a result such as this, or technically comparable and very close to it, is actually builtin into existing proofs of regularity/exact controllability/uniform stabilization of some (surely not all) Petrowskitype systems, rather than singled out per se and broadcast as a “relevant” abstract result. There are very good reasons for this apparent lack of an explicit statement, which is due to a sensible choice of exposition and treatment in the literature of PDE boundary stabilization of the past 15 years. Here is a first preview. (1) Claim 2.3 is very simple to prove within standard energy method settings, and thus its elevation to the rank of “theorem” is arguably unbecoming. See the short proof given in Section 3, which should be compared with the lengthier, more cumbersome time/frequency domain proof of [12, page 54]. (2) The key assumption of the abstract Claim 2.3 is, of course, assumption (2.14) that B∗ L ∈ ᏸ(L2 (0,T;U)). How general is it? And how can one verify it? Only a onedimensional EulerBernoulli beam is given in [12] as an illustrative example where assumption (2.14) is satisfied, and this after 6 pages of breathless eigenfunction computations for diagonal semigroups. Such tour de force in eigenfunction gimmickry can be spared, as we will show below in Section 3.2 that a few lines detailing a standard energy argument will do it. More to the point, assumption (2.8) is, yes, satisfied in some serious multidimensional hyperbolic and Petrowskitype systems (identified in Section 4, by essentially making reference to longpublished PDE and PDEcontrol literature); though it is I. Lasiecka and R. Triggiani 1073 also restrictive, as it is not fulfilled in other hyperbolic/Petrowski problems, also identified below in Sections 5, 6, 7, and 8. To add insult to injury, for these latter hyperbolic/Petrowskitype problems where assumption (2.14) fails, uniform boundary stabilization has been known to hold true for more than 15 years. In short, assumption (2.14) is far from being necessary, a further reason for dethroning Claim 2.3 from the rank of “theorem.” (3) We said above that assumption (2.14) is already known to hold true for some cases of hyperbolic/Petrowskitype systems, and just by relying on longpublished literature. But then, how is it verified in this published literature? Here is the “surprise”: the validity of assumption (2.14) on Claim 2.3 for some hyperbolic/Petrowskitype systems is verified (see Section 4) by precisely the same hard analysis PDE energy methods that are used to prove directly the final soughtafter result of regularity, exact controllability, and above all, uniform stabilization for these systems, save for the case of firstorder hyperbolic systems, where the proof of regularity via pseudodiﬀerential analysis is employed! Then, why does one need to go around the circle and artificially separate the desired conclusion on uniform stabilization into two suﬃcient building blocks—the properties of exact controllability (which is also necessary [65]) and the property (2.14) of regularity of B ∗ L (this second one, however, far from necessary)—if then the hard analysis PDE machinery that allows one to verify the assumption on B ∗ L is the very same that permits one to prove directly the soughtafter uniform stabilization property in one shot? No wonder that Claim 2.3 was not explicitly made in the PDEcontrol literature of the past 15 years! And no wonder if the actual proof of the soft Claim 2.3 is simple, the hard part to prove in order to reach the conclusion on uniform stabilization is buried in the hypotheses; one being far from necessary, but at any rate both verified by hard analysis energy methods. The lofty eyes of the traveller through Banach spaces do not wish to be perturbed by the hard machinery on the ground, where the serious computations take place. 3. The stabilization problem via B∗ L revisited 3.1. A simple (alternative) proof to a nonlinear generalization of Claim 2.3 We provide below a simple alternative proof of Claim 2.3, which, in fact, at no extra eﬀort, yields a new nonlinear generalization of Claim 2.3. In place of (2.8a) (hence (2.2)) we consider the following nonlinear version: yt = Ay − B f B ∗ y , y(0) = y0 ∈ Y (3.1.1) under the same assumptions (a1) for A and (a2) for B, where f is a monotone increasing, continuous function on U. It is known [19, 21] that A − B f (B∗ ) generates a nonlinear semigroup of contractions—say SF (t)—which yields the 1074 Regularity of B∗ L following variation of parameter formula for (3.1.1): y(t) = SF (t)y0 = eAt y0 − L f B ∗ SF (·)y0 (t) (3.1.2) U dt. (3.1.3) and obeys the energy identity y(T)2 = y(0)2 − 2 Y Y T 0 f B ∗ y ,B ∗ y Proposition 3.1. In addition to the standing assumption, we assume that (i) the operator B∗ L is continuous L2 (0,T;U) → L2 (0,T;U) as in (2.14); (ii) m u 2U ≤ ( f (u),u)U ; f (u) U ≤ M u U for all u ∈ U. Then, exact controllability of (A,B) implies exponential stability of SF (t), that is, there exist positive constants C,ω > 0 such that the solution of (3.1.1) satisfies y(t)2 ≤ Ce−ωt y0 2 . Y Y (3.1.4) Proof Step 1. We first show that for any y0 ∈ Y , we have via assumptions (i) and (ii) that ∗ A· B e y 0 L2 (0,T;U) ≤ 1 + kT M B ∗ SF (·)y0 L2 (0,T;U) , (3.1.5) where kT = B ∗ L in the uniform operator norm of ᏸ(L2 (0,T;U)). Indeed, (3.1.5) stems readily from (3.1.2), which yields B ∗ eAt y0 = B ∗ SF (t)y0 + B ∗ L f B ∗ SF (·)y0 (t). (3.1.6) Hence, invoking assumption (2.14) on B∗ L, we see that (3.1.6) along with the bound on f in (ii) at once implies (3.1.5). Step 2. The exact controllability assumption on the pair {A,B}, equivalently on the pair {A∗ ,B }, guarantees characterization (2.13). This combined with (3.1.5) yields then, for any y0 ∈ Y , 2 y0 ≤ cT Y T 0 ∗ At 2 B e y0 dt ≤ cT 1 + kT M U T 0 ∗ B SF (t)y0 2 dt. U (3.1.7) I. Lasiecka and R. Triggiani 1075 Step 3. The energy identity (3.1.3) when combined with (3.1.7) and (i) gives SF (T)y0 2 ≤ cT 1 + kT M Y T +2 0 T 0 ∗ B SF (t)y0 2 dt U B ∗ SF (t)y0 , f B ∗ SF (t)y0 ≤ cT 1 + kT M m−1 + 2 T 0 U dt B ∗ SF (t)y0 , f B ∗ SF (t)y0 U dt 2 2 = cT 1 + kT M m−1 + 2 SF (0)Y − SF (T)Y . (3.1.8) The above identity implies that SF (T) Y ≤ γ < 1 which, in turn, implies exponential decays for the semigroup. The proof of Proposition 3.1 is complete. 3.2. Example 2 in [12] revisited. In this section, we consider the 1dimensional beam problem with boundary control, proposed by [12]. This reference spends six tight pages of dreadful eigenfunction computations for diagonal semigroups to conclude that, in the beam example, property (2.14): B∗ L ∈ L2 (0, T;L2 (Γ)) holds true. However, the issue of exact controllability of this control problem is not addressed or even mentioned. Thus, [12] cannot actually invoke Claim 2.3 or its (weaker) version [12, Theorem 3, page 53], and conclude, as it does, that uniform stabilization holds true as well. By contrast, we provide here an elementary, short, energy method proof that, within the same unified setting, will readily yield in one shot the following properties: (i) B∗ L ∈ ᏸ(L2 (0,T;L2 (Γ))), that is, property (2.14) (as well as the implied L ∈ ᏸ(L2 (0,T;L2 (Γ));C([0,T];Y )), that is, property (1.3) with Y the space of finite energy defined below in (3.2.3)); (ii) uniform stabilization of the corresponding boundary dissipative problem on the finite energy space Y . See Theorem 3.3. Dynamics. Let Ω = (0,1), Σi = (0,T] × {i}, i = 0,1; Q = (0,T] × Ω. We consider the following 1dimensional beam problem with “shear” boundary control at x = 1 and its corresponding dissipative version: vtt + vxxxx = 0, v(0, ·) = v0 , vt (0, ·) = v1 ; vx=0 = vx x=0 ≡ 0; vxx x=1 ≡ 0, vxxx x=1 = g; wtt + wxxxx = 0 in Q; w(0, ·) = w0 , wt (0, ·) = w1 (3.2.1a) in Ω; wx=0 = wx x=0 ≡ 0 in Σ0 ; wxx x=1 ≡ 0, wxxx x=1 = wt x=1 (3.2.1b) (3.2.1c) in Σ1 . (3.2.1d) 1076 Regularity of B∗ L Abstract model of vproblem. We introduce the operators Ꮽψ = ∆2 ψ, ψ ∈ Ᏸ(Ꮽ) = ψ ∈ H 4 (Ω) : ψ x=0 = ψx x=0 = ψxx x=1 = wxxx x=1 = 0 , ϕ = G2 g ⇐⇒ ∆2 ϕ = 0 in Ω; ϕx=0 = ϕx x=0 = ϕxx x=1 = 0, ϕxxx x=1 = g . (3.2.2) The finite energy space of the above problems is Y ≡ Ᏸ Ꮽ1/2 × L2 (Ω) ≡ H 2 (Ω) × L2 (Ω), Ᏸ Ꮽ1/2 = ψ ∈ H 2 (Ω) : ψ x=0 = ψx x=0 = 0 . (3.2.3) Then the abstract model of the vproblem is [44, 45] vtt + Ꮽv = ᏭG2 g, 0 I , A= −Ꮽ 0 v d v + Bg; =A vt dt vt 0 Bg = , ᏭG2 g (3.2.4) B ∗ x1 = G∗ 2 Ꮽx2 , x2 (3.2.5) with obvious domains, where ∗ in B and G2 actually refers to diﬀerent topologies. With B ∗ defined by (Bg,x)Y = (g,B∗ x)L2 (Γ) with respect to the Y topology defined by (3.2.3), we readily find the expression in (3.2.5). The operator B∗ L. With y0 = {v0 ,v1 } = 0, we have via (3.2.5) that ∗ B Lg = B ∗ v t; y0 = 0 = G∗ 2 Ꮽvt t; y0 = 0 = −vt x=1 , vt t; y0 = 0 (3.2.6) recalling the usual property G∗2 Ꮽ· = −·x=1 via [44, 45], as well as the definition of L in (1.2b). Regularity of L, B ∗ L; uniform stabilization. We introduce the PDE problem which is dual to the vproblem: ψtt + ψxxxx = 0 in (0,T] × Ω; ψ(0, ·) = ψ0 , ψt (0, ·) = ψ1 in Ω = (0,1); (3.2.7a) (3.2.7b) ψ x=0 = ψx x=0 = 0 in (0,T] × {0}; (3.2.7c) ψxx x=1 = ψxxx x=1 = 0 in (0,T] × {1}; (3.2.7d) ψ(t) ψ0 ∈ C [0,T];Y = eAt ψ1 ψt (t) if ψ0 ,ψ1 ∈ Y, (3.2.8) I. Lasiecka and R. Triggiani 1077 where eAt is a s.c. group on Y . (Actually, the dual problem requires initial conditions at t = T, not t = 0; but, equivalently for what follows below, we may take initial conditions at t = 0 since the ψproblem is time reversible.) The above setting readily yields the following preliminary result. Lemma 3.2. (i) With reference to the ψproblem with {ψ0 ,ψ1 } ∈ Y , property (1.4) holds true, that is, ∗ T 0 2 2 ψt x=1 dt ≤ cT ψ0 ,ψ1 Y (3.2.9) ⇐⇒ B ∗ eA t : continuous Y −→ L2 (0,T) ⇐⇒ L : g −→ Lg = v,vt : continuous L2 (0,T) −→ C [0,T];Y ≡ H 2 (Ω) × L2 (Ω) , (3.2.10) where in (3.2.10), {v0 ,v1 } = 0 for the vproblem (3.2.1). (ii) With reference to the vproblem (3.2.1) again with y0 = {v0 ,v1 } = 0, B ∗ L : continuous L2 (0,T) −→ L2 (0,T) (3.2.11) if and only if the vproblem in (3.2.1) satisfies T 0 2 vt x=1 dt = ᏻ g 2 L2 (0,T) . (3.2.12) (iii) With reference to property (2.12) for the dissipative wproblem in (3.2.1), T 0 ∗ A t 2 B e F x dt ≥ cT eAF T x2 , U Y ⇐⇒ T 0 wt x=1 2 x ∈ Y, 2 dt ≥ cT w(T),wt (T) Y =H 2 (Ω)×L2 (Ω) . (3.2.13) Theorem 3.3. (i) The regularity of L in (3.2.10) holds true. (ii) The regularity of B∗ L in (3.2.11) holds true. (iii) With reference to the wproblem (3.2.1), (iii1) the map {w0 ,w1 } → {w(t),wt (t)} defines a s.c. contraction semigroup eAF t on Y ≡ Ᏸ(A1/2 ) × L2 (Ω), see (3.2.3); (iii2) with reference to (3.2.1d), wxxx x=1 = wt x=1 ∈ L2 (0, ∞) continuously in w0 ,w1 ∈ Y ; (3.2.14) (iii3) estimate (3.2.13) holds true, thus there exist constants M ≥ 1, δ > 0, such that 2 2 w(t) 2 AF t w0 −δt w0 = e ≤ Me , wt (t) w1 w1 Y Y Y t ≥ 0. (3.2.15) Regularity of B∗ L 1078 Proof. We will show, equivalently, inequalities (3.2.9) and (3.2.12). Step 1. Assume, at first, smooth data {v0 ,v1 ,g }. We multiply the vproblem (3.2.1) by the usual standard multiplier xvx and integrate by parts in t and x. We obtain 1 0 T vt xvx dx 0 − − T1 0 0 T1 0 0 T vt xvxt dx dt + 0 vxxx xvx x =1 x=0 dt (3.2.16) vxxx vx + xvxx dx = 0. Using the identities vt xvxt = 1 d 2 1 2 v x − vt , 2 dx t 2 1 0 vxxx xvxx = *1 vxxx vx dx =v xx vx 0 − 1 0 1 d 2 1 2 v x − vxx , 2 dx xx 2 (3.2.17) 2 vxx dx, in (3.2.16) as well as the boundary conditions (3.2.1c) and (3.2.1d), we obtain the preliminary desired identity 1 2 T 0 vt x=1 2 1 dt = 2 T1 0 0 2 vt2 + 3vxx 1 dx dt + 0 T T vt xvx dx + 0 0 gvx x=1 dt. (3.2.18) Step 2 (proof of (i)). We take g = 0, that is, we consider the corresponding specialization of the vproblem given by the ψproblem (3.2.7) with initial condition {ψ0 ,ψ1} ∈ Y . Thus, specializing identity (3.2.18) to the ψproblem (with g = 0) and using the generation result (3.2.8), we obtain 1 2 T 0 2 1 T 1 2 2 ψt + 3ψxx dx dt + 2 0 0 = ᏻ ψ0 ,ψ1 Y =H 2 (Ω)×L2 (Ω) , ψt2 x=1 dt = 1 0 T ψt xψx dx 0 (3.2.19) and (3.2.9) is proved. Thus, (3.2.10) for L is established. Step 3 (proof of (ii)). Now we consider the vproblem (3.2.1) with {v0 ,v1 } = 0 and regularity (3.2.10) for L just established. We return to identity (3.2.18) and using (3.2.10) we obtain 1 2 T 0 2 vt x=1 dt = ᏻ g 2 L2 (0,T) T + 0 gvx x=1 dt. (3.2.20) Next, we use here trace theory and again (3.2.10) to obtain vx x=1 ≤ C vx H 1 (Ω) ≤C v H 2 (Ω) =ᏻ g L2 (0,T) . Finally, substituting (3.2.21) in (3.2.20) yields (3.2.12), as desired. (3.2.21) I. Lasiecka and R. Triggiani 1079 Step 4 (proof of (iii3)). Parts (iii1) and (iii2) are very standard. Then, returning to identity (3.2.18) as specialized to the wproblem, hence with g = wt x=1 as in (3.2.14) and thus with T 0 2 E(t) = w(t),wt (t) Y , gwx x=1 dt ≥ −cT T 0 (3.2.22) 2 wt x=1 dt, recalling (3.2.21) with v replaced by w, and g = wt x=1 , we obtain T 0 2 wt x=1 dt ≥ c1 T 0 E(t)dt − c2 E(T) + E(0) (3.2.23) T ≥ c̃1 E(t)dt − c̃2 E(T) 0 ≥ c̃1 T − c̃2 E(T), (3.2.24) (3.2.25) and (3.2.25) is nothing but a rewriting of (3.2.13) with cT = c̃1 T − c̃2 > 0 for T suﬃciently large. To go from (3.2.23) to (3.2.24) and to (3.2.25), we have used the usual dissipativity identity. 4. Classes of PDE satisfying the regularity property (2.14): B ∗ L ∈ ᏸ(L2 (0,T;U)) Documenting and reinforcing the content of Section 1, our goal in the present paper is now twofold. (i) First, we provide (in the present section) several, multidimensional nontrivial hyperbolic and Petrowskitype mixed problems that indeed satisfy the regularity property (2.14) on B ∗ L. In this respect, our message is, in turn, that for each of the illustrations given below, the fact that B∗ L fulfills property (2.14) was either already noted explicitly in the literature or else is a builtin block in the proof of optimal regularity, exact controllability, and particularly, uniform stabilization of such systems—which is the ultimate goal in Claim 2.3. (ii) Second, we document in Sections 5, 6, 7, and 8 that property (2.14) fails to hold true for B∗ L in the case of several other hyperbolic Petrowskitype PDE systems where, however, uniform stabilization has long been proved, by PDE energy methods, in the literature. This says that property (2.14) for B∗ L is far from being necessary in Claim 2.3. That is to say, property (2.14) is not a precondition for either controllability or stabilization of these problems. Points (i) and (ii) call into question the “usefulness” of a result such as Claim 2.3, as elaborated before. Remark 4.1. Due to constraints on the overall length of the paper, to make our main point of the present section (Section 4)—singling out relevant classes of PDEs where the regularity (2.14) for B ∗ L holds true—it will be expedient to deemphasize generality. Thus, in our results below, we will deal primarily with 1080 Regularity of B∗ L canonical PDEs and with control acting, possibly, on the whole boundary, even though a much greater degree of generality is well known. In particular, we will not necessarily insist on the case of variable coeﬃcients PDEs, and refer instead to [3, 11, 47, 68, 80, 81], and so forth. 4.1. Firstorder hyperbolic systems with boundary control. This section considers a general firstorder hyperbolic system, which may be nonsymmetric and nondissipative, and is defined on a suﬃciently smooth bounded domain of arbitrary dimension. The control function acts through the boundary conditions. The treatment below follows closely [45, Chapter 10, Section 10.6]. The dynamics. Let Ω ⊂ Rm be an open bounded domain with smooth boundary Γ. In Ω, we consider a diﬀerential operator of the form A(x,∂)y ≡ m A j (x)∂ j y + A0 (x)y, (4.1.1) j =1 where y(x) is a kvector and ∂ j = ∂/∂x j . The coeﬃcients A j , A0 are smooth k × k matrixvalued functions defined on the open bounded domain Ω ⊂ Rm . We assume the following hypotheses throughout: (h1) A(x,∂) is strictly hyperbolic; that is, the matrix mj=1 A j (x)ξ j has k distinct real eigenvalues for all ξ = [ξ1 ,...,ξm ] ∈ Rm \ {0} and x ∈ Ω̄; (h2) the boundary Γ is noncharacteristic; that is, detAν (x) = 0 for x ∈ Γ, where Aν (x) ≡ mj=1 A j ν j (x); ν = (ν1 ,...,νm ) the inward unit normal. It follows from (h1) and (h2) that after a smooth change of coordinates, we may assume that Aν is of the following form: A− Aν = ν 0 a1 . . . 0 , A+ν .. a +1 . = .. 0 ··· .. . ··· < 0, . ··· 0 A+ν 0 .. . a2 A−ν = ··· a (4.1.2) 0 .. . > 0. ak Accordingly, any vector v ∈ Rk will be split consistently as v = [v− ,v+ ] with v− = [v1 ,...,v ] and v+ = [v +1 ,...,vk ]. Boundary conditions are imposed with the aid of a boundary operator M(x), which is a smooth × k matrixvalued function, where stands for the number of negative eigenvalues of Aν . We assume further the following hypotheses: I. Lasiecka and R. Triggiani 1081 (h3) rankM(x) = , x ∈ Γ; (h4) (Kreiss condition) the frozen (at the boundary point) mixed problem has no eigenvalues or generalized eigenvalues with nonnegative real parts. More specifically (h4) means that after making a local change of coordinates which maps Ω into the halfspace {x ∈ Rm ; x1 > 0}, the constant coeﬃcient problem that arises by freezing A j , j = 1,...,m, and M at the boundary point and setting A0 = 0, that is, yt − A1 yx1 − m A j yx j = 0, x1 > 0, (4.1.3a) j =2 M y = 0 at x1 = 0, (4.1.3b) has no eigenvalues or generalized eigenvalues with nonnegative real parts. For the halfspace problem (4.1.3), we have Aν = A1 , thus A1 is invertible by (h2). For a more detailed description of this condition we refer the reader to the fundamental papers [16, 64]. Convention. To streamline the notation, we will write L2 (Γ) and L2 (Ω) to mean, and L2 (Σ) and L2 (Q) to mean, respectively, L2 (Γ; R ) and L2 (Ω; Rk ), and so forth, respectively, L2 (0,T;L2 (Γ; R )) and L2 (0,T;L2 Ω; Rk )), without further mention, where Σ = (0,T] × Γ, Q = (0,T] × Ω, for a fixed 0 < T < ∞. The mixed problem for the firstorder hyperbolic system which we consider is then yt = A(x,∂)y in Q ≡ (0,T] × Ω, (4.1.4a) y(0, ·) = y0 (x) in Ω, M(x) y = g (4.1.4b) in Σ ≡ (0,T] × Γ, (4.1.4c) where the boundary control g ∈ L2 (Σ) = L2 (0,T;L2 (Γ; R )). Regularity theory for problem (4.1.4) with g ∈ L2 (Σ). A complete wellposedness theory for nonsymmetric, noncharacteristic firstorder hyperbolic systems as in (6.1.5) has been provided in [16], augmented by a note in [63], and completed in [64]. Theorem 4.2 [64, page 272]. Under hypotheses (h1), (h2), (h3), and (h4), for any T > 0, assume y0 ∈ L2 (Ω), g ∈ L2 0,T;L2 (Γ) . (4.1.5) Then, the unique solution of problem (4.1.4) satisfies y ∈ C [0,T];L2 (Ω) , continuously. y Γ ∈ L2 0,T;L2 (Γ) (4.1.6) 1082 Regularity of B∗ L Next, we single out the result of the homogeneous case g ≡ 0 for problem (4.1.4) in a form which will be useful in the sequel. To this end, we introduce the operator A, by setting Ah = A(x,∂)h : L2 (Ω) ⊃ Ᏸ(A) −→ L2 (Ω), (4.1.7a) Ᏸ(A) = h ∈ L2 (Ω) : A(x,∂)h ∈ L2 (Ω); MhΓ = 0 , (4.1.7b) where A(x,∂) is the diﬀerential operator in (4.1.1). Corollary 4.3. Under the above hypotheses (h1), (h2), (h3), and (h4), the operator A in (4.1.7), corresponding to problem (4.1.4) with g ≡ 0, is the generator of a s.c. semigroup eAt on L2 (Ω), t ≥ 0. Abstract setting for problem (4.1.4). To put problem (4.1.4), (4.1.5) into the abstract model (1.1), we need the following operators and spaces. First, we need the operator A defined by (4.1.7), which generates a s.c. semigroup eAt on the space Y = L2 (Ω). (4.1.8) Second, we introduce the “Dirichlet” map (natural extension from the boundary Γ into the interior Ω, which uniquely solves (a suitable translation of) the corresponding static problem), defined by A(x,∂)v − λv = 0 Dλ g = v ⇐⇒ MvΓ = g in Ω, in Γ, (4.1.9) for a suitably large constant λ ≥ 0, as justified by the following result. Lemma 4.4. With reference to problem (4.1.9), there exists a constant λ ≥ 0, henceforth kept fixed, such that problem (4.1.9) admits a unique solution v = Dλ g ∈ L2 (Ω) for g ∈ L2 (Γ). Moreover, the following estimate holds true: there is a constant Cλ > 0 depending on λ such that Dλ g L2 (Ω) + Dλ g Γ L2 (Γ) ≤ Cλ g L2 (Γ) . (4.1.10) Thus, Dλ : continuous L2 (Γ) −→ L2 (Ω), ∗ Dλ : continuous L2 (Ω) −→ L2 (Γ), where Dλ∗ is the adjoint (Dλ g,v)L2 (Ω) = (g,Dλ∗ v)L2 (Γ) . (4.1.11) (4.1.12) I. Lasiecka and R. Triggiani 1083 Third, we return to problem (4.1.4), and by virtue of definition (4.1.9) of Dλ , λ henceforth as in Lemma 4.4, we rewrite it as in (0,T] × Ω, yt = A(x,∂) − λ y − Dλ g + λy (4.1.13a) y(0,x) = y0 (x) in Ω, M y − Dλ g Γ = 0 (4.1.13b) in (0,T] × Γ, (4.1.13c) or abstractly, by (4.1.7), as yt = (A − λI) y − Dλ g + λy in L2 (Ω), (4.1.14) y(0) = y0 ∈ L2 (Ω). Moreover, extending the original operator A in (4.1.7) by A : L2 (Ω) → [Ᏸ(A∗ )] , that is, extending the original A in (4.1.7) to its double adjoint A∗∗ , we obtain, from (4.1.14), in Ᏸ A∗ yt = Ay − (A − λI)Dλ g , (4.1.15) y(0) = y0 ∈ L2 (Ω), which is precisely the abstract model (1.1), with A as in (4.1.7), and B = −(A − λI)Dλ : continuous U = L2 (Γ) −→ Ᏸ A∗ − λI , (4.1.16a) or equivalently, (A − λI)−1 B = −Dλ : continuous L2 (Γ) −→ L2 (Ω), (4.1.16b) as guaranteed by (4.1.11). Finally, with B ∈ ᏸ(U;[Ᏸ(A∗ − λI)] ) and so B∗ ∈ ᏸ(Ᏸ(A∗ );U) after identifying [Ᏸ(A∗ − λI)] with Ᏸ(A∗ ), we compute B∗ as B ∗ = −Dλ∗ A∗ − λI : continuous Ᏸ A∗ −→ U. (4.1.17) A more explicit representation of B∗ is given by the next result. Lemma 4.5. With reference to (4.1.17), B∗ y = −Dλ∗ A∗ − λI y = A−ν y − Γ , y ∈ Ᏸ A∗ , (4.1.18) where A−ν is defined in (4.1.2) and the component y − of y consisting of the first coordinates is likewise defined below (4.1.2). The main result of the present section is the following theorem. 1084 Regularity of B∗ L Theorem 4.6. With reference to the mixed problem (4.1.4) with y0 = 0, (recall the definition of L in (1.2b)) ∈ L2 0,T;L2 (Γ) continuously in g ∈ L2 0,T;L2 (Γ) . B ∗ Lg = B ∗ y t; y0 = 0 = A−ν y − t; y0 = 0 Σ (4.1.19) Proof. The regularity in (4.1.19) stems from (4.1.17) and (4.1.6) of Theorem 4.2. 4.2. Schrödinger equation with Dirichlet boundary control. The present section deals with the (multidimensional) Schrödinger equation with Dirichletboundary control. The main goal is threefold: (i) to recall from the literature of 1992 the main results of (optimal) regularity, exact controllability, and uniform stabilization; (ii) to point out that such literature also essentially contains the result that the operator B∗ L satisfies the required regularity assumption (2.14) which is, in fact, a builtin block into the process of studying the three related problems mentioned in point (i); (iii) to conclude, accordingly, that the use of Claim 2.3—based on exact controllability of {A,B} and regularity of B∗ L—to obtain uniform stabilization of {A,B} is neither enlightening nor technically and conceptually convenient. Openloop and closedloop feedback dissipative systems. Let Ω be an open bounded domain in Rn with suﬃciently smooth C 1 boundary Γ. We consider the following openloop problem of the Schrödinger equation defined on Ω, with Dirichletcontrol u ∈ L2 (0,T;L2 (Γ)) ≡ L2 (Σ) and its corresponding boundary dissipative version: yt = −i∆y, wt = −i∆w y(0, ·) = y0 , w(0, ·) = w0 y Σ = u ∈ L2 (Σ), wΣ = i in Q, (4.2.1a) in Ω, −1 ∂ A w ∂ν in Σ, (4.2.1b) (4.2.1c) with Q ≡ (0,T] × Ω, Σ ≡ (0,T] × Γ. Moreover, the operator A is defined below in (4.2.4) as Aw = −∆w, Ᏸ(A) = H 2 (Ω) ∩ H01 (Ω). Regularity, exact controllability of the yproblem, and uniform stability of the wproblem. Paper [39] gives a full account of the (optimal) regularity and exact controllability of the openloop yproblem in (4.2.1) as well as the uniform stabilization of the corresponding closedloop wproblem. Regularity issues of interest here are also contained in [20, pages 175–177] and [45, Chapter 10]. I. Lasiecka and R. Triggiani 1085 Theorem 4.7 (regularity [39, Theorem 1.2]). Regarding the yproblem (4.2.1) with y0 = 0, for each T > 0, the following interior regularity holds true (recall the definition of L in (1.2b)): the map L : u −→ Lu = y is continuous L2 (Σ) −→ C [0,T];H −1 (Ω) . (4.2.2) Theorem 4.8 (exact controllability [39, Theorem 1.3]). Let T > 0. Given y0 ∈ H −1 (Ω), there exists u ∈ L2 (0,T;L2 (Γ)) such that the corresponding solution to the yproblem (4.2.1) satisfies y(T) = 0. Theorem 4.9 (uniform stabilization [39, Theorems 1.4 and 1.5]). With reference to the wproblem in (4.2.1), (i) the map w0 ∈ H −1 (Ω) → w(t) defines a s.c. contraction semigroup on [Ᏸ(A1/2 )] ≡ H −1 (Ω); (ii) wΣ ∈ L2 (0, ∞;L2 (Γ)) continuously for w0 ∈ H −1 (Ω); (iii) there exist constants M ≥ 1, δ > 0 such that w(t) ≤ Me−δt w0 , t ≥ 0, (4.2.3) with · the H −1 (Ω)norm. Needless to say, in line with the content of Section 1, all three theorems above (as well as their generalizations alluded to in Remark 4.1) are obtained by PDE hard analysis energy methods (not by soft analysis methods). The most challenging result to prove is Theorem 4.9 on uniform stabilization; this, in addition, requires a shift of topology from H −1 (Ω) (the space of the final result) to H01 (Ω) (the space where the energy method works). This shift of topology is implemented by a change of variable; this is the same change of variable that is noted below in (4.2.8), and that is needed to establish the desired regularity of B∗ L. Abstract model of yproblem. We let Aψ = −∆ψ, Ᏸ(A) = H 2 (Ω) ∩ H01 (Ω); ϕ ≡ Dg ⇐⇒ ∆ϕ = 0 in Ω; ϕΓ = g on Γ . (4.2.4) Then, the abstract model (in additive form) of the yproblem (4.2.1) is [39, equation (1.2.2)] ẏ = iAy − iADu, y(0) = y0 ∈ Y ≡ Ᏸ A1/2 ≡ H −1 (Ω). (4.2.5) Comparing with (1.1), we have B = −iAD hence B∗ = iD∗ , (4.2.6) 1086 Regularity of B∗ L where the ∗ for B and D refer actually to diﬀerent topologies, as the following computation yielding B∗ in (4.2.6) shows: let u, y ∈ Y , then (Bu, y)Y = −i(ADu, y)[Ᏸ(A1/2 )] = −i(Du, y)L2 (Ω) = −i u,D∗ y = u,B ∗ y L2 (Γ) . L2 (Γ) (4.2.7) The operator B∗ L. With reference to the yproblem in (4.2.1), we will show that ∂z , B Lu = B y t; y0 = 0 = −i ∂ν Γ z(t) ≡ A−1 y t; y0 = 0 ∈ C [0,T];Ᏸ A1/2 ≡ H01 (Ω) , ∗ ∗ (4.2.8a) (4.2.8b) where z satisfies the following dynamics—abstract equation and corresponding PDEmixed problem: zt = −i∆z − iDu in Q; ż = iAz − iDu, (4.2.9a) z(0, ·) = z0 = 0 in Ω; in Σ. zΣ ≡ 0 (4.2.9b) (4.2.9c) Indeed, to obtain (4.2.8) and (4.2.9), one uses the definitions in (4.2.8) and (4.2.6), B∗ Lu ≡ B∗ y t; y0 = 0 = iD∗ AA−1 y t; y0 = 0 = iD∗ Az(t) = −i ∂z , ∂ν (4.2.10) as well as the usual property D∗ A = −∂/∂ν on Ᏸ(A1/2 ) = H01 (Ω) from [39, equation (1.21)]. The abstract zequation in (4.2.9) follows from the abstract yequation in (4.2.5) after applying A−1 and using the definition of z(t) in (4.2.8b). Since u(t) ∈ H01 (Ω), then the abstract zequation yields its PDE version in (4.2.9b). Theorem 4.10. With reference to (4.2.8), B∗ L : continuous L2 0,T;L2 (Γ) −→ L2 0,T;L2 (Γ) ; (4.2.11a) equivalently, with reference to (4.2.10), the map u −→ ∂z is continuous L2 0,T;L2 (Γ) −→ L2 0,T;L2 (Γ) . ∂ν (4.2.11b) This result (4.2.11) is explicitly stated and proved in [20, Proposition 4.2 and page 175ﬀ.], where the regularity (4.2.8) for z is established in [20, equation (4.14)] by energy methods (via the multiplier h · ∇z̄,hΓ = ν) without first establishing the yregularity (4.2.2) in Theorem 4.7. This result (4.2.11) also follows from [39, identity (2.1), Lemma 2.1] (built with the multiplier h · ∇z̄) I. Lasiecka and R. Triggiani 1087 with f = −iDu ∈ L2 (0,T;Ᏸ(A1/4− )) and the a priori regularity z ∈ C([0,T]; H01 (Ω)) in (4.2.8) for z; the latter uses, by contrast, the yregularity (4.2.2) in Theorem 4.7. The two avenues chosen in [20, 39] are very closely related and based on the same energy method and duality. The expression “double duality” was used in [20] as duality was used twice. Comparison between establishing Theorem 4.9(iii)—uniform stabilization—directly or else via Claim 2.3. (1) According to [39], in order to establish the exponential energy decay (4.2.3) directly, one needs the following ingredients: (1a) (easier step) the properties of generation and feedback regularity listed in Theorem 4.9(i) and (ii); this is a readily accomplished application of the LumerPhillips theorem; (1b) (harder step) application of energy methods by use of multipliers h · ∇ p̄ and p̄ div h to the pproblem, defined by p ≡ A−1 w ∈ C([0,T], H01 (Ω)) [39, equation (4.6)], to obtain—in the end—the estimate [39, equation (4.16)] T 2 ∂p dΣ ≥ cT E p (T), ∂ν 0 Γ (4.2.12) with E p (·) being the “energy” (square of H 1 (Ω)norm) of p. (2) In order to establish the exponential decay (4.2.3) by virtue of Claim 2.3, one needs the following ingredients: (2a) proof of the regularity property (2.8) for B∗ L. According to [39] or [20], this is accomplished as follows: (2aI) [20] either by applying energy methods (multiplier h · ∇z̄) to the z problem (4.2.9) to obtain first the a priori regularity z ∈ C([0,T];H01 (Ω)) and then the regularity trace inequality (specialization of (1.4)) T 2 ∂z dΣ ≤ cT Ez (0), ∂ν 0 Γ (4.2.13) (2aII) or else [39] by applying energy methods (multipliers h · ∇φ̄, φ̄ div h) to the dual homogeneous φproblem iφt = ∆φ in Q; φ(0, ·) = φ0 ∈ H01 (Ω), φΣ ≡ 0, (4.2.14) to obtain the same inequality (4.2.13) this time for φ, hence by duality y ∈ C([0,T];H −1 (Ω)) and hence z(t) = A−1 y(t; y0 = 0) ∈ C([0,T];H01 (Ω)) (as in (2aI)); and then read oﬀ inequality (4.2.13) from identity [39, equation (2.1)] in z, where one exploits the a priori regularity of z; 1088 Regularity of B∗ L (2b) establishing exact controllability of the yproblem, that is, continuous observability of the dual φproblem (4.2.14), again by energy methods, to obtain T 2 ∂φ dΣ ≥ cT Eφ (T), ∂ν 0 Γ (4.2.15) (specialization of (1.6)), where Eφ (·) is the energy (square of H 1 (Ω)norm) of φ. Conclusion. We submit that the direct approach in [39] is surely more desirable and amenable than the application of Claim 2.3. 4.3. EulerBernoulli plate with clamped boundary controls. Case 1: Neumann control. The present subsection deals with the EulerBernoulli plate equation with “clamped” boundary controls (in any dimension), while “hinged” boundary controls will be considered in Section 4.4. In either case, the corresponding results of optimal regularity, exact controllability, and uniform stabilization—all obtained by PDE energy methods—have been known for over 10 years. Moreover, we claim that the regularity result B∗ L ∈ ᏸ(L2 (0,T;U)) is also true for each of the aforementioned EB mixed problems. This result is contained in the treatments of the literature cited as a builtin block, rather than singled out in an explicit statement. Below we will extract the necessary details from the literature. Ultimately, the message of the present as well as of the next subsection is the same as that of Section 4.2 dealing with the Schrödinger equation: that verifying the key assumptions of Claim 2.3—the regularity B∗ L ∈ ᏸ(L2 (0,T;U)) and the exact controllability of {A,B}—is not any easier—on the contrary!—than establishing uniform stabilization of {A,B} directly. Thus, it pays oﬀ, possibly by much, to tackle uniform stabilization of {A,B} directly, rather than attempting to apply the tortuous route of Claim 2.3. At any rate, in all of these results, PDE (hard analysis) energy methods are the key and critical tools, not soft methods. For lack of space, and to limit repetitions, we will state the three fundamental results of optimal regularity, exact controllability, and uniform stabilization, and next establish the soughtafter regularity of B∗ L within the context of the treatments of the three aforementioned problems. Openloop and closedloop feedback dissipative systems. Let Ω be an open bounded domain in Rn (n = 2, in the physical case of plates) with suﬃciently smooth boundary Γ. We consider the following openloop problem of the EulerBernoulli equation defined on Ω, with Neumann boundary control g2 ∈ L2 (0,T; L2 (Γ)) ≡ L2 (Σ), as well as its corresponding boundary dissipative version: vtt + ∆2 v = 0; v(0, ·) = v0 , vt (0, ·) = v1 ; wtt + ∆2 w = 0 in Q; w(0, ·) = w0 , wt (0, ·) = w1 (4.3.1a) in Ω; (4.3.1b) I. Lasiecka and R. Triggiani wΣ ≡ 0 in Σ; vΣ ≡ 0; ∂v ∂w = g2 ; = ∆ Ꮽ−1 wt Σ ∂ν ∂ν Σ Σ 1089 (4.3.1c) in Σ, (4.3.1d) with Q = (0,T] × Ω, Σ = (0,T] × Γ. Moreover, the operator Ꮽ is defined below in (4.3.6) as Ꮽw = ∆2 w, Ᏸ(Ꮽ) ≡ H 4 (Ω) ∩ H02 (Ω). Regularity, exact controllability of the vproblem, and uniform stabilization of the wproblem. References for this subsection include [29, 53, 54] for the vproblem and [61] for the wproblem. These references give a full account of these three problems. We begin by introducing the (state) space (of optimal regularity) X ≡ L2 (Ω) × Ᏸ Ꮽ1/2 Ᏸ Ꮽ 1/2 Ᏸ Ꮽ 1/2 , ≡ H −2 (Ω), (4.3.2) ≡ H02 (Ω). Theorem 4.11 (regularity [53, 54]). Regarding the vproblem (4.3.1), with y0 = {v0 ,v1 } = 0, the following regularity result holds true for each T > 0 (recall the definition of L in (1.2b)): the map L : g2 −→ Lg2 = v,vt is continuous L2 (Σ) −→ C [0,T];X ≡ L2 (Ω) × H −2 (Ω) . (4.3.3) Theorem 4.12 (exact controllability [54, 55, 61]). Given any initial condition {v0 ,v1 } ∈ X and T > 0, there exists a g2 ∈ L2 (Σ) such that the corresponding solution of the vproblem (4.3.1) satisfies {v(T),vt (T)} = 0. Theorem 4.13 (uniform stabilization [61]). With reference to the wproblem (4.3.1), (i) the map {w0 ,w1 } ∈ X = L2 (Ω) × [Ᏸ(Ꮽ1/2 )] → {w(t),wt (t)} defines a s.c. contraction semigroup eAt on X; (ii) its Neumann trace satisfies ∂w = ∆ Ꮽ−1 wt continuously in w0 ,w1 ∈ X; Σ ∈ L2 0, ∞;L2 (Γ) ∂ν Σ (4.3.4) (iii) there exist constants M ≥ 1, δ > 0 such that w(t) At w0 −δt w0 = e ≤ Me , wt (t) w1 w1 X X X t ≥ 0. (4.3.5) Regularity of B∗ L 1090 Again, needless to say, in line with the content of Section 1, all the three theorems above are obtained by PDE hard analysis energy methods (not by soft analysis methods). As usual, the most challenging result to prove is Theorem 4.13 on uniform stabilization; this problem, in addition, requires a shift of topology from X ≡ L2 (Ω) × H −2 (Ω) (the space of the final result) to H02 (Ω) × L2 (Ω) (the space where the energy method works). This shift of topology is implemented by a change of variable: this is the same change of variable noted below in (4.3.10), that is needed to establish the desired regularity of B∗ L. Abstract model of vproblem. We let Ꮽψ = ∆2 ψ, Ᏸ(Ꮽ) = H 4 (Ω) ∩ H02 (Ω), G2 : H s (Γ) −→ H s+3/2 (Ω), ∂ϕ = g2 . ϕ = G2 g2 ⇐⇒ ∆ ϕ = 0 in Ω; ϕΓ = 0, ∂ν Γ 2 s ∈ R, (4.3.6a) (4.3.6b) Then, the secondorder, respectively, firstorder, abstract models (in additive form) of the vproblem (4.3.1) are [29, 61] vtt + Ꮽv = ᏭG2 g2 , A= 0 I , −Ꮽ 0 (4.3.7) 0 , ᏭG2 g2 Bg2 = v d v + Bg2 , =A vt dt vt x1 = G∗ 2 x2 , x2 B∗ (4.3.8) where ∗ for B and G2 refers actually to diﬀerent topologies. With B∗ defined by (Bg2 ,x)X = (g2 ,B ∗ x)L2 (Γ) with respect to the Xtopology, we readily find the expression in (4.3.8) since the second component of the space X is [Ᏸ(Ꮽ1/2 )] . The operator B ∗ L. With y0 = {v0 ,v1 } = 0, we will show that ∗ B Lg2 = B ∗ v t; y0 = 0 = G∗ 2 vt t; y0 = 0 = − ∆z(t) Γ , vt t; y0 = 0 z(t) ≡ Ꮽ−1 vt t; y0 = 0 ∈ C [0,T];Ᏸ Ꮽ1/2 ≡ H02 (Ω) (4.3.9) continuously in g2 ∈ L2 (Σ). (4.3.10) The new variable z(t) defined in (4.3.10) satisfies the following dynamics: abstract equation and corresponding PDEmixed problem ztt + Ꮽz = G2 g2t , ztt + ∆ z = G2 g2t 2 z(0, ·) = z0 = 0, zΣ ≡ 0, (4.3.11a) in Q, zt (0, ·) = z1 ∂z ≡0 ∂ν Σ (4.3.11b) in Ω, in Σ. (4.3.11c) (4.3.11d) I. Lasiecka and R. Triggiani 1091 Indeed, to establish (4.3.9) (right), (4.3.10), one uses the definition in (4.3.9) (left), followed by (4.3.8) for B∗ , to obtain B∗ Lg2 = G∗2 vt t; y0 = 0 = G∗2 ᏭᏭ−1 , (4.3.12) vt t; y0 = 0 = G∗2 Ꮽz(t) = −∆z(t)Γ , where, in the last step, we have recalled the usual property G∗2 Ꮽ = −∆·Γ on Ᏸ(Ꮽ1/2 ) ≡ H02 (Ω) [61, equation (1.11)], [4, equation (1.20), page 49]. The abstract zequation is readily obtained from the abstract vequation after applying throughout Ꮽ−1 and d/dt to it and using the definition of z(t) in (4.3.10), whose a priori regularity in (4.3.10) follows from (4.3.3) and (4.3.2). Since z(t) ∈ H02 (Ω), both boundary conditions are satisfied and the abstract zequation leads to its corresponding PDE version. By (4.3.19) below, and within the class (4.3.20), we can take z1 = 0. Remark 4.14. As already noted, the change of variable vt → z in (4.3.10) and the resulting zproblems in (4.3.11a) are precisely the same that were used in [61, Section 2.1] in obtaining the uniform stabilization, Theorem 4.13, directly; the only diﬀerence is the specific form of the righthand side term (thus, the letter p was used in [61, equation (2.11)], while the letter z is used now for a closely related, yet not identical system). In both cases, however, a timederivative term occurs (in our case G2 g2t ), which will require—in [61] as well as in Step 6 in the proof of Lemma 4.16 below—an integration by parts in t to obtain the soughtafter estimate. Theorem 4.15. With reference to (4.3.9), B∗ L : continuous L2 0,T;L2 (Γ) −→ L2 0,T;L2 (Γ) ; (4.3.13a) equivalently, with reference to (4.3.11a), the map g2 −→ ∆zΣ is continuous L2 0,T;L2 (Γ) −→ L2 0,T;L2 (Γ) . (4.3.13b) We will see below in the proof that this result, though not explicitly stated, is builtin in the treatments of [61] to prove Theorem 4.13. Proof Step 1 (basic energy identity). We return to the basic identity of the energy methods [61, equation (2.24), page 287], which we use with a vector field h satisfying (as usual in obtaining trace regularity results [22]) the additional condition hΓ = ν. Thus, with h · ν = 1 on Γ, for the solution z of a priori regularity 1092 Regularity of B∗ L z ∈ C([0,T];H02 (Ω)) as in (4.3.10), we have 1 2 RHS1 = Q Σ (∆z)2 dΣ = RHS1 + RHS2 + b0,T , ∆z div H + H T ∇z dQ + RHS2 = − Q b0,T = zt ,h · ∇z T Ω 0 1 2 G2 g2t h · ∇z dQ − 1 2 Q z∆z∆(div h)dQ, (4.3.15) G2 g2t z div hdQ, (4.3.16) T 1 zt ,z div h Ω . 0 2 (4.3.17) Q + (4.3.14) Step 2 (estimate for RHS1 ). From the a priori regularity (4.3.10) for z, we immediately find that 2 RHS1 = ᏻ g2 L2 (Σ) ∀g2 ∈ L2 (Σ). (4.3.18) Step 3 (regularity of zt ). To handle RHS2 (by integration by parts in t, precisely as in the proof of the uniform stabilization theorem (Theorem 4.13) given in [61, pages 283–289]), we need the regularity of zt . By (4.3.10) and the vequation (4.3.7), we obtain zt (t) = Ꮽ−1 vtt = Ꮽ−1 − Ꮽv + ᏭG2 g2 = −v + G2 g2 ∈ L2 0,T;L2 (Ω) continuously in g2 ∈ L2 (Σ), (4.3.19) by recalling that v ∈ C([0,T];L2 (Ω)) (see (4.3.3)) and that G2 g2 ∈ L2 (0,T;H 3/2 (Ω)), by virtue of (4.3.6a) with s = 0 on G2 and g2 ∈ L2 (Σ). Step 4 (estimates for RHS2 and b0,T for smoother g2 ). Henceforth, to estimate both RHS2 and b0,T , we will at first take g2 within the smoother class g2 ∈ C [0,T];L2 (Γ) , g2 (0) = g2 (T) = 0. (4.3.20) This initial restriction is dictated by the fact that zt in (4.3.19) is only in L2 in time. Lemma 4.16. In the present setting, 2 RHS2 = ᏻ g2 L2 (Σ) , 2 b0,T = ᏻ g2 L2 (Σ) , (4.3.21) for all g2 in the class (4.3.20). Step 5 (proof of (4.3.21) for b0,T ). First from (4.3.10), (4.3.3), and (4.3.2), we have, since vt (0) = v1 = 0, z(0) = 0, z(T) = Ꮽ−1 vt T; y0 = 0 ∈ Ᏸ Ꮽ1/2 ≡ H02 (Ω) continuously in g2 ∈ L2 (Σ). (4.3.22) I. Lasiecka and R. Triggiani 1093 Next, for g2 in the class (4.3.20) used in (4.3.19), we compute, since v(0) = v0 = 0, zt (0) = 0, zt (T) = −v(T) ∈ L2 (Ω) continuously in g2 ∈ L2 (Σ), (4.3.23) where the regularity follows from (4.3.3). Using (4.3.22) and (4.3.23) in (4.3.17), we readily obtain, as desired, b0,T = zt (T),h · ∇z(T) Ω+ 2 1 zt (T),z(T)div h Ω = ᏻ g2 L2 (Σ) 2 (4.3.24) for all g2 in the class (4.3.20). Thus, (4.3.21) (right) is proved. Step 6 (proof of (4.3.21) for RHS2 ). The most critical term of RHS2 to estimate is the first term in (4.3.16). As in the direct proof of the uniform stabilization theorem (Theorem 4.13) given in [61, page 287], we integrate by parts in t, with g2 in the class (4.3.20), thus obtaining Q G2 g2t h · ∇z dQ = z dΩ * G2 g2 h ·∇ Ω T 0 − Q G2 g2 h · ∇zt dQ, (4.3.25) where the first term on the righthand side vanishes since g2 (0) = g2 (T) = 0. Moreover, the usual divergence theorem [61, equation (2.31), page 288] yields, with h · ν = 1, T 0 Ω G2 g2 h · ∇zt dΩdt = T 0 − *· νdΓdt − G2 g2 zt h Γ T Ω 0 T 0 Ω zt h · ∇ G2 g2 dΩdt (4.3.26) 2 G2 g2 zt div hdΩdt = ᏻ g2 L2 (Σ) for all g2 in the class (4.3.20). The indicated estimate in terms of g2 in (4.3.26) follows by virtue of zt ∈ L2 (0,T;L2 (Ω)) (see (4.3.19)), G2 g2 ∈ L2 (0,T;H 3/2 (Ω)) by (4.3.6a) with s = 0 on G2 and thus ∇(G2 g2 ) ∈ L2 (0,T;H 1/2 (Ω)), all bounded by the L2 (Σ)norm of g2 . A similar estimate as (4.3.26) holds true, a fortiori, for the more regular second term in the definition of RHS2 in (4.3.16). Accordingly, we obtain (4.3.21) for RHS2 . Step 7. We can then extend estimates (4.3.21) for RHS2 and b0,T to all g2 ∈ L2 (Σ), by density, starting from the class (4.3.20). Using these extended estimates as well as (4.3.18) in (4.3.14), we finally obtain Σ 2 (∆z)2 dΣ = ᏻ g2 L2 (Σ) ∀g2 ∈ L2 (Σ), and (4.3.13b) is proved. The proof of Theorem 4.15 is complete. (4.3.27) 1094 Regularity of B∗ L 4.4. EulerBernoulli plate with clamped boundary controls. Case 2: Dirichlet control Openloop and closedloop feedback dissipative systems. In the notation of Case 1 above, we consider the EulerBernoulli equation defined on Ω, with Dirichlet boundary control g1 ∈ L2 (0,T;L2 (Γ)), in both openloop and closedloop dissipative form: vtt + ∆2 v = 0; v(0, ·) = v0 , vt (0, ·) = v1 ; vΣ = g1 ; ∂v ∂ν Σ wtt + ∆2 w = 0 in Q, w(0, ·) = w0 , wt (0, ·) = w1 (4.4.1a) in Ω, ∂∆ Ꮽ wt wΣ = − in Σ, ∂ν Σ ∂w = 0 in Σ. = 0; ∂ν (4.4.1b) −3/2 Σ (4.4.1c) (4.4.1d) Here the operator Ꮽ is the same as in Section 4.3, (4.3.6). Regularity, exact controllability of the vproblem, and uniform stabilization of the wproblem. References for this subsection are [4, 29, 54, 55]. These references give a full account of these three problems. We begin by introducing the (state) space (of optimal regularity) × Ᏸ Ꮽ3/4 ≡ H −1 (Ω) × V , ∂f 3 V = f ∈ H (Ω) : f Γ = =0 . Y ≡ Ᏸ Ꮽ1/4 ∂ν (4.4.2) Γ Theorem 4.17 (regularity [54] and [29, Theorem 1.0, page 331]). Regarding the vproblem (4.4.1), with y0 = {v0 ,v1 } = 0, the following regularity result holds true for each T > 0 (recall the definition of L in (1.2b)): the map L : g1 −→ Lg1 = v,vt is continuous L2 (Σ) −→ C [0,T];Y ≡ H −1 (Ω) × V . (4.4.3) Theorem 4.18 (exact controllability [29, Theorems 1.1 and 1.4], [4, Theorem 1.3, Remark 1.1]). Assume that there exists a coercive vector field h(x) ∈ [C 2 (Ω)]n (in particular, a radial vector field h(x) = x − x0 , for some x0 ∈ Rn ), such that h·ν ≥ 0 on Γ. (4.4.4) Given any initial condition {v0 ,v1 } ∈ Y and T > 0, there exists a g1 ∈ L2 (Σ) such that the corresponding solution of the vproblem (4.4.1) satisfies {v(T),vt (T)} = 0. Theorem 4.19 (uniform stabilization [4, Theorem 1.3, page 51]). With reference to the wproblem (4.4.1), (i) the map {w0 ,w1 } ∈ Y ≡ H −1 (Ω) × V → {w(t),wt (t)} defines a s.c. contraction semigroup eAt on Y ; I. Lasiecka and R. Triggiani 1095 (ii) the following trace result holds true: wΣ = − ∂∆ Ꮽ−3/2 wt ∈ L2 0, ∞;L2 (Γ) ∂ν Σ continuously in w0 ,w1 ∈ Y ; (4.4.5) (iii) moreover, assume the geometrical condition of Theorem 4.18. Then, there exist constants M ≥ 1, δ > 0 such that w(t) At w0 −δt w0 = e ≤ Me , wt (t) w1 w1 Y Y t ≥ 0. (4.4.6) Y We stress again, in line with the content of Section 1, that all three theorems above are obtained by PDE hard analysis energy methods (not by soft analysis methods). As usual, the most challenging result to prove is Theorem 4.19 on uniform stabilization; this problem, in addition, requires a shift of topology from Y ≡ H −1 (Ω) × V ≡ [Ᏸ(Ꮽ1/4 )] × [Ᏸ(Ꮽ3/4 )] (the space of the final result) to Ᏸ(Ꮽ3/4 ) × Ᏸ(Ꮽ1/4 ) (the space where the energy method works). This shift of topology is implemented by a change of variable; this is the same change of variable noted below in (4.4.10b), that is needed to establish the desired regularity of B∗ L. Abstract model of vproblem. We let Ꮽψ = ∆2 ψ, Ᏸ(Ꮽ) = H 4 (Ω) ∩ H02 (Ω), G1 : H s (Γ) −→ H s+1/2 (Ω), ϕ = G1 g1 ⇐⇒ ∆2 ϕ = 0 in Ω; ϕΓ = g1 , ∂ϕ =0 . ∂ν Γ s ∈ R, (4.4.7a) (4.4.7b) Then, the secondorder, respectively, firstorder abstract models (in additive form) of the vproblem (4.4.1) are [4, 29] vtt + Ꮽv = ᏭG1 g1 , 0 I , A= −Ꮽ 0 v d v + Bg1 , =A vt dt vt 0 Bg1 = , ᏭG1 g1 (4.4.8) B ∗ x1 −1/2 x2 , = G∗ 1Ꮽ x2 (4.4.9) where ∗ for B and G1 refers to diﬀerent topologies. With B∗ defined by (Bg1 ,x)Y = (g1 ,B ∗ x)L2 (Γ) with respect to the Y topology, we readily find the expression in (4.4.9) since the second component of the space Y is [Ᏸ(Ꮽ3/4 )] . 1096 Regularity of B∗ L The operator B ∗ L. With y0 = {v0 ,v1 } = 0, we will show that B ∗ Lg1 = B ∗ v t; y0 = 0 ∂∆z(t) −1/2 , = G∗ vt t; y0 = 0 = 1Ꮽ ∂ν Γ vt t; y0 = 0 z(t) ≡ Ꮽ−3/2 vt t; y0 = 0 ∈ C [0,T];Ᏸ Ꮽ3/4 ≡ V (4.4.10a) continuously in g1 ∈ L2 (Σ). (4.4.10b) The new variable z(t) defined in (4.4.10) satisfies the following dynamics: abstract equation and corresponding PDEmixed problem ztt + ∆2 z = Ꮽ−1/2 G1 g1t ztt +Ꮽz = Ꮽ −1/2 z(0, ·) = z0 = 0, G1 g1t zΣ ≡ 0, zt (0, ·) = z1 ∂z ≡0 ∂ν Σ in Q, (4.4.11a) in Ω, in Σ. (4.4.11b) (4.4.11c) Indeed, to obtain (4.4.10a) (right) and (4.4.11), one uses the definition in (4.4.9) (left), followed by (4.4.8) for B∗ , to obtain B ∗ Lg1 = G∗1 Ꮽ−1/2 vt t; y0 = 0 = G∗1 ᏭᏭ−3/2 vt t; y0 = 0 = G∗ 1 Ꮽz(t) = ∂∆z(t) , ∂ν Γ (4.4.12) where, in the last step, we have recalled the usual property G∗1 Ꮽ = ∂∆/∂νΓ on V [4, equation (1.19), page 49], [29, equation (2.4)]. The abstract zequation is readily obtained from the abstract vequation after applying throughout Ꮽ−3/2 and d/dt to it and using the definition of z(t) in (4.4.10b), whose a priori regularity in (4.4.10b) follows from (4.4.3) and (4.4.2). Since z(t) ∈ Ᏸ(Ꮽ3/4 ) = V (see (4.4.2)), both boundary conditions are satisfied and the abstract zequation leads to its corresponding PDEversion. By (4.4.19) below, and within the class (4.4.20), we can take z1 = 0. Remark 4.20. As already noted, the change of variable vt → z in (4.4.10) and the resulting zproblems in (4.4.11) are precisely the same that were used in [4] in obtaining the uniform stabilization, Theorem 4.21, directly; the only diﬀerence is the specific form of the righthand side term (thus, the letter p was used in [4, equation (3.12), page 55], while the letter z is used now for a closely related, yet not identical system). In both cases, however, a timederivative term occurs (in our case Ꮽ−1/2 G1 g1t ), which will require—in [4] as well as in Step 3 below—an integration by parts in t to obtain the soughtafter estimate. Theorem 4.21. With reference to (4.4.10), B∗ L : continuous L2 0,T;L2 (Γ) −→ L2 0,T;L2 (Γ) ; (4.4.13a) I. Lasiecka and R. Triggiani 1097 equivalently, with reference to (4.4.11), the map g1 −→ ∂∆z is continuous L2 0,T;L2 (Γ) −→ L2 0,T;L2 (Γ) . ∂ν Γ (4.4.13b) We will see below in the proof that this result, though not explicitly stated, is builtin in the treatments of [4, 29, 54] of Theorems 4.17, 4.18, and 4.19. This situation is the exact counterpart of what was noted in Section 4.3, in the paragraph just below Theorem 4.15. Proof Step 1 (basic energy identity). We return to the basic identity of the energy method [29, equation (2.24), page 340], [4, equation (3.31), page 58, with β = 0 and values at t = T], which we use with a vector field h satisfying (as usual in obtaining trace regularity results [22]) the additional condition hΓ = ν. Thus, with h · ν = 1 on Γ, for the solution z of a priori regularity z ∈ C([0,T];Ᏸ(Ꮽ3/4 ) ≡ V ) as in (4.4.10), ∂∆z 1 h · ∇(∆z)dΣ − 2 Σ ∂ν RHS1 = RHS2 = Q Σ ∇(∆z)2 h · νdΣ + 1 ∂∆z ∆z div hdΣ 2 Σ ∂ν (4.4.14) = RHS1 + RHS2 + β0,T , Q H ∇(∆z) · ∇(∆z)dQ + Ꮽ−1/2 G1 g1t h · ∇(∆z)dQ + 1 βo,T = 2 div h∇z · ∇zt dΩ 0 Q H ∇zt · ∇zt dQ, − (4.4.15) Ꮽ−1/2 G1 g1t ∆z div hdQ, T Ω Q (4.4.16) T Ω zt h · ∇(∆z)dΩ . (4.4.17) 0 Step 2 (estimate for RHS1 ). From the a priori regularity (4.4.10) for z and V as in (4.4.2), we immediately find that 2 RHS1 = ᏻ g1 L2 (Σ) ∀g1 ∈ L2 (Σ). (4.4.18) Step 3 (regularity of zt ). To handle RHS2 (by integration by parts in t, precisely as in the proof of the uniform stabilization theorem (Theorem 4.19) given in [4, page 59]), we need the regularity of zt . By (4.4.10b) and the vequation (4.4.8), we obtain zt (t) = Ꮽ−3/2 vtt = Ꮽ−3/2 − Ꮽv + ᏭG1 g1 = −Ꮽ−1/2 v + Ꮽ−1/2 G1 g1 ∈ L2 0,T;Ᏸ Ꮽ1/4 ≡ H01 (Ω) continuously in g1 ∈ L2 (Σ), (4.4.19) by recalling that v ∈ C([0,T];[Ᏸ(Ꮽ1/4 )] ) (see (4.4.3), (4.4.2)) and that G1 g1 ∈ L2 (0,T;H 1/2 (Ω)), by virtue of (4.4.7a) with s = 0 on G1 , hence (conservatively) Ꮽ−1/2 G1 g1 ∈ L2 (0,T;Ᏸ(Ꮽ1/2 ) ≡ H02 (Ω)) for g1 ∈ L2 (Σ). Regularity of B∗ L 1098 Step 4 (estimates for RHS2 and b0,T for smoother g1 ). Henceforth, to estimate both RHS1 and β0,T , we will at first take g1 within the smoother class g1 (0) = g1 (T) = 0. g1 ∈ C [0,T];L2 (Γ) , (4.4.20) This initial restriction is dictated by the fact that zt in (4.4.19) is only in L2 in time. Lemma 4.22. In the present setting, 2 RHS2 = ᏻ g1 L2 (Σ) , 2 β0,T = ᏻ g1 L2 (Σ) , (4.4.21) for all g1 in the class (4.4.20). Step 5 (proof of (4.4.21) for β0,T ). First from (4.4.10), (4.4.3), and (4.4.2), we have, since vt (0) = v1 = 0, z(0) = 0, z(T) = Ꮽ−3/2 vt T; y0 = 0 ∈ Ᏸ Ꮽ3/4 ≡ V (4.4.22) continuously in g1 ∈ L2 (Σ). Next, for g1 in the class (4.4.20) used in (4.4.19), we compute, since vt (0) = v1 = 0, zt (T) = −Ꮽ−1/2 vt T; y0 = 0 ∈ Ᏸ Ꮽ1/4 ≡ H01 (Ω) zt (0) = 0, continuously in g1 ∈ L2 (Σ), (4.4.23) where the regularity follows from (4.4.3) and (4.4.2). Using (4.4.22) and (4.4.23) in (4.4.17), we readily obtain, as desired, β0,T = 1 2 Ω div h∇z(T) · ∇zt (T)dΩ − 2 = ᏻ g1 L2 (Σ) Ω zt (T)h · ∇ ∆z(T) dΩ (4.4.24) for all g1 in the class (4.4.20). Thus, (4.4.21) (right) is proved. Step 6 (proof of (4.4.21) for RHS2 ). The most critical term of RHS2 to estimate is the first term in (4.4.16). As in the direct proof of the uniform stabilization theorem (Theorem 4.19) in [4, page 59], we integrate by parts in t, with g1 in the class (4.4.20), thus obtaining Q Ꮽ−1/2 G1 g1t h · ∇∆z dQ = Ω Ꮽ −1/2 * G1 g1 h · ∇∆z dΩ T 0 − (4.4.25) Q Ꮽ −1/2 G1 g1 h · ∇∆zt dQ, I. Lasiecka and R. Triggiani 1099 where the first term on the righthand side of (4.4.25) vanishes since g1 (0) = g1 (T) = 0. Moreover, we will see that Q Ꮽ−1/2 G1 g1 h · ∇∆zt dQ = T 0 G1 g1 ,Ꮽ−1/2 h · ∇ ∆zt Ω dt 2 = ᏻ g1 L2 (Σ) . (4.4.26) In fact, by [4, Lemma 3.5, page 50], the second term in the inner product satisfies (as Ᏸ(Ꮽ1/2 ) ≡ H02 (Ω)) −1/2 Ꮽ h · ∇∆zt L2 (Ω) = h · ∇∆zt [Ᏸ(Ꮽ1/2 )] ≤ C1 h · ∇∆zt H −2 (Ω) ≤ Ch ∇ ∆zt H −2 (Ω) ≤ Ch zt H 1 (Ω) = Ch Ꮽ1/4 zt L2 (Ω) , (4.4.27) where, in the last step, we have used zt Γ = 0. Recalling (4.4.19), 1/4 Ꮽ zt L2 (0,T;L2 (Ω)) = ᏻ g1 L2 (Σ) ; (4.4.28) we then see that (4.4.27) and (4.4.28), used in the integral term of (4.4.26), produce the indicated estimate. From (4.4.26) used in (4.4.25), we conclude that Q 2 Ꮽ−1/2 G1 g1t h · ∇∆z dQ = ᏻ g1 L2 (Σ) (4.4.29) for all g1 in the class (4.4.20), as desired. A similar estimate as the one in (4.4.29) holds true, a fortiori for the more regular second term in the definition of RHS2 in (4.4.16). Accordingly, we obtain (4.4.21) for RHS2 . Step 7. We can then extend estimates (4.4.21) for RHS2 and β0,T to all g1 ∈ L2 (Σ), by density, starting from the class (4.4.20). Using these extended estimates as well as (4.4.18) in (4.4.14), we obtain for the righthand side of (4.4.14), 2 RHS1 + RHS2 + β0,T = ᏻ g1 L2 (Σ) ∀g1 ∈ L2 (Σ). (4.4.30) Step 8. It remains to handle the lefthand side (boundary terms) of identity (4.4.14). We first note that since hΓ = ν ⊥ Γ, then as usual, on Γ : h · ∇(∆z) = ∂∆z , ∂ν 2 ∇(∆z)2 = ∂∆z + ∇σ (∆z)2 , ∂ν (4.4.31) where ∇σ denotes the tangential gradient on Γ. Hence, regarding the first two terms on the lefthand side of (4.4.14), we have by (4.4.31), on Γ, 2 1 ∂∆z h · ∇(∆z) − ∇(∆z) h · ν = ∂ν 2 2 1 ∂∆z − 1 ∇σ (∆z)2 . 2 ∂ν 2 (4.4.32) Regularity of B∗ L 1100 Hence, (4.4.32) yields for the lefthand side of (4.4.14), LHS of (4.4.14) = 1 2 ∂∆z 2 ∂∆z dΣ − 1 ∇σ (∆z)2 + 1 ∆z div hdΣ ∂ν 2 2 ∂ν Σ Σ (4.4.33) " ∂∆z 2 1 dΣ − Ch ∆z2 dΣ ≥ − 2 4 Σ ∂ν 4 Σ 2 1 ∇σ (∆z) dΣ, − ! 2 T 0 Γ Σ ∆z2 dΣ ≤ C T z 0 2 H 3 (Ω) dt =ᏻ z 2 L2 (0,T;V ) 2 = ᏻ g1 L2 (Σ) (4.4.34) (4.4.35) by (4.4.10) . (4.4.36) In the last step in (4.4.35) we have recalled that z satisfies the two boundary conditions (4.4.11c) as well as the space V in (4.4.2). To go from (4.4.35) to (4.4.36), we have invoked (4.4.10). Finally, substituting estimate (4.4.36) in (4.4.34) and recalling (4.4.30), we obtain ! 1 − 2 4 " ∂∆z 2 1 dΣ = ᏻ g1 2 ∇σ (∆z)2 dΣ. + ∂ν L2 (Σ) 2 Σ Σ (4.4.37) Step 9. We now estimate in terms of g1 ∈ L2 (Σ) the last integral term in the righthand side of (4.4.37). Lemma 4.23. With reference to problem (4.4.11) and to (4.4.37), Σ ∇σ (∆z)2 dΣ = ᏻ g1 2 L2 (Σ) , g1 ∈ L2 (Σ). (4.4.38) Proof. As in [22, 29] and [45, page 970], we introduce the following operator: Ꮾ ≡ firstorder diﬀerential operator on Ω, tangential to Γ (i.e., without transversal derivatives to Γ, when expressed in local coordinates) and with smooth coeﬃcients on Ω. (4.4.39) We next define a new variable y ≡ Ꮾz ∈ C [0,T];H 2 (Ω) , yt ≡ Ꮾzt ∈ L2 0,T;L2 (Ω) continuously in g1 ∈ L2 (Σ), (4.4.40a) yt ∈ C [0,T];L2 (Ω) for g1 in the class (4.4.20) continuously in the L2 (Σ)norm of g1 , (4.4.40b) I. Lasiecka and R. Triggiani 1101 where the indicated regularity of { y, yt } in (4.4.40a) stems from (4.4.10b) and (4.4.19), respectively. Moreover, (4.4.19) yields (4.4.40b) if g1 belongs to the class (4.4.20). Thus, applying Ꮾ to the PDE zproblem (4.4.11) yields the corresponding yproblem ytt + ∆2 y = F y(0, ·) = 0, y Σ ≡ 0, in (0,T] × Ω ≡ Q, y1 (0, ·) = y1 = Ꮾz1 ∂y =u ∂ν Σ (4.4.41a) in Ω, (4.4.41b) in (0,T] × Γ ≡ Σ, (4.4.41c) where F ≡ ∆2 ,Ꮾ z+Ꮽ−1/2 G1 g1t , KI z ≡ ∆2 ,Ꮾ z ∈ C [0,T];H −1 (Ω) , (4.4.42) ∂ 3/2 u≡ ,Ꮾ z ∈ C [0,T];H (Γ) . ∂ν Γ (4.4.43) Both regularity properties in (4.4.42) and (4.4.43) are continuous in g1 ∈ L2 (Σ). Moreover, if g1 is in the class (4.4.20), we can take y1 = 0. The regularity of the fourthorder commutator in (4.4.42) and of the firstorder commutator in (4.4.43) follows from the regularity of z in (4.4.10b) as well as trace theory in the former case. Further, we notice that by (4.4.39) and (4.4.40a), we have Γ ∇σ ∆zΓ 2 dΓ = Ꮾ ∆zΓ 2 dΓ Γ 2 = ∆(Ꮾz) Γ dΓ + l.o.t. Γ 2 = ∆y Γ dΓ + l.o.t., (4.4.44) Γ where l.o.t stands for “lowerorder terms.” Thus, by (4.4.44), instead of establishing (4.4.38), we seek to prove equivalently that Σ ∆y Γ 2 dΣ = ᏻ g1 2 L2 (Σ) g1 ∈ L2 (Σ). , (4.4.45) Furthermore, since u in (4.4.41c) is smooth, see (4.4.43), we replace the yproblem (4.4.41) with the following boundary homogeneous ηproblem: ηtt + ∆2 η = F η(0, ·) = 0, wΣ ≡ 0, in Q, ηt (0, ·) = y1 ∂η ≡0 ∂ν Σ (4.4.46a) in Ω, in Σ, (4.4.46b) (4.4.46c) 1102 Regularity of B∗ L where F is defined by (4.4.42) and where η is subject to the same a priori regularity as y (compare with (4.4.40)): η ∈ C [0,T];H02 (Ω) , ηt ∈ C [0,T];L2 (Ω) ηt ∈ L2 0,T;L2 (Ω) continuously in g1 ∈ L2 (Σ), (4.4.47a) for g1 in the class (4.4.20) continuously in the L2 (Σ)norm of g1 , in which case we can take y1 = 0. (4.4.47b) Accordingly, we now seek to establish that ∆ηΓ 2 dΣ = ᏻ g1 2 L2 (Σ) Σ , g1 ∈ L2 (Σ), (4.4.48) which is equivalent to (4.4.45), hence to the original soughtafter estimate (4.4.38). Proof of (4.4.48). We take, at first, g1 in the class (4.4.20), prove estimate (4.4.48), and then extend it to all g1 ∈ L2 (Σ). Thus, below, we may assume the regularity (4.4.47b). To establish (4.4.48), we recall the energy method based on the multiplier h · ∇η for problem (4.4.46), where h is a smooth vector field such that h = ν on Γ, and hence h · ν = 1 on Γ. We can thus invoke the usual identity, see, for example, [61, equation (2.20), page 286], for the ηproblem (4.4.46): 1 RHS1 = 2 1 2 Q − Q Σ (∆η)2 h · νdΣ = RHS1 + RHS2 + b0,T , ηt2 − (∆η)2 div hdQ + Q ∆η div H + H T ∇η dQ (4.4.49) ∆η∇η · ∇(div h)dQ, RHS2 = − Q Fh · ∇η dQ, b0,T = ηt (t),h · ∇η(t) T Ω 0. From the a priori regularity of {η,ηt } in (4.4.47), we have 2 RHS1 = ᏻ g1 L2 (Σ) b0,T 2 = ᏻ g1 L2 (Σ) ∀g1 ∈ L2 (Σ), (4.4.50) for g1 in the class (4.4.40). (We are taking g1 in the class (4.4.40) since b0,T requires continuity in time of ηt as in (4.4.47b), which is not available in (4.4.47a). Alternatively, as in [22], we could apply the multiplier (T − t)h · ∇η to problem (4.4.46) to eliminate the I. Lasiecka and R. Triggiani 1103 terms in [·]T0 .) It remains to show that RHS2 = − Q 2 Fh · ∇η dQ ≡ ᏻ g1 L2 (Σ) , g1 ∈ L2 (Σ). (4.4.51) We now establish (4.4.51). Since F = KI z + Ꮽ−1/2 G1 g1t by (4.4.42), where KI is the interior commutator in (4.4.42), we proceed for each term separately. We have Q 2 KI zh · ∇η dQ = ᏻ g1 L2 (Σ) , g1 ∈ L2 (Σ). (4.4.52) This is so for the following reasons. First, we have KI z ∈ C([0,T];H −1 (Ω)) continuously in g1 ∈ L2 (Σ) by (4.4.42), while preliminarily ∇η ∈ C([0,T];H 1 (Ω)). Next, the latter combined with ηΣ = 0, hence ∇η ⊥ Γ and ∂η/∂ν = ∇η · ν = 0 on Σ, hence ∇η = 0 on Σ, yields finally ∇η ∈ C([0,T];H01 (Ω)) continuously in g1 ∈ L2 (Σ), and (4.4.52) is proved. (We could also use the divergence theorem [61, equation (2.3.1), page 288] to reach the same conclusion.) Similarly, T Ω 0 Ꮽ−1/2 G1 g1t h · ∇η dt dΩ = Ω T * Ꮽ−1/2G g h · ∇ η dΩ − 1 1 0 Q 2 Ꮽ−1/2 G1 g1 h · ∇ηt dQ = ᏻ g1 L2 (Σ) (4.4.53) since Ꮽ−1/2 G1 g1 ∈ L2 (0,T;Ᏸ(Ꮽ1/2 ) ≡ H02 (Ω)) for g1 ∈ L2 (Σ) as noted below (4.4.19) and ∇ηt  ∈ L2 (0,T;H −1 (Ω)) for g1 ∈ L2 (Σ) by (4.4.47a). Thus, (4.4.53) is proved. Then, estimates (4.4.52) and (4.4.53) as well as F ≡ KI z + Ꮽ−1/2 G1 g1t yield estimate (4.4.51), as desired. Thus, estimate (4.4.48) is proved. Equivalently, estimate (4.4.45) and the soughtafter estimate (4.4.38) are established as well. Step 10. We use (4.4.38) in (4.4.37) and obtain ∂∆z 2 2 ∂ν dΣ = ᏻ g1 L2 (Σ) Σ and Theorem 4.21 is finally proved. ∀g1 ∈ L2 (Σ), (4.4.54) 4.5. EulerBernoulli plate with hinged boundary controls. Case 1: control in the “moment” boundary condition Openloop and closedloop feedback dissipative systems. We let, again, Ω be an open bounded domain in Rn (n = 2 in the physical case of plates) with suﬃciently smooth C 2 boundary Γ. We consider the following openloop problem 1104 Regularity of B∗ L of the EulerBernoulli equation defined on Ω, with boundary control g2 ∈ L2 (0, T;L2 (Γ)) ≡ L2 (Σ), in the “moment” boundary condition as well as its corresponding boundary dissipative version: vtt + ∆2 v = 0; v(0, ·) = v0 , vt (0, ·) = v1 ; wtt + ∆2 w = 0 in Q, w(0, ·) = w0 , wΣ ≡ 0 in Σ, vΣ ≡ 0; ∆vΣ = g2 ; wt (0, ·) = w1 ∆wΣ = ∂ ∂ν Ꮽ−1 wt (4.5.1a) in Ω, (4.5.1b) (4.5.1c) in Σ, (4.5.1d) with Q = (0,T] × Ω; Σ = (0,T] × Γ. Moreover, the operator Ꮽ is defined below in (4.5.6) as Ꮽ f = −∆ f ; Ᏸ(Ꮽ) = H 2 (Ω) ∩ H01 (Ω). Regularity, exact controllability of the vproblem, and uniform stabilization of the wproblem. References for this subsection include [20, 31, 33, 36, 50, 54, 55]. We begin by introducing the (state) space of optimal regularity Y ≡ Ᏸ Ꮽ1/2 × Ᏸ Ꮽ1/2 ≡ H01 (Ω) × H −1 (Ω). (4.5.2) Theorem 4.24 (regularity [31, Theorem 1.3, equations (1.22), (1.23), page 203]). Regarding the vproblem (4.5.1) with y0 = {v0 ,v1 } = 0, the following regularity result holds true for each T > 0 (recall the definition of L in (1.2b)): the map L : g2 −→ Lg2 = v,vt is continuous L2 (Σ) (4.5.3a) −→ C [0,T]H01 (Ω) × H −1 (Ω) −→ vtt continuous L2 (Σ) −→ L2 0,T; Ᏸ Ꮽ3/2 ≡ V , (4.5.3b) V = Ᏸ Ꮽ3/2 = h ∈ H 3 (Ω) : hΓ = ∆hΓ = 0 . (4.5.4) (Note that the operator A in [31, Theorem 1.3] is A = Ꮽ2 in our present notation for Ꮽ, see [31, equations (1.5), (1.6)]). Theorem 4.25 (exact controllability [20, 50]). Given any initial condition {v0 , v1 } ∈ Y and T > 0, there exists a g2 ∈ L2 (Σ) such that the corresponding solution of the vproblem (4.5.1) satisfies {v(T),vt (T)} = 0. Remark 4.26. Exact controllability of the vproblem (4.5.1) with two boundary controls vΣ = g1 and ∆vΣ = g2 , g1 ∈ H01 (0,T;L2 (Γ)), g2 ∈ L2 (Σ), was previously obtained in [33, Theorem 1.2], [54, 55]. A diﬀerent exact boundary controllability result with g1 = 0 and g2 ∈ L2 (0,T;H 1/2 (Γ)), however, in the space [H 2 (Ω) ∩ H01 (Ω)] × L2 (Ω) was obtained in [36, Theorem 1.1]. Theorem 4.27 (uniform stabilization [20]). With reference to the wproblem (4.5.1), (i) the map {w0 ,w1 } ∈ Y = Ᏸ(Ꮽ1/2 ) × [Ᏸ(Ꮽ1/2 )] → {w(t),wt (t)} defines a s.c. contraction semigroup eAt on Y ; I. Lasiecka and R. Triggiani 1105 (ii) the following trace result holds true: ∆wΣ = ∂Ꮽ−1 wt ∈ L2 0, ∞;L2 (Γ) ∂ν (4.5.5) continuously in {w0 ,w1 } ∈ Y . (iii) there exist constants M ≥ 1, δ > 0, such that w(t) At w0 −δt w0 = e ≤ Me , wt (t) w1 w1 Y Y t ≥ 0. (4.5.6) Y As in Sections 4.1, 4.2, 4.3, and 4.4 and in line with the content of Section 1, we stress once more that all three theorems above are obtained by PDE hardanalysis energy methods (not by soft analysis methods). As usual, the most challenging result to prove is Theorem 4.27 on uniform stabilization. Abstract model of vproblem. We let Ꮽψ = −∆ψ, Ᏸ(Ꮽ) = H 2 (Ω) ∩ H01 (Ω), s ∈ R, (4.5.7) G2 : H s (Γ) −→ H s+5/2 (Ω), ϕ = G2 g2 ⇐⇒ ∆2 ϕ = 0 in Ω; ϕΓ = 0, ∆ϕΓ = g2 on Γ (4.5.8) and we recall the Dirichlet map D : H s (Γ) → H s+1/2 (Ω) defined in (4.2.4): ϕ = Dg2 ⇐⇒ ∆ϕ = 0 in Ω; ϕΓ = g2 on Γ , G2 = −Ꮽ−1 D, (4.5.9) where the last relationship is taken from [31, Remark 3.2, page 211]. Then, the secondorder, respectively, firstorder abstract models (in additive form) of the vproblem (4.5.1) are [31, 33] vtt + Ꮽ v = Ꮽ G2 g2 = −ᏭDg2 , 2 A= 0 −Ꮽ2 2 I , 0 Bg2 = v d v + Bg2 , =A vt dt vt 0 , Ꮽ 2 G2 g 2 (4.5.10) B∗ x1 ∗ = G∗ 2 Ꮽx2 = −D x2 , x2 (4.5.11) where ∗ for B, and G2 and D, refer to diﬀerent topologies. With B∗ defined by (Bg2 ,x)Y = (g2 ,B ∗ x)L2 (Γ) with respect to the Y topology defined in (4.5.2), we readily find the expression in (4.5.11) also by virtue of G2 = −Ꮽ−1 D. Regularity of B∗ L 1106 The operator B ∗ L. With y0 = {v0 ,v1 } = 0, we will show that v t; y0 = 0 ∗ = G∗ B Lg2 = B 2 Ꮽvt t; y0 = 0 = −D vt t; y0 vt t; y0 = 0 (4.5.12) ∂ −1 ∂ Ꮽ vt t; y0 = 0 = zt (t), = ∂ν ∂ν continuously in g2 ∈ L2 (Σ). z(t) = Ꮽ−1 v t; y0 = 0 ∈ C [0,T];Ᏸ Ꮽ3/2 ≡ V (4.5.13) ∗ ∗ Indeed, to obtain (4.5.12), one uses the definition in (4.5.11) for B ∗ , followed by the usual property that G∗2 Ꮽ2 = ∂/∂ν on Ᏸ(Ꮽ1/2 ) [31, Lemma 3.1, equation (3.7), page 212] or D∗ Ꮽ = −∂/∂ν on Ᏸ(Ꮽ1/2 ) = H01 (Ω) [39, equation (1.21)]. The regularity of z(t) noted in (4.5.13) follows from (4.5.3a) for v, and Ᏸ(Ꮽ1/2 ) ≡ H01 (Ω). The new variable z(t) defined in (4.5.13) satisfies the following dynamics: abstract equation and the corresponding PDEmixed problem ztt + Ꮽ2 z = ᏭG2 g2 = −Dg2 , ztt + ∆ z = ᏭG2 g2 = −Dg2 2 z(0, ·) = 0, zΣ ≡ 0, (4.5.14a) in Q; (4.5.14b) zt (0, ·) = 0 in Ω; (4.5.14c) ∆zΣ ≡ 0 (4.5.14d) in Σ. The abstract zequation in (4.5.14) (left) is readily obtained from the abstract vequation in (4.5.10) after applying Ꮽ−1 and using the definition of z(t) in (4.5.13). Since z(t) ∈ Ᏸ(Ꮽ3/2 ) ≡ V (see (4.5.4)), both boundary conditions are satisfied and the abstract zequation leads to its corresponding PDEversion. Remark 4.28. As already noted, the change of variable v → z in (4.5.13) and the resulting zproblems in (4.5.14) are precisely the same that were used in [20, equations (2.7), (2.8), and (4.3)] in obtaining there the uniform stabilization, Theorem 4.27, directly; the only diﬀerence is that in [20, equations (2.8), (4.3)] g2 is expressed in feedback form: g2 = D∗ Ꮽpt = (∂/∂ν)pt ∈ L2 (0, ∞;L2 (Γ)) in the notation of [20]. Thus, the letter p was used in [20], while the letter z is used now. Thus, the techniques in the proof of the next soughtafter result are contained in [20] and indeed in [33, 54]. Theorem 4.29. With reference to (4.5.12), B∗ L : continuous L2 0,T;L2 (Γ) −→ L2 0,T;L2 (Γ) ; (4.5.15) equivalently, with reference to (4.5.14), ∂z is continuous L2 0,T;L2 (Γ) −→ L2 0,T;L2 (Γ) . the map g2 −→ t ∂ν Σ (4.5.16) I. Lasiecka and R. Triggiani 1107 We will see below in the proof that this result, though not explicitly stated, is builtin in the treatments of [20, 31, 33, 54, 55] to prove Theorem 4.24. Proof Step 1 (basic energy identity). We return to the basic identity of the energy method [20, 31, 33, 54], which we use with a vector field h satisfying (as usual in obtaining trace regularity results [22]) the additional condition hΓ = ν. Thus, with h · ν = 1 on Γ, for the solution z of a priori regularity z ∈ C([0,T];Ᏸ(Ꮽ3/2 ) ≡ V ) as in (4.5.13), we have (see, e.g., [33, equations (2.29), (2.32)], [31, equations (2.1), (2.4)]) 1 2 ! Σ RHS1 = Q + ∂∆z ∂ν "2 "2 ! ∂zt + ∂ν dΣ = RHS1 + RHS2 + b0,T , H ∇∆z · ∇∆z dQ + 1 2 Q (4.5.17) Q H ∇zt · ∇zt dQ (4.5.18) ∇zt 2 − ∇∆z2 div hdQ + zt ∇(div h) · ∇zt dQ, Q RHS2 = − Dg2 ∇∆z dQ, (4.5.19) Q b0,T = − zt ,h · ∇∆z T L2 (Ω) 0 . (4.5.20) Step 2 (regularity of zt ). To handle RHS1 , we need the a priori regularity of zt , zt = Ꮽ−1 vt t; y0 = 0 ∈ C [0,T];Ᏸ Ꮽ1/2 ≡ H01 (Ω) continuously in g2 ∈ L2 (Σ), (4.5.21) as it follows from (4.5.13), (4.5.3a), and H −1 (Ω) = [Ᏸ(Ꮽ1/2 )] , see (4.5.2). Step 3 (estimate of RHS1 ). By (4.5.13) for z and (4.5.21) for zt , we obtain ∇∆z, ∇zt ∈ C [0,T];L2 (Ω) continuously in g2 ∈ L2 (Σ). (4.5.22) Using (4.5.22) in (4.5.18) readily yields 2 RHS1 = ᏻ g2 L2 (Σ) ∀g2 ∈ L2 (Σ). (4.5.23) Step 4 (estimates of RHS2 and b0,T ). From (4.5.19) and (4.5.20), by virtue of (4.5.21) and (4.5.22), we readily obtain 2 RHS2 + b0,T = ᏻ g2 L2 (Σ) ∀g2 ∈ L2 (Σ). (4.5.24) Step 5 (final estimate). Using (4.5.23) and (4.5.24) in (4.5.17) yields 1 2 ! Σ ∂∆z ∂ν "2 ! ∂zt + ∂ν "2 2 dΣ = ᏻ g2 L2 (Σ) ∀g2 ∈ L2 (Σ), (4.5.25) 1108 Regularity of B∗ L and (4.5.25) a fortiori proves (4.5.16), as desired. The proof of Theorem 4.29 is complete. Remark 4.30. In this case, the proof of Theorem 4.29 is easier than the proof of uniform stabilization in [20]. But Claim 2.3 requires also exact controllability. 4.6. EulerBernoulli plate with hinged boundary controls. Case 2: control in the Dirichlet boundary condition Openloop and closedloop feedback dissipative systems. In the notation for Ω, Γ, Ꮽ of Section 4.5, we consider now the following openloop problem of the EulerBernoulli equation with boundary control g1 ∈ L2 (0,T;L2 (Γ)) ≡ L2 (Σ) and its corresponding boundary dissipative version: vtt + ∆2 v = 0; v(0, ·) = v0 , vt (0, ·) = v1 ; wtt + ∆2 w = 0 in Q, w(0, ·) = w0 , vΣ = g1 ; wt (0, ·) = w1 (4.6.1a) in Ω, ∂ −2 Ꮽ wt in Σ, ∂ν ∆wΣ ≡ 0 in Σ. wΣ = ∆vΣ = 0; (4.6.1b) (4.6.1c) (4.6.1d) Regularity, exact controllability of the vproblem, and uniform stabilization of the wproblem. References for this subsection include [20, 31, 33]. We begin by introducing the (state) space of optimal regularity X ≡ Ᏸ Ꮽ1/2 3/2 × Ᏸ Ꮽ ≡ H −1 (Ω)×V , (4.6.2) with the space V defined in (4.5.4). Theorem 4.31 (regularity [31, Theorem 1.3, equations (1.20), (1.21), page 203]). Regarding the vproblem (4.6.1) with y0 = {v0 ,v1 } = 0, the following regularity result holds true for each T > 0 (recall (1.2b)): the map L : g1 −→ Lg1 = v,vt is continuous L2 (Σ) −→ C [0,T];X ≡ H −1 (Ω) × V . (4.6.3) Theorem 4.32 (exact controllability [20]). Given any initial condition {v0 ,v1 } ∈ X and T > 0, there exists a g1 ∈ L2 (Σ) such that the corresponding solution of the vproblem (4.6.1) satisfies {v(T),vt (T)} = 0. Remark 4.33. Exact controllability of the vproblem (4.6.1) with two boundary controls vΣ = g1 ∈ L2 (Σ) and ∆vΣ = g2 ∈ [H 1 (0,T;L2 (Γ))] was previously obtained in [33, Theorem 1.1], [54]. Theorem 4.34 (uniform stabilization [20]). With r