Luenberger compensator theory for heat- Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls

Abstract: An optimal, complete, continuous theory of the Luenberger dynamic compensator (or state estimator or state observer) is obtained for the recently studied class of heat-structure interaction partial differential equation (PDE) models, with structure subject to high Kelvin-Voigt damping, and feedback control exercised either at the interface between the two media or else at the external boundary of the physical domain in three different settings. It is a first, full investigation that opens the door to numerous and far reaching subsequent work. They will include physically relevant fluid-structure models, with waveor platestructures, possibly without Kelvin-Voigt damping, as explicitly noted in the text, all the way to achieving the ultimate discrete numerical theory, so critical in applications. While the general setting is functional analytic, delicate PDE-energy estimates dictate how to define the interface/boundary feedback control in each of the three cases.

On this geometry in Figure 1, we thus consider the following heat-structure PDE model in solution variables = u u t x u t x u t x , , , , …, , , ,˜, 0 ; , , , ,˜, 1. ( ) ( ) is endowed with the gradient norm. Relevant properties of this model, obtained in [1], will be reviewed in Section 1.3.
In particular, it was shown in [1] and reproduced in Section 1.3 that homogeneous problems (1.1a)-(1.1f) can be rewritten as the abstract model ′ = = y y y w w u , , , Three controlled systems. We consider three cases of boundary/interface control g, as applied to systems (1.1a)-(1.1f).
• CASE 1 (Section 2): the control g acts on the matching of the stresses condition (2.1e) at the interface between the two media, thus as a Neumann control as follows:  • CASE 2 (Section 4): the control g acts this time as a Dirichlet control on the matching of the "velocity" condition (1.1d) also at the interface between the two media as follows: See the entire system (4.1a)-(4.1f) in Section 4.
• CASE 3 (Section 5): the control g acts, still as a Dirichlet control, but this time as a boundary control on the external boundary of the heat domain in (1.1c) as follows: See the entire system (5.1a)-(5.1f) in Section 5.
In each case, the control g introduces a control operator , highly unbounded. Thus, the resulting model is now with the highly unbounded control operator depending on the three cases, see equation (2.2) for CASE 1; equation (4.2) for CASE 2; and equation (5.13) for CASE 3. The selection of the operators entering into the development of the Luenberger theory, as described in Section 1.2.2, depends on the three cases.
Objective of the present article, as a template for future work. The initial objective in the present work is to offer a "continuous theory" of the long-standing, highly relevant topic corresponding to the Luenberger compensator program. The deliberate goal is to have this first contribution serve as a basis for further development of investigations in various directions. Among them, we cite: (i) replacing the heat component with a fluid component, thus accounting for the pressure variable while keeping the Kelvin-Voigt structure (model as in [2] (wave) or as in [3] (plate)). The new technique inspired by the boundary control theory [4], which is required for the extension from the heat component to the fluid component is described in Appendix A, with a focus on the present uncontrolled model, following [2]. (ii) replacing the present Kelvin-Voigt damped-wave with a Kelvin-Voigt damped-plate in modeling the structure. This in turn opens up a variety of different, physically relevant, coupling conditions at the interface between the two media (models as in [3] and [5]); (iii) obtain a corresponding "discrete theory" or (rigorous) numerical analysis theory, as done in past genuine PDE models of different types in [4, pp. 495-504], [6][7][8] and also [9,10], to name just a few references. This is a very challenging and technical direction of investigation, and yet highly important in engineering applications); (iv) analysis of the same models as in (i) and (ii), and this time, however, with no Kelvin-Voigt damping, such as in [11][12][13][14] (wave), where the original stability properties are different. Here, in contrast with the heatstructure case, = λ 0 is a simple eigenvalue of the uncontrolled model. Heat-viscoelastic plates are studied in [15,16]; (v) further extension of both the continuous and the discrete analysis to nonlinear models, with static interface [17][18][19][20][21][22][23][24], and even with moving interface [25,26].

Historical orientation on Luenberger's compensator theory
The Luenberger theory of "observers" was introduced for lumped (finite-dimensional) linear systems in 1971 [27], and it was met with great success [28, p. 48]. It subsequently stimulated investigations for PDE problems with boundary controls/boundary observations (infinite-dimensional systems with "badly unbounded" control and observation operators) of both parabolic and hyperbolic types [6][7][8], [4, pp. 495-504]. It consists, in its first phase, of a continuous theory, followed next by a rigorous numerical implementation, as in the aforementioned references. At the level of numerical implementation, it was in a sense rediscovered with the more recent topic of "data assimilation" that shares the same philosophy as the Luenberger discrete theory. Recent references include [29][30][31][32]. A more detailed description of these topics is given below.
Step 1. The continuous theory. Here, in a purely informal manner, we shall provide the special setting that we shall select in our application of the continuous Luenberger's dynamic compensator theory to heat-structure models. For a preliminary conceptual understanding, we may regard the operators below as being all finitedimensional, in line with Luenberger's original contribution [27]. Its standard representation is as follows: ( ) The basic idea behind is that the full state y is inaccessible, unknown, beyond any measurement, as is often the case in applications. What we have instead at our disposal is the partial observation (Cy), where C is the known observation operator. Examples abound: (i) the actual state within a furnace or (ii) the true distribution of "noise" within an acoustic chamber are not exactly accessible, and only some information from the boundary may be available in each case. Thus, the (compensator) z-equation (1.4b) is fed, or determined by, only the available partial observation Cy ( ). Subtracting (1.4b) from (1.4a) with = Bg BFz, we obtain after a cancellation of the term BFz:  (1.6) and the dynamic compensator z t ( ), which is fed only by the known partial observation (Cy) of the inaccessible state y, asymptotically approximates such state y t ( ), at an exponential rate. This is the key of Luenberger's theory in the lumped case where the state of the system is a finite dimensional vector. Nontrivial extensions were subsequently introduced and studied in the case of distributed parameter systems modeled by partial differential equations with boundary control/boundary observation [4, pp. 495-504], [6][7][8].
Step 2. The numerical theory. Particularly in the case of PDEs dynamics, it is critically important to provide a (finite element) approximation theory of dynamic compensators of Luenberger's type for partially observed systems. The aforementioned PDE references include also the discrete/numerical Luenberger theory based on finite element method. The analysis is very technical.
Connections with data assimilation. In recent years, a numerical procedure called "data assimilation" has been introduced, particularly with emphasis on nonlinear dissipative PDE dynamics with finite degrees of freedom, which in spirit is closely related in terms of goals to the discrete Luenberger's compensator theory. In common with the Luenberger's theory, in the presence of inadequate knowledge of the original system, a suitable data assimilation algorithm is introduced to force its corresponding solution to approach the original solution at an exponential rate in time. This is done by having access "to data from measurements of the system collected at much coarser spatial grid than the desired resolution of the forecast" [30]. As expected, the efficiency of data assimilation relies also on the finite dimensionality of the proposed algorithm. Inspiration comes from a rigorous result on the 2D Navier-Stokes equations (NSE) given in [33], where it is proved that if a number of Fourier modes of two different solutions of the NSE have the same asymptotic behavior as t goes to infinity, then the remaining infinite number of modes also have the same asymptotic behavior.
It seems unfortunate that data assimilation theory introduced in 2014 has not apparently been aware of the large body of works in Luenberger theory, which was introduced in 1971, to include PDE parabolic and hyperbolic problems as in [4, pp. 495-504], [6][7][8]. Luenberger's theory in these references emphasizes control/ observation on the boundary, unlike the literature of data assimilation. The original Luenberger theory was for linear models, but it was later introduced for nonlinear models, which are the core of data assimilation. Moreover, data assimilation is passive in the sense that there is no control action. Mutual awareness and knowledge of the two communities' research effort may well benefit both. In this spirit, being this an article on the Luenberger theory in systems of coupled PDEs with control/observation at "the boundary," we are pleased to provide specific references in data assimilation.
The first main data assimilation was done in [29] in 2014. This gives the AOT algorithm. It describes the interpolation operators (nodes, modes, and volume averages) and uses the "nudging" algorithm, which is essentially interior control for the data assimilated problem. The technique utilizes the existence of finitely many determining functional to capture the essential asymptotic dynamics of the system.
The initial article that does data assimilation for 3D NSE without assuming any regularity of the solution is [34]. All previous works critically utilized the regularity of the 2D NSE to show asymptotic convergence of the data assimilated solution to the reference solution. In the absence of global regularity in the 3D case, the previous article [34] achieved the exact same result for the 3D NSE by imposing conditions on the observed (model) data. Next, we quote paper [30], which provides results on the Boussinesq system (and also, therefore, for 3D NSE). While [34] does data assimilation for the 3D NSE with the assumption that the reference solution is obtained via the Galerkin procedure, the article [30] makes no such assumptions and does data assimilation for a general (Leray-Hopf) weak solution, which obeys the energy identity corresponding to the system. Also, the article only uses velocity measurements to perform data assimilation.
Luenberger problems for three interface/boundary feedback controlled models: system (1.1a)-(1.1f) subject to control action g as in (1.1eN) (CASE 1); or (1.1dD) (CASE 2); or (1.1cD) (CASE 3), in the final form ′ = + y y g as in equation (1.3-controlled). The goal of the present article is to investigate the Luenberger's dynamic compensator theory (continuous version) as applied to a class of fluid-structure interaction models, in the particular setting where the structure is subject to visco-elastic (Kelvin-Voigt) damping, as in (1.1a)-(1.1f). How to handle the corresponding fluid-wave model is described in Appendix A. This focuses on the new tricky technique that is required in the present homogeneous case (1.1a)-(1.1f) following [2]. In summary, in the present article, we consider a fluid (heat)-structure interaction model with high Kelvin-Voigt damping under three different scenarios: (1) in Part I, with Neumann control g at the interface Γ s as in (1.1eN); (2) in Part II, with Dirichlet control g at the interface Γ s as in (1.1dD); (3) in Part III, with Dirichlet control g at the external boundary Γ f as in (1.1cD).

Orientation on the contributions of the present article. Conceptual description of the mathematical setting and ultimate results
To ease the reading of this article, we find it most appropriate to provide a focused, synthetic orientation regarding both the mathematical setting of the article and its ultimate, sought-after Luenberger-type results.

Uncontrolled, homogeneous model
The uncontrolled model is a coupled heat-structure interaction, where the structure is modeled by a wave with strong Kelvin-Voigt damping, which interacts with a heat component through the interface between the two media, see the linear, coupled PDE system (1.1a)-(1.1f) and Figure 1. The state of the system is the triple = y w w u , , t { }, displacement, velocity of the elastic structure, and temperature. A comprehensive study of this model was carried out in [1]. Selected results to be used in the present article are reviewed in Section 1.3. The uncontrolled coupled system is described by an operator , which is the generator of a s.c. contraction semigroup e t on a natural finite energy functional setting. From the purpose of the Luenberger theory to be here investigated, its main feature is that such semigroup is uniformly (exponentially) stable, Theorem 1.3(ii). This is due to the Kelvin-Voigt damping. An additional property of such semigroup is that it is analytic in its natural setting, also Theorem 1.3(ii). This analyticity propertyalso due to the Kelvin-Voigt dampingadds a positive feature to the uncontrolled dynamics. One then seeks, successfully, to retain it and propagate it to the corresponding Luenberger feedback problem, that is, the dynamics of the observer variable z, expressed in feedback-form with respect to the partial observation Cy ( ) of the original unknown state y. But analyticity is not critical for the key Lueberger's goal to recover asymptotically the originally unknown full state y by using the observer z.

Controlled systems
As already noted, we consider three cases of boundary/interface control g : • Case 1 (Section 2): The control g acts on the matching of the stresses condition (1.1f) at the interface between the two media (Neumann control). See the entire system (1.1a)-(1.1f) in Section 2. • Case 2 (Section 4): The control g acts this time as a Dirichlet control on the matching of the "velocity" condition (3.1d) also occurring at the interface between the two media. See the entire system (3.1a)-(3.1f) in Section 4. • Case 3 (Section 5): The control g acts, still as a Dirichlet control, but this time as a boundary control on the external boundary of the heat domain as in (4.1c). See the entire system (4.1a)-(4.1f) in Section 5.
• CASE 1. Here, the following analysis selects the feedback form = g Fz for the Neumann boundary control g at the interface, by taking the Luenberger operators as follows: , . This way, the observer equation becomes: , whose proof is given in Section 5.2.4. The operator * is again a trace operator (cf. equation (5.15)).
• Insight on the choice = − * F versus = * F in the various cases. For CASE 2, this insight is given in (4.15), and in CASE 3, this insight is given in equation (5.23). In short, this is a purely PDE problem related to the operator * for the purpose to achieve the feedback generator still dissipative. Thus, while the setting of the analysis is functional analytic, the key technical parts are based on PDE estimates.
• Finally, to ease the reading, each case is dealt individually. In other words, one may read CASE 3 without knowledge of Cases 1 or 2.
1.3 Review of homogeneous heat-structure interaction model with Kelvin-Voigt damping: The constant b in (1.1b) will take up either the value = b 0, or else the value = b 1, as explained later. Accordingly, the space of well-posedness is taken to be the finite energy space (We are using the common notation , ,˜, 0 ; , , , ,˜, 1. is endowed with the gradient norm. Abstract model of the homogeneous PDE problem (1.1a)-(1.1f). The operator b and its adjoint * b , = b 0, 1. Basic results [1]. The abstract version of the homogeneous PDE model (1.1a)-(1.1f) is given as a firstorder equation by where the operator is given by Γ .
in the ⋅ L 2 ( ) norms of Ω s and Ω f . (ii) Thus, b and * b are maximal dissipative on H b . Then [35] gives that b generates a s.c. C 0 ( )-contraction semigroup e t b on H b , which gives the unique solution of problems (1.1a)-(1.1f): (1.14) The same generation results hold also for * b on H b , with * e t b solving system (1.10c)-(1.10g).
Thus, taking the real part of the aforementioned expression, one obtains (1.12) for = b 0 and = b 1.
(1) Section 1.3 (a subset of [1]) shows that the natural functional setting for problems (1.1a)-(1.1f) is: the energy In each such case, = b 0 and = b 1, the free dynamic operator is maximal dissipative, it defines a corresponding expression for the adjoint * b and the resulting contraction semigroups e t b and * e t b are analytic and uniformly stable. Analyticity in Theorem 1.3(ii) above is consistent with abstract results [36][37][38], in view of the Kelvin-Voigt damping.
(2) If, however, one insists in considering problems (1.1a)-(1.1f) with = b 0 in the energy space = H b 1 with full H 1 -norm for the position variable, then stability is lost: more precisely, one can readily prove or verify that: 0 is a simple eigenvalue of the free dynamics operator with 0 with corresponding eigenvector In fact, setting equal to zero equation (1. (3) In this case, we may view the problem with = b 1 on = H b 1 as having "stabilized" (and regularized) the same 2 CASE 1. Heat-structure interaction with Kelvin-Voigt damping: Neumann control g at the interface Γ s The present article begins with this section. In the present CASE 1, we consider problem (1.1a)-(1.1f) subject this time to control g acting in the Neumann interface condition (1.1e); that is, with Neumann boundary control g acting at the interface Γ s . The constant b in (1.1b) will take up either the value = b 0, or else the value = b 1, as explained earlier. Accordingly, the space of well-posedness is taken to be the finite energy space defined in (1.2a), or (1.2b).

Abstract model on
of the nonhomogeneous PDE model (2.1a)-(2.1f) with Neumann control g acting at the interface Γ s This topic was duly treated in [39], at least for = b 0. This will be reviewed below and complemented by the case = b 1. In either case, the abstract version of the nonhomogeneous PDE model (2.1a)-(2.1f) is given by with the full dynamic operator b given by (1.8) and (1.9). The definition of the Neumann control N b ( ) depends on the two cases = b 0 and = b 1, on the respective space H b . We shall provide a unified treatment covering the two cases = b 0 and = b 1, which will recover the case = b 0 in [39].
We first define the positive self-adjoint operator A N s b , The following regularity holds true for D f s , : for any r, continuously. Then, as usual [1], we rewrite the u-problem in (2.1a) via (2.10) as follows: The operator in (2.14) acting on , except that in (2.14) the relevant BCs (1.9) are included in the operator entries.
Thus, in conclusion, the abstract model for the nonhomogeneous PDE model (2.1a)-(2.1f) is given again by (2.2), where now we compute as a duality pairing via (1.2a)-(1.2b) and (2.15): where we shall establish that Proof of (2.20). [see also [4, pp. 195-196].] Take initially ∈ f We compute by means of the second Green's theorem, where we recall that on Γ s , the normal ν is inward. We obtain by ( , f o r a n y Γ (2.24) can be extended to all [4,Chapter 3]. □ In conclusion: Then the abstract model of the nonhomogeneous PDE problem (2.1a)-(2.1f) on the respective energy space with b is given by (1.8), (1.9), or alternatively by (2.14), N b ( ) is given by (2.15), and for ∈ = Henceforth, in light of Theorem 2.1, we shall omit the qualifying parameter "b" for CASE 1, as the model

Special selection of the data
With reference to the representation (1.4) in Step 1 of the Orientation in Section 1.1, we take in our present case "exponentially stable": , 0, 0, 3.1a ; , Thus, the special setting becomes, in this case, leading to the Luenberger's dynamics .
As noted in Section 1, Step 2, it is the PDE argument to be carried out in Section 3.1.3 in the present case of Neumann control on the interface Γ s that will determine that the infinite dimensional version of − * e A BB t ( ) is exponentially stable, as desired, as well as analytic.
Insight. How did we decide that = − * F B in (3.1b); that is, that the preassigned control = g Fz is given by  .1b) is the correct one in our present CASE 1.

The counterpart of
[ ] , the Luenberger's compensator variable, in line with (3.2), we select the Neumann control g in (2.1e) in the form where we have critically invoked the trace result (2.26) of Theorem 2.1 in both cases ] , the PDE version of (3.3a) corresponding to the abstract feedback problem ( from problem (2.25), b as in (1.8), (1.9), alternatively in (2.14), with g as in (3.5), is

The counterpart of the dynamic compensator equation
With partial observation as in (3.2) according to (2.26) 2 , and thus (3.10) is re-written as follows: The PDE version of the abstract z-model (3.11) with partial observation −w t Γs | in the Neumann condition at the interface Γ s (see (3.8)) is We are omitting the superscript "b" on N , * N . The main result of the present section is as follows: is the infinitesimal generator of a s.c. contraction semigroup e on H b , which moreover is analytic and exponentially stable on H b : there exist constants ≥ > C ρ 1, 0, possibly depending on "b," such that The proof of Theorem 3.1 is by PDE methods, which consist of analyzing the corresponding PDE system (3.16). We proceed through a series of steps.
corresponds to the following PDE system, where we relabel the variable  and w t  have the same sign across the equality sign in (3.16d). This is due to the normal vector ν being inward with respect to Ω s as shown in Figure 1 and as noted in (2.20). This is consistent with the fluid-structure interaction model in [12], [39, equation 2.16.1e, p. 128], also with ν being inward with respect to Ω s . This model without the Kelvin-Voigt term is known in these references to be uniformly stable by PDE-techniques.

Description of
The adjoint operator if and only if the same conditions The PDE corresponding to [35]. Explicitly in terms of the corresponding PDE systems, we have: Proof of (3.22). Step 3. This step provides the key PDE-energy estimate, = b 0 and = b 1, of the entire present section. The case = b 1 is more challenging.

Remark 3.2. Given
. Then: (i) the following estimate holds true: given > ε 0 sufficiently small, there exists a constant > C 0 ε such that: , which in turn is equivalent to Step 1. Return to (3.29) re-written for Step 2. Take the L Ω f 2 ( )-inner product of equation ( Similarly, we take the L Ω s 2 ( )-inner product of (3.33b) against  (3.17)), and we rewrite (3.35) as follows: Using, via (3.33a), the identities we obtain from (3.37a) the final identity Step 3. We take the real part of identity (3.38), thus obtaining the new identity: (3.40) Step 4. We now take the imaginary part of identity (3.38), thus obtaining the new identity hence, for ω | | as in (3.44) Summing up estimate (3.49) with estimate (3.50) finally yields for ω | | as in (3.44): The analyticity of the s.c. contraction semigroup Step 5. We proceed now with the proof of analyticity in the case = b 1 on = H b 1 . This case is more challenging and requires the following additional result (in substitution of the Poincare inequality, which does not hold true for v 2 on Ω s ).
where ∼ Γ is any fixed portion of the boundary = ∂ Γ Ω of Ω of positive measure.
We now return to inequality (3.46) 17a) on its left-hand side (LHS) and then invoke Lemma 3.5(a) for = Φ v 2 to obtain ‖ ‖ , which along with the RHS inequality in (3.53a) gives the desired estimate for = b 1: The LHS in (3.55) estimates the two terms ∇ (3.40). We may now proceed as in going from (3.47) to (3.51) in Remark 3.3 for = b 0.
Step 6. We substitute the new estimate (3.55) on the RHS of (3.40). We obtain Step 7. Summing up estimates (3.58) with estimate (3.55) finally yields for ω | | as in (3.44):  In Proposition 3.6, we shall prove, in both cases = b 0 and = b 1, that we have is given explicitly by where the positive self-adjoint operator A D f , and the Dirichlet map D f s , from Γ s into Ω f were defined in (2.9) and (2.10), and are repeated as follows:     Figure 1) and notation w w u , , t { } as in Section 1.3 for the uncontrolled problem, in the present CASE 2, we consider the following controlled problem: where the operator is of course the same as given by (1.8) and (1.9). Instead, the (boundary) control operator D is given by ( ) (4.3) Here, −A D f , is the negative, self-adjoint operator on L Ω f 2 ( ) defined by (2.9) = (3.66), i.e., by while D f s , is the Dirichlet map from Γ s to Ω f defined by (2.10) = (3.66), i.e., by The following regularity holds true for D f s , [40,41], [4,Chapter 3]: ( ), we have: : continuous Γ in the following sense. For ∈ g L Γ s 2 ( ) and , ,  where we have recalled (the normal ν is outward with respect to Ω f ) x 0 3 Ω f | , and proceeds analogously to the path (2.21)-(2.23) via Green's second theorem to obtain (4.11) in this case. Next, we extends (4.11) to ∈ ≡

Special selection of the data
For the present heat-structure interaction problem with Dirichlet control at the interface Γ s , we shall modify the special selection made in CASE 1 of Neumann control at the interface Γ s , on the basis of the representation (1.4) in Step 1 of the orientation in Section 1.1. In fact, in the present case, we now take, in the notation of (1.4)-(1.6): .
This is the setting that will be selected in the study of the Luenberger's theory below, as applied to heat (fluid)-structure interaction models with Dirichlet control g at the interface Γ s , as in (4.1d).
Insight. How did we decide that = * F B in (4.12b); that is, that the preassigned control = g Fz is given by from problem (4.2) with operator b as in (1.8) and with g as in (4.16), is    .19) is given as follows:  The main result of the present section is as follows: in the infinitesimal generator of a s.c. contraction semigroup e t F D b ,

( )
on H b , which moreover is analytic and exponential stable on H b : there exist constants ≥ > C ρ 1, 0 possibly depending on "b" such that given in (4.28a)-(4.28b). The proof of Theorem 4.1 is by PDE methods, which consists of analyzing the corresponding PDE system (4.27).
, the abstract equation 1 , corresponds to the following PDE system, where we relabel the variable The adjoint operator recalling D in (4.3).
if and only if the same conditions The PDE version corresponding to the adjoint operator Step 2.
similarly, for the adjoint * Proof of (4.33). For ∈ v v h , , where recalling (4.3) for D and (4.7) for * , Γs from (4.11). On the other hand, recalling (1.12)    (i) the following estimate holds true: there exists a constant > C 0 ε such that: , in turn equivalent to estimate We indicate the relevant changes.
Step 1. Return to (4.40), re-written for Step 2. Take the L Ω f 2 ( )-inner product of equation ( Similarly, we take the L Ω s 2 ( )-inner product of (4.44b) against , recalling that the normal vector ν is inward with respect to Ω s , and obtain as in (3.35) We now invoke the B.C.
Step 5. We proceed now with the proof of the case = b 1 on the space as in (4.56). We finally obtain the desired estimate also for = b 1 to include also v 2 2 ‖ ‖ : as in (4.56). Substituting the estimate for v 2 2 ‖ ‖ from (4.2.5) into the RHS of (4.58) with ≤ bε b yields Now, add the estimate for v 2 2 ‖ ‖ in (4.72 1 ) to (4.73 1 ) and obtain Estimate (4.73 1 ) is the counterpart of estimate (4.58) for = b 0.
Step 6. The rest of the proof for = b 1 now proceeds as in the case = b 0. In ( * which is a counterpart of (4.63 0 ) for = b 0. By substituting (4.75 1 ) into the RHS of ( Finally, summing up (4.75 1 ) and (4.76 1 ) yields , as in (4.56). In Proposition 4.5, we shall prove that, in both cases = b 0 and = b 1, we have so that there exists a disk r0 centered at the origin and of suitable radius ( ) in (4.78) allows one to conclude that the resolvent is uniformly bounded on the imaginary axis i : c o n s t , . ( ) , we have: given is given explicitly by In the operator form, we have , , , where the operators A N , , is the Robin Laplacian on Ω f : and R f s , is the Robin map from Γ s to Ω f : Proof. Identity (4.81) and the characterization of

Abstract model on
of the nonhomogeneous PDE model (5.1a)-(5.1f) with Dirichlet control g acting at the external boundary Γ f This is the counterpart of the treatment in [39], where an interior Neumann or Dirichlet control acts at the interface Γ s . To this end, we define two boundary → interior maps, with interior Ω f : the map D f s , (introduced in (2.10) = (3.67) = (4.5)) acting from Γ s , and the map D f f , acting from Γ f : Similarly, we recall from ( To obtain the abstract model of problem (5.1), we proceed as usual [4,39]. We re-write the u-problem in (5.1) as follows: where Ã D f , is the isomorphic extension of the operator A D f , in (5.3): ) with respect to L Ω f 2 ( ) as a pivot space. Similarly, we re-write the w-problem in (5.1b) via the RHS of (5.1e) as (compare with (2.6) of CASE 1): ) and (5.9a) has an extra term + b w w t ( )and their combination produces a cancellation of the term Combining (5.10) for the w-problem with (5.7) for the u-problem, we obtain the corresponding first order system The operator in (5.12) on ≡ w w u g , , 0 ) is of course the same operator b in (1.8)-(1.9), except that in (5.12) the relevant BCs in (1.9a)-(1.9b) are included in the operator entries. Equation (5.12) can be rewritten as follows: where the operator is of course the same as given by (1.8) and (1.9), while the (boundary) control operator D is given by ( ) (5.14) The adjoint operator * D of D in (5.14) is given by in the following sense. For ∈ g L Γ s 2 ( ) and , , D f ε 1 2 3 , 3 4 { } , we compute as a duality pairing via (5.14). ,

Special selection of the data
For the present heat-structure interaction problem with Dirichlet control at the external boundary Γ f in the notation of (1.4)-(1.6), we take recalling D in (5.14).  (i) the following estimate holds true: there exists a constant > C 0 ε such that:

Description of
, which in turn is equivalent to Thus, the s.c. contraction semigroup e t We indicate the relevant changes: Step 1. Return to (5.45), re-written for ∈ v v h , ,

[ ( ) ]
Step 2. Take the L Ω f 2 ( )-inner product of equation ( We now use the identities from (5.49a) ,Δ ,Δ . Step 5. Case = b 1. In the present case, as in CASE 2 of Dirichlet control on Γ s , there are additional challenges in establishing the sought-after estimate (5.46) with = b 1, i.e., in the space = H b 1 . The required argument is provided in the present step, which is the counterpart of Step 5 in Section 3 (CASE 1) or of Step 5 in Section 4 (CASE 2). In line with these two cases, it will rely on Lemma 3.5.
We return to (5.59a) and add the term Step 6. The rest of the proof for = b 1 now proceeds as in the case = b 0. In ( which is the counterpart of (3.58) in CASE 1, and (4.75 1 ) in CASE 2. By substituting (5.70 1 ) into the RHS of (5.68 1 ), we finally obtain