Luenberger compensator theory for heat-Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls
-
Roberto Triggiani
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 wave- or plate-structures, 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.
1 Introduction
The present article was conceived in response to the intents of the special issue of Open Mathematics. In fact, it provides a new, optimal, rather complete and comprehensive theory on a control theory topic of long standing relevance to applications, with the focus on a recently introduced, by necessity special, class of coupled partial differential equation (PDE) models: heat-structure interaction (HSI) models with high, Kelvin-Voigt damping for the wave-structure, subject to feedback control exercised on the boundary.
Homogeneous (uncontrolled) model. Throughout,
The fluid-structure interaction.
On this geometry in Figure 1, we thus consider the following heat-structure PDE model in solution variables
The constant
for the variable
In (1.2a), the space
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
with
Three controlled systems. We consider three cases of boundary/interface control
In each case, the control
with the highly unbounded control operator
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:
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].
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], [6–8] 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);
analysis of the same models as in (i) and (ii), and this time, however, with no Kelvin-Voigt damping, such as in [11– 14] (wave), where the original stability properties are different. Here, in contrast with the heat-structure case,
1.1 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 [6–8], [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 [29–32]. 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 finite-dimensional, in line with Luenberger’s original contribution [27]. Its standard representation is as follows:
The basic idea behind is that the full state
One next assumes the detectability condition for the pair
and the dynamic compensator
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
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], [6–8]. 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
1.2 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.
1.2.1 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
1.2.2 Controlled systems
y
′
=
A
y
+
ℬ
g
As already noted, we consider three cases of boundary/interface control
CASE 1. Here, the following analysis selects the feedback form
CASE 2. Here, the following analysis selects the feedback form
CASE 3. Here, the following analysis selects the feedback form
Insight on the choice
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:
b
=
0
,
b
=
1
[1]
We return to the homogeneous problem (1.1a)–(1.1f), which for easy reading, we reproduce here:
The constant
for the variable
In (1.2a), the space
Abstract model of the homogeneous PDE problem (1.1a)–(1.1f). The operator
where the operator
A description of
- (1.9a)
- (1.9b)
Remark 1.1
The aforementioned description of
Theorem 1.1
(The adjoint
The PDE version of
is given by
compared with equation (1.1a)–(1.1e). The domain
- (1.11a)
- (1.11b)
Actually, [1, Appendix A] gives the detailed proof for
Theorem 1.2
(Generation by
The operator
(1.12)(1.13)in the
Thus,
(1.14)The same generation results hold also for
Again, [1, Section 2] gives the proof only for
Theorem 1.3
With reference to the operator
- (1.15)
with explicit expression of
[1, Theorem 1.4] The contraction semigroups
(1.16)
Remark 1.2
Section 1.3 (a subset of [1]) shows that the natural functional setting for problems (1.1a)–(1.1f) is: the energy space
If, however, one insists in considering problems (1.1a)–(1.1f) with
(1.17)In fact, setting equal to zero equation (1.8) with
In this case, we may view the problem with
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
with Neumann boundary control
2.1 Abstract model on
H
b
,
b
=
0
,
1
,
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
with the full dynamic operator
We first define the positive self-adjoint operator
so that
Next we return to (2.1b) and rewrite it via (2.4) and (2.1e) [40], [4, Chapter 3]
By (2.1), (2.1e), the term
where now
while
The following regularity holds true for
continuously. Then, as usual [1], we rewrite the
where
By combining (2.8) and (2.13), we obtain
The operator in (2.14) acting on
Thus, in conclusion, the abstract model for the nonhomogeneous PDE model (2.1a)–(2.1f) is given again by (2.2), where now
recalling (2.5). The adjoint operator
with
where we shall establish that
for the operator
Proof of (2.20)
[see also [4, pp. 195–196].] Take initially
In the aforementioned computation we have used (2.4) and
initially for
In conclusion:
Theorem 2.1
Let
with
Henceforth, in light of Theorem 2.1, we shall omit the qualifying parameter “
3 The Luenberger’s compensator model for the heat-structure interaction model (1.1a)–(1.1f) with Neumann boundary control
g
at the interface
Γ
s
,
b
=
0
, and
b
=
1
3.1 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
Thus, the special setting becomes, in this case,
leading to the Luenberger’s dynamics
and hence,
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
Insight. How did we decide that
3.1.1 The counterpart of
y
˙
=
A
y
–
B
B
∗
z
in (2.3a) for the heat structure interaction (2.1a)–(2.1f), with
F
→
−
ℬ
N
∗
Accordingly, for
where we have critically invoked the trace result (2.26) of Theorem 2.1 in both cases
from problem (2.25),
3.1.2 The counterpart of the dynamic compensator equation
z
˙
=
(
A
‒
2
B
B
∗
)
z
+
B
(
B
∗
y
)
in (2.3b) for the heat-structure interaction (2.1a)–(2.1f)
With partial observation as in (3.2) according to (2.26)
the compensator equation (3.3b) in
or, via (1.8) for
whereby
The PDE version of the abstract
3.1.3 The dynamics
d
˙
=
(
A
b
‒
ℬ
N
(
b
)
ℬ
N
(
b
)
∗
)
d
,
d
(
t
)
=
y
(
t
)
‒
z
(
t
)
, corresponding to (3.4): analyticity,
b
=
0
and
b
=
1
We are omitting the superscript “
Theorem 3.1
Let
is the infinitesimal generator of a s.c. contraction semigroup
A more detailed description of
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.
Step 1. Identification of the
Lemma 3.2
Let
recalling (1.8) for
so that
Remark 3.1
We note that the term
Description of
- (3.17a)
- (3.17b)
The adjoint operator
recalling also
Description of
The PDE corresponding to
(where
Step 2. (Analysis of the PDE problem (3.16): the operator
Proposition 3.3
Let
in the
for the
for the
Proof of (3.22)
For
where
recalling (2.26). On the other hand, the first term on the right-hand side (RHS) of (3.26) is equal to
Step 3. This step provides the key PDE-energy estimate,
Remark 3.2
Given
in terms of
Theorem 3.4
Let
for
the following estimate holds true: given
(3.30)(Analyticity) In view of Remark 3.2, Estimate (3.30) without the term
(3.31)(3.32)Thus, the s.c. contraction semigroup
Proof
(i) The proof of estimate (3.30) follows closely the technical proof of [1, Section 3] for the operator
Step 1. Return to (3.29) re-written for
Step 2. Take the
Similarly, we take the
We now invoke the B.C.
Summing up (3.34) and (3.36) yields after a cancellation of the boundary term
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:
or
Step 4. We now take the imaginary part of identity (3.38), thus obtaining the new identity
or
Take now
so that (3.43) yields for
hence, for
Remark 3.3
In the case
or
or
Estimate (3.48) coincides with estimate [1, equation (3.26)], case
Next, substitute estimate (3.49) into the RHS of estimate (3.46) with
Summing up estimate (3.49) with estimate (3.50) finally yields for
for all points
The analyticity of the s.c. contraction semigroup
Step 5. We proceed now with the proof of analyticity in the case
Lemma 3.5
[44, p. 260] On a sufficiently smooth bounded domain
hence for positive constant
We now return to inequality (3.46) with
hence,
by taking
The LHS in (3.55) estimates the two terms
Step 6. We substitute the new estimate (3.55) on the RHS of (3.40). We obtain
or
or
Estimate (3.58) for
Step 7. Summing up estimates (3.58) with estimate (3.55) finally yields for
for all points
3.1.4 Exponential stability of
e
A
F
,
N
(
b
)
t
and
e
A
F
,
N
(
b
)
∗
t
on
H
b
,
b
=
0
,
1
In Proposition 3.6, we shall prove, in both cases
Hence, [45] the s.c. analytic semigroup
Similarly for the adjoint
Proposition 3.6
Statement (3.60) holds true. Hence, the exponential stability for
is given explicitly by
where the positive self-adjoint operator
While
Proof
Identity (3.63) yields by (3.17b)
and the
Moreover, (3.63),
from which the expression for
4 CASE 2. Heat-structure interaction with Kelvin-Voigt damping: Dirichlet control
g
at the interface
Γ
s
: abstract model
We return to the homogeneous heat-structure interaction model (1.1a)–(1.1f) with Kelvin-Voigt damping. In this CASE 2, we insert a control
this time with Dirichlet control
4.1 Abstract model on
H
b
,
b
=
0
,
1
of the nonhomogeneous PDE model (4.1a)–(4.1f) with Dirichlet control
g
acting at the interface
Γ
s
This topic was duly treated in [39, Section 6]. Here, it was shown that the abstract version of the nonhomogeneous PDE model (4.1a)–(4.1f) is given by
where the operator
Here,
while
The following regularity holds true for
Thus, with
in the following sense. For
where we have recalled (the normal
from [1, p. 181], [4, Chapter 3]. Thus (4.7) is established. The proof initially takes
As noted in Theorem 0.3 from [1], the operator
4.2 The Luenberger’s compensator model for the heat-structure interaction model (4.1a)–(4.1f) with Dirichlet control
g
at the interface
Γ
s
,
b
=
0
,
1
4.2.1 Special selection of the data
For the present heat-structure interaction problem with Dirichlet control at the interface
Thus, the special setting becomes, in this case,
leading to the Luenberger’s dynamics
(as
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
Insight. How did we decide that
4.2.2 The counterpart of
y
˙
=
A
y
+
B
B
∗
z
in (4.14a) for the HSI model (4.1a)–(4.1f) with
F
→
ℬ
D
∗
Accordingly, for
i.e., with
from problem (4.2) with operator
4.2.3 The counterpart of the dynamic compensator equation
z
˙
=
A
z
+
B
(
B
∗
y
)
in the special setting (4.14b) for the HSI model (4.1a)–(4.1f)
With partial observation as in (4.13),
the compensator equation (4.14b) in
or via (1.8) on
whereby
4.2.4 The dynamics
d
˙
=
(
A
b
–
ℬ
D
ℬ
D
∗
)
d
,
d
(
t
)
=
y
(
t
)
–
z
(
t
)
corresponding to (4.15): analyticity and exponential decay,
b
=
0
,
1
The main result of the present section is as follows:
Theorem 4.1
Let
in the infinitesimal generator of a s.c. contraction semigroup
A more detailed description of
Step 1.
Lemma 4.2
Let
or via (1.8) on
so that
Description of
- (4.28a)
- (4.28b)
recalling
Description of
The PDE version corresponding to the adjoint operator
(where
Step 2. (Analysis of the PDE problem (4.27): the operator
Proposition 4.3
The operator
in the
for the
Proof of (4.33)
For
where recalling (4.3) for
recalling
Thus, (4.38) and (4.39), used in (4.36), yield (4.33), as desired. Similarly for (4.34) starting from (1.10a) for
4.2.5 Analyticity of
e
A
F
,
D
(
b
)
t
and
e
A
F
,
D
(
b
)
∗
t
on
H
b
,
b
=
0
,
1
Remark 4.1
Given
in terms of
Theorem 4.4
Let
for
the following estimate holds true: there exists a constant
(4.41)In view of Remark 4.1, estimate (4.41) is equivalent to
(4.42)(4.43)Thus, the s.c. contraction semigroup
Proof
(i) The proof of estimate (4.41) follows closely the technical proof of estimate (3.30) for the operator
Step 1. Return to (4.40), re-written for
Step 2. Take the
Similarly, we take the
We now invoke the B.C.
Summing up (4.45) and (4.47) yields after a cancellation of the boundary terms
By using via (4.44a), the identities
and we obtain the final identity
Step 3. We take the real part of identity (4.50), thus obtaining the new identity:
or
Step 4. We now take the imaginary part of identity (4.50), thus obtaining the new identity
or
Take now
so that (4.55) yields for
hence for
Remark 4.2
In the case
or
or
Estimate (
Substitute estimate (
Summing up estimate (
for all points
Step 5. We proceed now with the proof of the case
Thus, with
where
For
Next from
and hence,
Using (
Next we invoke Lemma 3.5(a) on the term
where going from (
which is legal since
Now we select
and
(we can always take
recalling
We finally obtain the desired estimate also for
for
Now, add the estimate for
Estimate (
Step 6. The rest of the proof for
or, since
which is a counterpart of (
Finally, summing up (
for all
Estimate
4.2.6 Exponential stability of
e
A
F
,
D
(
b
)
t
and
e
A
F
,
D
(
b
)
∗
t
on
H
b
,
b
=
0
,
1
In Proposition 4.5, we shall prove that, in both cases
so that there exists a disk
Hence, [45] the s.c. analytic semigroup
It was similar for the adjoint
Proposition 4.5
Statement (4.78) holds true. Hence, the exponential stability for
is given explicitly by
In the operator form, we have
where the operators
and
Proof
Identity (4.81) and the characterization of
and the
Then, the solution of problem (4.88) is given by (4.82) via (4.83), where the operators
where
Proposition 4.5 regarding (4.79) and (4.80) is proved.□
5 CASE 3: Heat-structure interaction with Kelvin-Voigt damping: Dirichlet control
g
at the external boundary
Γ
f
We return to the homogeneous heat-structure interaction model (1.1a)–(1.1f) with Kelvin-Voigt damping. In the present CASE 3, we apply a Dirichlet control
with Dirichlet boundary control
5.1 Abstract model on
H
b
,
b
=
0
,
1
,
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
[40,41], [4, p. 181], [39], where
Similarly, we recall from (2.4) the Neumann map
with regularity given by (2.5), as well as the Neumann-Laplacian in
To obtain the abstract model of problem (5.1), we proceed as usual [4,39]. We re-write the
where
The term
where
ultimately via (5.5) with
Combining (5.10) for the
The operator in (5.12) on
where the operator
The adjoint operator
in the following sense. For
where we have recalled from [4, Chapter 3] (same technique as in obtaining (2.24), CASE 1 or (4.11), CASE 2)
Thus, (5.15) is established. As shown in (4.11), the proof initially takes
5.2 The Luenberger’s compensator for the heat-structure interaction model (5.1a)–(5.1f) with Dirichlet control
g
at the external boundary
Γ
f
,
b
=
0
,
1
5.2.1 Special selection of the data
For the present heat-structure interaction problem with Dirichlet control at the external boundary
as in CASE 2, and unlike CASE 1. Thus, the special setting becomes, in this case,
leading to the Luenberger’s dynamics
(as
Insight. How did we decide that
5.2.2 The counterpart of
y
˙
=
A
y
+
B
B
∗
z
in (5.22b) for the HSI model (5.1a)–(5.1f) with
F
→
B
D
∗
Accordingly, for
by (5.15). With
recalling
5.2.3 The counterpart of the dynamic compensator equation
z
˙
=
A
z
+
B
(
B
∗
y
)
in (5.22b) for the HSI model (5.1a)–(5.1f)
With partial observation as in (5.21) according to (5.24) or (5.15)
the compensator equation (5.22b) in
or via (1.8) for
whereby
5.2.4 The dynamics
d
˙
=
(
A
b
‒
B
D
B
D
∗
)
d
,
d
(
t
)
=
y
(
t
)
‒
z
(
t
)
, corresponding to (5.23): analyticity and exponential decay,
b
=
0
,
1
The main result of the present section is as follows:
Theorem 5.1
Let
is the infinitesimal generator of a s.c. contraction semigroup
A more detailed description of
The proof of Theorem 5.1 is by PDE methods, which consist of analyzing the corresponding PDE system (5.30). We proceed through a series of steps.
Step 1.
Lemma 5.2
Let
or via (1.8) on
so that
Description of
- (5.35a)
- (5.35b)
recalling
Description of
The PDE version corresponding to the adjoint operator
(where
Step 2. (Analysis of the PDE problem (5.34): the operator
Proposition 5.3
The operator
in the
for the
Proof of (5.3)
For
recalling (1.12) and (5.24). Thus, (5.40) is established. Similarly for (5.41) recalling (1.13) for
5.2.5 Analyticity of
e
A
F
,
D
(
b
)
t
and
e
A
F
,
D
(
b
)
∗
t
on
H
b
,
b
=
0
,
1
Remark 5.1
A remark such as Remark 3.2 (CASE 1) or Remark 4.1 (CASE 2) applies in the present case, to justify (5.47) from (5.46).
Theorem 5.4
Let
for
the following estimate holds true: there exists a constant
(5.46)In view of Remark 5.1, estimate (5.46) without the term
(5.47)(5.48)Thus, the s.c. contraction semigroup
Proof
(i) The proof of estimate (5.46) follows closely the technical proof of estimate (3.30) for the operator
Step 1. Return to (5.45), re-written for
Step 2. Take the
Similarly, we take the
Now we invoke the B.C. in (5.35a) and (5.35b) for
and re-write (5.50) and (5.51) as follows, respectively
Summing up (5.53) and (5.54) yields after a cancellation of the boundary terms
We now use the identities from (5.49a)
to obtain the final identity from (5.55)
Identity (5.57) is the counterpart of identity (4.50) for Dirichlet control
Step 3. We proceed as in Step 3 of CASE 1 or of CASE 2. We take the real part of identity (5.57) and next obtain
the counterpart of estimate (4.52) CASE 2, or estimate (3.40), CASE 1.
Step 4. We now take the imaginary part of identity (5.57), and proceeding as in Step 4 of CASE 1 or CASE 2, we arrive at the following estimate
Equation (5.59a) and (5.59b) are the counterpart of (3.46) (CASE 1) or (
Case
Step 5. Case
We return to (5.59a) and add the term
We obtain
for
In going from (
Invoking (
We now select
(we can always assume
for
for
Substituting the estimate for
Next add to (
recalling
Step 6. The rest of the proof for
or, since
which is the counterpart of (3.58) in CASE 1, and (
Summing up (
which is the sought-after estimate (5.15),
5.2.6 Exponential stability of
e
A
F
,
D
(
b
)
t
and
e
A
F
,
D
(
b
)
∗
t
on
H
b
,
b
=
0
,
1
In Proposition 5.5, we shall prove that, in both cases
so that there exists a disk
Hence, [45] the s.c. analytic semigroup
It is similar for the adjoint
Proposition 5.5
Statement (5.73) holds true. Hence, the exponential stability for
is given explicitly by
Here,
In operator form, we have
where the operators
Proof
Identity (5.76) and the characterization of
and the
Then, the solution of problem (5.81) is given by (5.77a) via (5.77b).□
Remark 5.2
A recent contribution of a heat-plate interaction with the plate subject to a (formal) “square root” damping in [46].
Acknowledgments
The authors wish to thank Yongjin Lu, Oakland University, MI, for reading the manuscript and providing useful comments; and Justin Webster, University of Maryland, Baltimore County, for providing insight and references on data assimilation.
-
Funding information: The research of R.T. was partially supported by the National Science Foundation under Grant NSF DMS-2205508.
-
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
-
Conflict of interest: The authors state no conflict of interest.
Appendix A The genuine fluid-structure interaction model with Kelvin-Voigt (viscoelastic) damping [2,12,14]
In this article, we have deliberately chosen to consider the simpler heat-viscoelastic structure model, as a first step of an entirely new investigation regarding the corresponding Luenberger theory. To be sure, replacing the heat equation with a fluid equation introduces conceptual and technical difficulties. These however have already been dealt with and ultimately resolved in prior work by one co-author, for a structure model originally without viscoelastic damping [12], and next with viscoelastic damping [2]. A first serious obstacle is faced at the very outset: because of the coupled nonhomogeneous boundary conditions involving the linearized Navier-Stokes equations, it is not possible to use the classical, by now standard idea of N-S problems with no-slip boundary conditions to eliminate the pressure by applying the Leray projector on the equation from
We thus consider the following fluid-plate PDE model in solution variables
and
while the scalar-valued
The constant
for the variable
The norm-including inner product on
Abstract model for the free dynamics (A.1a)–(A.1g)
The previous article [12] (as well as paper [48], where the
Elimination of p, by expressing p in terms of u, w, and
In fact, (A.4a) is obtained by taking the divergence div across equation (A.1a), and using div
Explicit solution of problem (A) for p. We set
where
Accordingly, define the following “Dirichlet" and “Neumann" maps
Elliptic theory gives that
Accordingly, in view of problems (A.7), we write the solutions
where
hence via (A.10a) and (A.10b):
where
The linear maps
only in terms of
Abstract model of system (A.15). The abstract model of system (A.15) is given by
where the matrix form for
The operator
The finite energy space
The scalar harmonic function
(compare with (A.10b) for the dynamic problem). According to the definition of the Dirichlet map
It then follows from
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- The construction of nuclei for normal constituents of Bπ-characters
- Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem
- Mean square exponential stability of stochastic function differential equations in the G-framework
- Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents
- Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow
- The dual index and dual core generalized inverse
- Study on Birkhoff orthogonality and symmetry of matrix operators
- Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
- Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
- On semigroups of transformations that preserve a double direction equivalence
- Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
- A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
- Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
- Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
- General Randić indices of a graph and its line graph
- On functional reproducing kernels
- On the Waring-Goldbach problem for two squares and four cubes
- Singular moduli of rth Roots of modular functions
- Classification of self-adjoint domains of odd-order differential operators with matrix theory
- On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
- Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
- Remarks on hyponormal Toeplitz operators with nonharmonic symbols
- Complete decomposition of the generalized quaternion groups
- Injective and coherent endomorphism rings relative to some matrices
- Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
- Continued fractions related to a group of linear fractional transformations
- Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
- Approximate controllability for a stochastic elastic system with structural damping and infinite delay
- On extremal cacti with respect to the first degree-based entropy
- Compression with wildcards: All exact or all minimal hitting sets
- Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
- Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
- Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
- On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
- Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
- On the quadratic residues and their distribution properties
- On average theta functions of certain quadratic forms as sums of Eisenstein series
- Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
- Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
- Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
- On some spaces via topological ideals
- Linear maps preserving equivalence or asymptotic equivalence on Banach space
- Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
- On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
- Entire solutions of two certain Fermat-type ordinary differential equations
- Generalized Lie n-derivations on arbitrary triangular algebras
- Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
- Notes on pseudodifferential operators commutators and Lipschitz functions
- On Graham partitions twisted by the Legendre symbol
- Strong limit of processes constructed from a renewal process
- Construction of analytical solutions to systems of two stochastic differential equations
- Two-distance vertex-distinguishing index of sparse graphs
- Regularity and abundance on semigroups of partial transformations with invariant set
- Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
- Spin(8,C)-Higgs pairs over a compact Riemann surface
- Properties of locally semi-compact Ir-topological groups
- Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
- Ordering stability of Nash equilibria for a class of differential games
- A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
- About a dubious proof of a correct result about closed Newton Cotes error formulas
- Ricci ϕ-invariance on almost cosymplectic three-manifolds
- Schur-power convexity of integral mean for convex functions on the coordinates
- A characterization of a ∼ admissible congruence on a weakly type B semigroup
- On Bohr's inequality for special subclasses of stable starlike harmonic mappings
- Properties of meromorphic solutions of first-order differential-difference equations
- A double-phase eigenvalue problem with large exponents
- On the number of perfect matchings in random polygonal chains
- Evolutoids and pedaloids of frontals on timelike surfaces
- A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
- The 𝔪-WG° inverse in the Minkowski space
- Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
- Approximate solvability method for nonlocal impulsive evolution equation
- Construction of a functional by a given second-order Ito stochastic equation
- Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
- On pomonoid of partial transformations of a poset
- New fractional integral inequalities via Euler's beta function
- An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
- Eigenfunctions in Finsler Gaussian solitons
- On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
- Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
- A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
- A digital Jordan surface theorem with respect to a graph connectedness
- A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
- The structure fault tolerance of burnt pancake networks
- Average value of the divisor class numbers of real cubic function fields
- Uniqueness of exponential polynomials
- An application of Hayashi's inequality in numerical integration
