The controllability of a thermoelastic plate problem revisited

In this paper, the controllability for a thermoelastic plate problem with a rotational inertia parameter is considered under two scenarios. In the first case, we prove the exact and approximate controllability when the controls act in the whole domain. In the second case, we prove the interior approximate controllability when the controls act only on a subset of the domain. The distributed controls are determined explicitly by the physical constants of the plate in the first case, while this is no longer possible in the second case as the relation (79) is no longer valid. In this case, we propose an approximation of the control function with an error that tends to zero. By means of a powerful and systematic approach based on spectral analysis, we improve some already existing results on the optimal rate of the exponential decay and on the analyticity of the associated semigroup.


(Communicated by Irena Lasiecka)
Abstract. In this paper, the controllability for a thermoelastic plate problem with a rotational inertia parameter is considered under two scenarios. In the first case, we prove the exact and approximate controllability when the controls act in the whole domain. In the second case, we prove the interior approximate controllability when the controls act only on a subset of the domain. The distributed controls are determined explicitly by the physical constants of the plate in the first case, while this is no longer possible in the second case as the relation (79) is no longer valid. In this case, we propose an approximation of the control function with an error that tends to zero. By means of a powerful and systematic approach based on spectral analysis, we improve some already existing results on the optimal rate of the exponential decay and on the analyticity of the associated semigroup.
The function w represents the vertical displacement of the mid-plane of the plate from its equilibrium position, while θ the temperature at this middle plane. The vibrations of the plate are described by the Kirchhoff model which takes into account the rotational inertia through the second term in the first equation of system (1). The rotational inertia parameter γ ≥ 0 related with the thickness of the plate and therefore it is usually small. The coupling term ν∆w t (ν ≥ 0) takes into account the heat induced by the high frequency vibrations of the plate.
The above system is derived in [20] and references therein. Many authors discussed the exponential stability (with or without rotational inertia parameter) and analyticity of the system (1) under different types of boundary conditions (see, e.g. [19,21,30,31,34,35,37]). If γ > 0 ( Kirchhoff type), the system (1) has a hyperbolic character and hence the corresponding semigroup is not analytic, but the exponential stability of solutions is kept; see for example [35]. In fact, the rotational inertia parameter makes the plate more realistic physically but more difficult mathematically. Avalos and Lasiecka [6] proved the uniform exponential stability of system (1) with both clamped and simply supported boundary conditions, with Newton's law of cooling applied to the thermal component on the boundary. Additional stability and analyticity results that apply to a class of abstract systems are given in Russell [36] and Ammar Khodja and Benabdallah [17].
In the particular case γ = 0 (Euler-Bernoulli type), which corresponds to neglecting rotational inertia in the vibration of the plate, Kim [19] proved that the energy of the plate decays exponentially fast with a certain dissipative boundary condition. Liu and Renardy [31] improved Kim's result, showing that the semigroup associated to the elliptic part of the system is of analytic type, which in particular implies that the solution decays uniformly as time goes to infinity and reveals the parabolic character of the system. Chang and Triggiani [11] performed a spectral analysis of an abstract thermoelastic plate equations with different boundary conditions. They proved that the resulting semigroup of contractions is neither compact nor differentiable, which contradicts the case γ = 0. They also provided interesting spectral properties without giving the explicit expressions of the corresponding eigenvalues. Control problems for system (1) have been studied intensively in the last years, since such systems combine the conservative effect of the motion equation with the dissipative effect of the heat equation.
In the case γ = 0, the null controllability of system (1) with hinged boundary conditions is proved by Lasiecka and Triggiani [22] with a single control either on the mechanical component or in the thermal one. These results were extended to other types of boundary conditions in [4,5]. Some problems were studied by Benabdallah and Naso [8]. They proved that when the control functions (u 1 = 0 and u 2 = f ) act on the whole domain (ω ≡ Ω), for any T > 0 and for any initial data (w 0 , w 1 , θ 0 ) ∈ (H 2 (Ω)∩H 1 0 (Ω))×L 2 (Ω)×L 2 (Ω), there exists a control function f ∈ L 2 ((0, T ) × Ω) such that the problem (1) is null controllable. This results was found by Lasiecka and Triggiani [22] by a different method. The same result was obtained when the control function f acts on a small subset ω of Ω [8]. The proof of this theorem follows from the procedure developed by Lebeau and Robbiano in [24] (see also [25,29]).
In [38], de Terasa and Zuazua proved that system (1), is controllable in the following sense: If the control time is large enough and we act in the equation of displacement by means of a control supported in a neighborhood of the boundary of the plate, then we may control exactly the displacement and simultaneously the temperature in an approximate way. The method of proof is inspired from techniques elaborated by Zuazua in [40]. Leiva [27] proved a necessary and sufficient algebraic condition for the approximate controllability for thermoelastic plate equations with Dirichlet boundary conditions. The controllability analysis is based on Kalman criterion.
In the case γ > 0, Lasiecka and Seidman [23] proved observability inequalities either for the thermal component or the elastic one with hinged boundary conditions. The observation is in the whole set Ω and it is valid for any T > 0. Avalos [4] proved the null controllability with a single control acting in the thermal under clamped boundary conditions. Another null controllability result was given by Castro and Teresa in [10] for system (1) when two different controls are written in series form. The result is then obtained by combining the null controllability of these new systems with the convergence of the series. Hansen and Zhang [16] proved that a linear thermoelastic beam may be controlled exactly to zero in a finite time by a single boundary control. Moreover, they showed that the optimal time of controllability becomes arbitrarily small if γ = 0, as in the case of Euler-Bernoulli beam.
Regularity, stability and controllability are the most interesting properties for the solutions to evolution equations that attract people's attention. In spite of the importance of these results, the explicit expressions of the optimal decay rate and the distributed controls, have not yet been addressed in the literature for thermoelastic plates. So, we study this topic here to fill this gap.
In this paper, and through a simple, powerful and systematic approach based on the spectral analysis of semigroups, we prove the exponential stability, analyticity, approximate and exact controllability for a system (1)-(3) when the controls act on the whole domain (ω ≡ Ω) and the interior approximate controllability results when the controls act on an open subset of the domain (ω Ω). Moreover, we provide the explicit expressions of the optimal decay rate and the distributed controls (ω ≡ Ω) by the physical constants of the plate. In fact, these expressions have never been given explicitly in the literature. In the case (ω Ω), it is no possible to explicitly determine the control function since (79) is no longer valid. The analyticity of the thermoelastic plate was proved in [27,28] by a spectral analysis also. But our approach is more detailed and complete since it based on the explicit expressions of the eigenvalues of the corresponding linear operator. The spirit of our approach is very different from the traditional methods used to study these topics for thermoelastic plates. In fact, our spectral analysis gives more precise bounds that allow us to provide the explicit expressions of the optimal decay rate and the distributed controls.
This paper is organized as follows. In Section 2, we perform the spectral analysis of the problem. In Section 3, we prove that the uncontrolled problem decays exponentially to zero at a rate determined explicitly. In the particular case γ = 0, we prove that the associated semigroup is analytic. In Section 4, we prove the approximate and the exact controllability of the problem and we provide the explicit expressions of the distributed controls, when (ω ≡ Ω). In the other case, when (ω Ω), we prove the interior approximate controllability and we propose an approximate expression of the control function with an error which tends to zero.
2. Spectral analysis of the problem. Before starting the analysis of the problem (1)-(3), let us summarize the main properties of some operators that will be useful to us. Consider the positive operators A and A 2 on X = L 2 (Ω) defined by Aφ = −∆φ and The operator A has the following very well-known properties (see [12]).
have finite multiplicity equal to the dimension of the corresponding eigenspace.
where ·, · is the inner product in X = L 2 (Ω) and So {E n } is a complete family of orthogonal projections in X and (f ): The fractional power spaces X r are given by and Setting z = (z 0 , z 1 , z 2 ), we have Let the state space is Let us introduce the inertia operator (I − γ∆) with the domain (H 2 ∩ H 1 0 )(Ω). It is well known that the operator (I − γ∆) is a positive and self-adjoint in L 2 (Ω). Then, one has (1 + γλ n ) E n z 2 < ∞}, endowed with the norm The system (1)- (3) and (4) can be written respectively as a linear evolution equation of the form where with the domain Computing Az yields where {P n } n≥1 is a complete family of orthogonal projections in Z γ satisfying and We can also easily verify that A n P n = P n A n , n ≥ 1.
The characteristic equation of R n is given by where c n = 1 + γλ n .
(i): The characteristic equation (17) is the same that obtained in [2,16]. Moreover, Hansen and Zhang [16] proved that the roots of (17) are simple if 0 < ν ≤ 1/ √ 2. In Proposition 2.3, we shall improve this result. (ii): The eigenvalues of R n are given by where i (n) are the roots of the following equation 3 + c n 2 + (1 + ν 2 ) + c n = 0.
Proof. The discriminant of the characteristic polynomial in (20) reads as For ∆ < 0, we know that (20) possesses three distinct roots: one real and two (non-real) complex conjugate ones. Taking into account 1 + ν 2 ≥ 1 and using Young's inequality Therefore, for ∆ to be negative, it suffices that Then From this and Lemma 2.2 we deduce for c > 0 that To prove (v) we used the fact that ν < 2 and a contradiction argument.
By (v) we know that the branches {σ 0 (c n )}, {σ 1 (c n )} and {σ 2 (c n )} are distinct. Thus we only need to show that the roots are distinct within each of these branches. It will therefore suffice to show that the functions |σ i (c)|, i = 0, 1, 2, are monotone on [1, ∞) (This is since c n = √ 1 + γλ n ≥ 1). First note that since 0 (c), 1 (c) and 1 (c) are distinct roots of P c ( ) = 0, the Implicit Function Theorem implies that 0 (c), 1 (c) and 1 (c) are analytic functions of c (locally) for each c ≥ 1. In particular, these functions are differentiable for Eliminating e 1 and 0 from (26) gives Implicitly solving for r (c) we obtain Note that by (ii), N (c) is negative for all c > 0.
On the other hand, the discriminant of D(c) = 3r We are going to show now that R(c), given by (27) .
where √ . and 3 √ . stand for the main branch of complex square and cubic roots. The three roots of (20) are given by where I is the imaginary unit (I 2 = −1).
Otherwise, compute By (30), the expression of B becomes Following the same procedure as in [33] (page 179), one can get (29).
(i): Note that C in (29) cannot be zero due to the simplicity of the roots. Moreover, we already know that ∆, defined by (23) Also, it is easy to prove that ∆ 1 , given by (28), is positive for 0 ≤ ν < 2 √ 19 − 8. Consequently, one can easily conclude that C > 0 for 0 ≤ ν < 2 √ 19 − 8.
We shall prove that the supremum of n −→ max i=0,1,2 e σ i (n) over n ≥ 1 is attained at n = 1.
Let σ i (λ) ∈ C, i = 1, 2, 3, denote the roots of (32) for arbitrary λ > 0. To prove the minimum of eσ i (λ) is attained at the smallest λ, for i = 1, 2, 3, we need to show To compute the later derivative, we apply the Implicit Function Theorem to (32).
In the following we prove that max i=0,1,2 e σ i (1) is attained at i = 1 (or i = 2).
Proof. From (29) and Lemma 2.2, we have Subtracting the above expressions, we obtain The sign of this difference depends on the sign of C + ∆0 C . In the following we study this issue.
Recall that, according to Remark 2.5, ∆ 1 > 0 and C > 0 for 0 ≤ ν < 2 √ 19 − 8. From (31) and (30) which leads together with (28) to We distinguish now two cases: Adding both expressions of (38), we get C has the same sign that C is positive in both cases, (36) follows from (37) and (19). 3. Exponential decay and analyticity of the uncontrolled system. In this section we study the exponential decay and analyticity of the uncontrolled system (u 1 = u 2 = 0). Let us before recall some preliminaries concerning analytic semigroups on X , a complex Banach space with dual space X . Let L (X ) be the Banach space of all linear bounded operators on X . For 0 < ϕ ≤ π, let S(ϕ) be the sector S(ϕ) = {z ∈ C \ {0} : | arg(z)| < ϕ}.
(i): Definition 3.1 is equivalently to the definition 3.7.1 in [3]. In fact, a semigroup is analytic if it has an analytic extension to a sector.
The following assertions are equivalent. (i) (A, D(A)) is the generator of an analytic semigroup in S(ϕ 0 ) on X .
In the case of uncontrolled problem, it is well known that the operator A defined by (12) generates a C 0 -semigroup {T (t)} t≥0 = {e −tA } t≥0 (see [35]). In view of (13), the C 0 -semigroup has the following representation where P n and A n are defined by (14) and (16), respectively. Following our procedure in [1], one can prove Lemma 3.4. The operator R n defined by (16) can be written as where {q n i } 2 i=0 ∈ R 3 is a complete family of complementarily projections defined by where c n is defined by (18), σ i (n) is defined by (19) and Moreover, we have where P ni = q n i P n , i = 0, 1, 2 is a complete family of orthogonal projections in Z γ .
(ii): Since R n given by (16) is diagonalizable, it can be written as where and where 3.1. Optimal rate of decay. Here, we are interested in determining explicitly the optimal decay rate of (1)-(3) with γ ≥ 0 and u 1 = u 2 = 0.
Theorem 3.6. For 0 ≤ ν < 2 √ 19 − 8, the semigroup {T (t)} t≥0 given by (39) decays exponentially to zero, where N is a positive constant and µ 1 is the optimal decay rate given by where Proof. Since J n is diagonal, (39) can be written as K −1 n e tJn K n P n z (use P 2 n = P n ) = ∞ n=1 K −1 n P n e tJn P n K n P n z, z ∈ Z γ which, written in norm, reads We define the following two linear bounded operators K n P n : H → Z γ and K −1 n P n : Z γ → H, where H = X × X × X, Z γ = X × V γ × X and X = L 2 (Ω). Let us find bounds for K −1 n P n and K n P n . Consider z = (z 0 , z 1 , z 2 ) ∈ Z γ , such that z Zγ = 1, it follows from (9) that It follows from (14) and (47) that Using (52) we infer that From (5) and (29), we infer for i = 0, 1, 2 and n ≥ 1, that which implies the existence of a positive constant Γ 1 depending on γ and ν such that

THE CONTROLLABILITY OF THERMOELASTIC PLATES PROBLEM 15
To find a bound for K n P n L (H,Zγ ) , we consider z = (z 0 , z 1 , z 2 ) ∈ H, such that z H = 1. As a result and E j z i ≤ 1, i = 0, 1, 2, j ≥ 1.
By applying Rouché's Theorem to (20), we can prove eσ i (n) → −∞ as n → +∞ for i = 0, 1, 2. Since σ i (n) is always negative and continuously depends on λ n (roughly speaking), sup n∈N σ i (n) must be attained at some finite n.
where z ∈ Z 0 = X 3 = L 2 (Ω) 3 and λ ∈ S(ϕ + π 2 ). Proof. In view of (13), the resolvent of A becomes On the other hand, direct calculations give Similarly, we can obtain that n≥1 (λI − A n P n ) −1 P n λI − n≥1 A n P n = n≥1 P n = I.
Combining the two last identities, we conclude that Taking into consideration Eqs.  Proof. Thanks to Theorem 3.3, it suffices to prove the assertion (ii). To this end, we define the following two linear bounded operators K n P n : Z 0 → Z 0 and K −1 n P n : Z 0 → Z 0 .
Using (61) and P 2 n = P n , we get Using the same argument to get the estimations (54) and (57), one can prove that Let λ ∈ S(ϕ + π 2 ), to find a bound for (λI − J n P n ) −1 L (Z0) , we consider w = (w 0 , w 1 , w 2 ) ∈ Z 0 and z = (z 0 , z 1 , z 2 ) ∈ Z 0 such that Moreover, it is not difficult to check that which, written in norm, reads Since {E n } is a complete family of orthogonal projections in Z 0 satisfying it follows from (67) and (69) that Since S(ϕ + π 2 ) ⊂ ρ(A), by substituting (70) into (68) we arrive at
Following the standard approach (see, e.g., Curtain and Zwart [13]), we define the following concepts: (a): The controllability map G : whose adjoint operator G * : Z γ → L 2 (0, τ ; L 2 (Ω)), is given by The Gramian mapping W : Z γ → Z γ is given by W = GG * that is to say where T (t) is the C 0 -semigroup defined by (39).
Moreover, the operator B is given by (11) and its adjoint is given by the following Proposition 4.3. The operator J γ : X → V γ is a continuous (bounded) in X. Moreover, the operator B is bounded in Z γ and Proof. First we will show that J γ : X → V γ is bounded. From (9) 1 , we have Since Therefore J γ is bounded and consequently the operator B is bounded. Now, we shall give B * : V γ → X for ω ≡ Ω, Analogously we obtain B * for ω Ω. This completes the proof.
(ii): In the case ω ≡ Ω, we have P n BB * = BB * P n , n ≥ 1, whereas in the case ω Ω, the relation (79) does not hold, P n B ω B * ω = B ω B * ω P n , n ≥ 1. 4.1. Approximate and exact controllability when ω ≡ Ω. In the following we show that the operator W can be written in the form of series thanks to (79).
From condition (75) and the representation (39) of T (t) we obtain e Ant P n BB * e A * n t P n ds. Now using (79), we obtain e Ant BB * P n e A * n t P n ds = ∞ n=1 τ 0 e Ant BB * e A * n t P n ds. Hence where W n (τ ) : R(P n ) → R(P n ) is defined by while R(P n ) = Range(P n ) and B n (τ ) = τ 0 e Rns BB * e R * n s ds.
The following result is proved in [27].
Theorem 4.5. The system (10) is approximately controllable on [0, τ ] if and only if the finite dimensional systems y = A n P n y + P n Bu, y ∈ R(P n ); n ≥ 1 (83) Our main tool is the following result proved in [26] (see Lemma 1) and in [27] (see Proposition 4.1).  4.6 holds for general operators B * , P n , W n and A * n (see [21] for more details). We remark that the assertion (b) means the approximate controllability of (10). Now we prove the approximate controllability of (1)-(3).
Proof. According to Theorem 4.5 and Proposition 4.6, it suffices to show that the assertion (b) holds. In view of (16), we observe that A * n = R * n P n where A * n and R * n are the adjoint operators of A n and R n , respectively Following the same argument of our previous work (see Lemma 3.4

of [1]), one can prove that there exists a family of complete complementary projections
and Υ is defined by (41). Combining (84) and (85) we get From (44), we infer that Multiplying (88) 2 by B * P n and using (69), we arrive at Now, suppose for z ∈ Z γ that B * P n e A * n t z = 0, for all t ∈ [0, τ ]. We need to show that z = 0. From (89), for i = 0, 1, 2 and n ≥ 1, we obtain Solving this last equation, we easily obtain that E n z i = 0 which readily implies that z i = 0.
The following result (see [9,13]) proves to be helpful in showing the exact controllability of the problem (1)-(3) when ω ≡ Ω.  (i) The operator W n is invertible and its inverse is given by and (93) (ii) Moreover, there exist a positive constant C 1 depending on ν and γ such that Proof. (i) Thanks to the equivalence between assertions (a) and (c) of Proposition 4.6, we obtain that the operator W n is invertible. By (81), B n (τ ) is then invertible on R(P n ) and we have We have used (15) and the fact that P n is invertible on R(P n ) and its inverse is equal to itself. First we give the expression of B −1 n (τ ). We infer from (78) and (87) that By (82) we get and a ij are given by (92). Since B n (τ ) is invertible, the determinant ∆ n of B n (τ ) given by (93) is nonzero, then
Since {P n } n≥1 is a complete family of orthogonal projections in Z γ and P n W n (τ ) = W n (τ )P n , so {W n (τ )P n z} n≥1 is a family of orthogonal vector in Z γ . Then, from (80) we have Therefore from (60), (97) follows immediately.
Remark 4.11. We remark that the operator W −1 can not be obtained explicitly from the expression W because the projection P n as a mapping from Z γ to Z γ is not invertible. In fact only the restriction of P n on R(P n ) is invertible from Z γ to Z γ . Now we are ready to prove that W is invertible in Z γ .
Lemma 4.12. The operator W is invertible in Z γ and its inverse W −1 (τ ) : Z γ → Z γ is defined by where ∆ n and b ij are given by (93) and (91), respectively.
Proof. Since W −1 n is bounded, we set U (τ ) = ∞ n=1 W −1 n (τ )P n . By (60) and (94) we infer that the operator U is bounded. For all z ∈ Z γ , from (80) we have Similarly one can prove that W (τ )U (τ )z = z for all z ∈ Z γ . Then W −1 (τ ) = U (τ ). By (90) our conclusion follows. Now, we are ready to formulate the exact controllability of the problem (1)-(3) by a control u = (u 1 , u 2 ) acting on the whole domain Ω that will be determined explicitly by the physical coefficients of the plate.

4.2.
Interior approximate controllability when ω Ω. The previous approach can not be applied here since the relation (79) does not hold when ω Ω. Consequently the operator W can not be written in the form of series. However the following property of analytic functions proves to be helpful (see Theorem 1.23 from [7], pg. 20).  tends to zero as α tends to zero. Proof.
(i): According to Theorem 4.5 and Proposition 4.6, it suffices to show that the assertion (b) holds. In view of (16), we observe that A * n = R * n P n where A * n and R * n are the adjoint operators of A n and R n , respectively with Combining (111) and (112), we get This completes the proof of the theorem.

5.
Conclusion. In the case ω ≡ Ω, the control u is determined explicitly by the physical parameters by writing the expressions of W and W −1 in the form of series (see (109) where (103) is used) thanks to the relation (79). When ω Ω, the relation (79) is no longer possible and consequently the control u can not be expressed explicitly. As R(α, −W ) converges to W −1 as α → 0, we propose an approximation of the control functions by replacing W −1 in (107) by R(α, −W ). Then we prove that the error of this approximation given by (117) tends to zero as α → 0.