Exact controllability for nonlinear thermoviscoelastic plate problem

1 Introduction

In the last decades, the theory of control has gained great importance as a discipline for engineers, mathematicians, and other scientists. Examples of control problems range from simple cases, such as driving from heat through a bar, to more complex cases, such as the landing of a vehicle on the Moon, the control of the economy of a nation, and the control of epidemics, among others. There is an extensive literature on control theory for continuous systems.

We study the exact controllability of the following nonlinear damped thermoelastic Kirchhoff-Love plate with controls acting in the whole domain Ω

with Dirichlet boundary conditions

and initial conditions

where Ω is a bounded domain in R d and Δ = ∑ i = 1 d ∂ x i 2 is the Laplace operator.

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 (1.1). The rotational inertia parameter γ ≥ 0 , related to the thickness of the plate, and therefore, it is usually small. The coupling term ν Δ w t ( ν ≠ 0 ) takes into account the heat induced by the high-frequency vibrations of the plate, and α Δ w t ( α > 0 ) is a viscoelastic damping term.

In (1.1), the distributed controls u 1 , u 2 ∈ L 2 ( 0 , τ ; L 2 ( Ω ) ) and the nonlinear functions f i : [ 0 , τ ] × R 3 → R , ( i = 1 , 2 ) are continuous functions on t and globally Lipschitz in the other variables, i.e., there exist constants l i > 0 such that for all x 1 , x 2 , y 1 , y 2 , u 1 , u 2 ∈ R , we have

Control problems for system (1.1) have attracted considerable attention in the literature over the past several years. In particular, many efforts have been devoted to study the controllability problem under varying boundary conditions, with different choices of control domain and with different values of γ .

In the case γ = 0 , which corresponds to neglecting rotational inertia in the vibration of the plate, the null controllability of the linear thermoelastic plate system with hinged boundary conditions was proved by Lasiecka and Triggiani [1] with a single control either on the mechanical component or in the thermal one. The controls are assumed to be supported on the whole set Ω with regularity L 2 ( ( 0 , T ) × Ω ) . These results were extended to other types of boundary conditions in [2]. Some problems were studied by Benabdallah and Naso [3]. They proved that when the control functions act on the whole domain for any T > 0 and for any initial data ( w 0 , w 1 , θ 0 ) ∈ ( H 2 ( Ω ) ∩ H 0 1 ( Ω ) ) × L 2 ( Ω ) × L 2 ( Ω ) , there exists a control function f ∈ L 2 ( ( 0 , T ) × Ω ) such that the problem of linear thermoelastic plate is null controllable. This result was found by Lasiecka and Triggiani [1] by a different method. The same result was obtained when the control function f acts on a small subset ω of Ω [3]. The proof of this theorem follows from the procedure developed by Lebeau and Robbiano in [4]. De Terasa and Zuazua [5] proved the exact approximate controllability of linear thermoelastic plate system subject to clamped boundary conditions with interior control acting only on the Kirchoff plate component. Based on the Kalman criterion, Leiva and Zambrano [6] established a necessary and sufficient algebraic condition for the approximate controllability for thermoelastic plate equations with Dirichlet boundary conditions. Chang and Triggiani [7] performed a spectral analysis of an abstract thermoelastic plate equations with different boundary conditions. They provided interesting spectral properties and proved that the resulting semigroup of contractions is neither compact nor differentiable, which contradicts the case γ = 0 . Recently, Triggiani and Zhang [8] proved analyticity, sharp location of the spectrum, and exponential decay with sharp decay rate for a two-dimensional heat-damping viscoelastic plate interaction.

In the case γ > 0 , Lasiecka and Seidman [9] proved observability inequalities for either 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 [2] proved the null controllability with a single control acting in the thermal component only under clamped boundary conditions. Another null controllability result was given by Castro and Teresa [10] for linear thermoelastic plate systems with controls 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 [11] 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. Eller et al. [12,13] studied the exact-approximate boundary controllability of thermoelastic plates with variable thermal coefficient, under three boundary controls active in the hinged mechanical/Dirichlet thermal boundary conditions and clamped mechanical/Dirichlet thermal boundary conditions, respectively. Yang and Yao [14] studied the exact-approximate controllability of system (1.1) by the multiplier technique and the Riemannian geometry approach inspired from the work by Avalos and Lasiecka [15].

Because of the smoothing effects associated with analyticity, the null controllability is a more natural question than for the hyperbolic problems. On the contrary, in this article, we consider a hyperbolic model, and we want to show that the nonlinear system (1.1) is exact controllable. The exact controllability is a hard problem even for linear systems; for that reason, there are few results on nonlinear systems. The difficulty here lies in the nonlinear functions that depend on all variables appearing in the considered system, including the control. For exact controllability of non-coupled semilinear equation, one can see, for example, the articles credited by Leiva [16–18]. In this work, we generalize Lieva’s works [16–18] on semilinear single equations to coupled thermoviscoelastic plate equations. Our procedure is quite different. In particular, we do not assume that the operator K (defined by (5.25)) is a contraction; but we prove it under a suitable condition, which makes the control holds (see the condition (5.9)). In addition, we show the surjectivity of the controllable mapping differently. In fact, thanks to a result due to Carrasco et al. [19], we prove the exact controllability of the nonlinear problem by showing that the nonlinear controllability mapping G F (given by (5.3)) is surjective. For this end, we first need to prove that the linear controllability mapping G (given by (5.1)) is surjective. According to Curtain and Zwart [20], this is equivalent to the exact controllability of the linear problem. This is done by proving that the Gramian mapping (defined by (5.4)) is invertible. Nevertheless, our work is very general and can be applied to many systems of coupled partial differential equations. For a more careful review of some known results on analogous problems, we refer to our previous study [21], where the linear thermoelastic model (1.1) is studied for α = f 1 = f 2 = 0 . When the controls act in the whole domain, we proved the exact and approximate controllability. 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.

In this article, and through a simple, powerful, and systematic approach based on the spectral analysis of semigroups, we prove the exponential stability, approximate, and exact controllability of the nonlinear thermoviscoelastic plate system given by (1.1)–(1.3). Moreover, we provide the explicit expressions of the optimal decay rate and the distributed controls by the physical constants of the plate. In fact, these expressions have never been given explicitly in the literature. The spirit of this approach is very different from the traditional methods used to study these topics for thermoviscoelastic plate. Our approach is more detailed and complete since it is 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 thermoviscoelastic plate. 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 article is organized as follows. In Section 2, we describe briefly a global well-posedness of the nonlinear problem. In Section 3, we perform the spectral analysis of the linear and uncontrolled problem. In particular, we establish conditions on the physical constants of the plate to guarantee that the roots of the characteristic equation are simple. In Section 4, we prove the exponential stability of the solutions at an optimal rate determined explicitly by the physical constants of the plate. In Section 5, we prove the exact controllability of the considered problem and the explicit expressions of the distributed controls.

2 Preliminaries and well-posedness

Before starting the analysis of problems (1.1)–(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 A ϕ = − Δ ϕ and A 2 ϕ = Δ 2 ϕ with domains D ( A ) = ( H 2 ∩ H 0 1 ) ( Ω ) , D ( A 2 ) = { w ∈ H 4 ( Ω ) with w = Δ w = 0 on ∂ Ω } . The operator A has the following very well-known properties [22].

1. The spectrum of A = − Δ consists of only eigenvalues

   (2.1) 0 < λ 1 < λ 2 < … < λ n < ∞ ,

   with finite multiplicity equal to the dimension of the corresponding eigenspace.

2. The eigenfunctions of A with Dirichlet boundary condition are real analytic functions.

3. There exists a complete orthonormal set { ϕ n } of eigenvectors of A .

4. For all x ∈ D ( A ) , we have

   A x = ∑ n = 1 ∞ λ n ⟨ x , ϕ n ⟩ ϕ n = ∑ n = 1 ∞ λ n E n x ,

   where ⟨ ⋅ , ⋅ ⟩ is the inner product in X = L 2 ( Ω ) and

   E n x = ⟨ x , ϕ n ⟩ ϕ n .

   So { E n } is a complete family of orthogonal projections in X and

   (2.2) x = ∑ n = 1 ∞ E n x , x ∈ X .

5. The operator − A generates an analytic semigroup { e − A t } t ≥ 0 given by

   e − A t x = ∑ n = 1 ∞ e − λ n t E n x .

6. The fractional power spaces X r are given by

   X r = D ( A r ) = x ∈ X , ∑ n = 1 ∞ ( λ n ) 2 r ‖ E n x ‖ 2 < ∞ , r ≥ 0 ,

   with the norm

   (2.3) ‖ x ‖ X r = ‖ A r x ‖ = ∑ n = 1 ∞ λ n 2 r ‖ E n x ‖ 2 1 ⁄ 2 , x ∈ X r ,

   and

   A r x = ∑ n = 1 ∞ λ n r E n x .

Setting Z = ( z 0 , z 1 , z 2 ) , we have

Let the state space be

equipped with the inner product

where Z 1 = ( y 0 , y 1 , y 2 ) and Z 2 = ( z 0 , z 1 , z 2 ) . In view of (2.4), the corresponding norm in Z γ is given by

Let us introduce the inertia operator ( I − γ Δ ) with the domain ( H 2 ∩ H 0 1 ) ( Ω ) , where H 0 1 ( Ω ) = { u ∈ H 1 ( Ω ) ∣ u = 0 on ∂ Ω } . It is well known that the operator ( I − γ Δ ) is positive and self-adjoint in L 2 ( Ω ) . Then, one has

endowed with the norm

System (1.1)–(1.3) can be written, respectively, as a nonlinear evolution equation of the form

where the nonlinear function F : [ 0 , τ ] × Z γ × Y → Z γ is given by

where Y = L 2 ( 0 , τ ; L 2 ( Ω ) ) × L 2 ( 0 , τ ; L 2 ( Ω ) ) and the function F : [ 0 , τ ] × Z γ × Y → Z γ is given by

where J γ = ( I − γ Δ ) − 1 : X → V γ . We assume that F is a strict contraction on [ 0 , τ ] × Z γ × Y , i.e., there exists a constant L < 1 such that for all Z 1 , Z 2 ∈ Z γ corresponding to the controls U 1 , U 2 ∈ Y , we have

The operator B : X × X → Z γ in (2.8) is defined by

The linear operator A : D ( A ) ⊂ Z γ → Z γ is given by

with the domain

The adjoint operator of A is easily calculated and given by

for z ∈ D ( A ∗ ) = D ( A ) .

Our first result is the following.

The proof of Theorem 1 will be completed through several lemmas.

Now, we show that the operator B given by (2.11) is bounded in Z γ .

The following result shows to be useful to prove Theorem 1.

3 Spectral analysis

In this section, we are interested to study some spectral properties of the operator A defined by (2.12). This proves to be useful in the following sections, in particular in determining explicitly the optimal decay rate.

Let Z ∈ D ( A ) , computing A Z yields

where { P n } n ≥ 1 is a complete family of orthogonal projections in Z γ

satisfying

and

We can also easily verify that

The characteristic equation of R n is given by

where

where ϱ i ( n ) are the roots of the following equation:

The roots of (3.8), ϱ i ( n ) , i = 0 , 1 , 2 , are given by the following.