Controllability of a thermoelastic system

Abstract

In this paper we present a null controllability result for a thermoelastic Rayleigh system. Instead of working directly with the control system, we obtain the controlled system as the modulus of elasticity in shear tends to infinity in the corresponding thermoelastic Mindlin–Timoshenko system. Our results follow the seminal book of Lagnese and Lions (Rech. Math. Appl. 6(1988)) where the controllability of a Kirkhhoff model is proposed as the limit of a controlled Mindlin–Timoshenko one. We use estimates for some eigenvalues of the beam model that were obtained in (SIAM J. Control Optim. 47 (2008) 1909–1938) and the recent paper of Komornik and Tenenbaum (Evolution Equations and Control Theory 4(3) (2015) 297–314) where explicit estimates for systems with real and complex eigenvalues are proposed.

Keywords

Thermoelastic Rayleigh beam Mindlin–Timoshenko system controllability

1. Introduction and main result

The aim of this paper is to study the null controllability properties of a thermoelastic Rayleigh beam: $\begin{matrix} (1) & \{\begin{matrix} b v_{t t} - a v_{x x t t} + v_{x x x x} + (ξ + μ) θ_{x x} = 0, & x \in (0, π), t \in (0, T), \\ θ_{t} - θ_{x x} = 0, & x \in (0, π), t \in (0, T), \\ v_{x} (0, t) = v_{x} (π, t) = 0, & t \in (0, T), \\ v_{x x x} (0, t) = m (t), v_{x x x} (π, t) = 0, & t \in (0, T), \\ θ_{x} (0, t) = δ (t), θ_{x} (π, t) = 0, & t \in (0, T), \\ v (x, 0) = v_{0} (x), v_{t} (x, 0) = v_{1} (x), & x \in (0, π), \\ θ (x, 0) = θ_{0} (x), & x \in (0, π) . \end{matrix} \end{matrix}$ The unknowns $v = v (x, t)$ and $θ = θ (x, t)$ represent, respectively, the vertical displacement and temperature at time t of the cross-section located x units from the end-point $x = 0$ . The functions $m = m (t)$ and $δ = δ (t)$ are controls acting at the extreme $x = 0$ , and T, ξ, μ, a and b are positive constants. More precisely, $a = σ τ^{3} / 12$ and $b = σ τ$ , where τ represents the thickness of the beam, considered to be small ( $τ ≪ π$ ), and σ stands for the mass density per unit volume. More details about the model can be found in [15].

Our main result is the following:

Theorem 1.1.
Suppose $T > 2 π \sqrt{a}$ . Then, for any data $(v_{0}, v_{1}, θ_{0}) \in H^{2} (0, π) \times H^{1} (0, π) \times L^{2} (0, π)$ , there exist a pair of controls $(m, δ) \in H^{- 1} (0, T) \times L^{2} (0, T)$ such that the solution of ( 1 ) satisfies $\begin{matrix} (2) & v (\cdot, T) = v_{t} (\cdot, T) = θ (\cdot, T) = 0 in (0, π) . \end{matrix}$
Remark 1.2.
Notice that, according to [10], the time of controllability is optimal and depends on the moment of inertia parameter of the beam.

In order to study the controllability of system (1), we analyze the limit properties of a thermoelastic Mindlin–Timoshenko system as the modulus of elasticity in shear tends to infinity. That is, we follow the approach used in [3], where the Kirchhoff system is analyzed as a limit of Mindlin–Timoshenko beam systems. The connection between similar systems through some singular perturbation have been intensively investigated. We refer, for instance, [1,2,18–20], where this issue is addressed for a number of nonlinear models and under various boundary conditions. As far as we know, this is the first result obtaining the null controllability of the thermoelastic Rayleigh beam as a limit of a thermoelastic Mindlin–Timoshenko system. Related results can be found in [10], where the authors study the null controllability of a thermoelastic Rayleigh beam by means of solving a problem of moments, but different boundary conditions than those of (1) are considered. The literature concerning the distributed null controllability of thermoelastic plates (that is a similar equation but in $R^{n}$ ) is vast, see e.g. [4–8] and references therein.

In order to perform our approach, let us consider the thermoelastic Mindlin–Timoshenko system given by $\begin{matrix} (3) & \{\begin{matrix} a u_{t t} - u_{x x} + k (u + v_{x}) + ξ θ_{x} = 0, & x \in (0, π), t \in (0, T), \\ b v_{t t} - k {(u + v_{x})}_{x} + μ θ_{x x} = 0, & x \in (0, π), t \in (0, T), \\ θ_{t} - θ_{x x} = 0, & x \in (0, π), t \in (0, T), \\ u (0, t) = 0, & t \in (0, T), \\ μ θ_{x} (0, t) - k v_{x} (0, t) = m (t), & t \in (0, T), \\ θ_{x} (0, t) = δ (t), & t \in (0, T), \\ u (π, t) = v_{x} (π, t) = θ_{x} (π, t) = 0, & t \in (0, T), \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & x \in (0, π), \\ v (x, 0) = v_{0} (x), v_{t} (x, 0) = v_{1} (x), & x \in (0, π), \\ θ (x, 0) = θ_{0} (x), & x \in (0, π) . \end{matrix} \end{matrix}$ For this model, in addition to the components of system (1), we have the angle of rotation given by $u = u (x, t)$ and the parameter $k > 0$ which denotes the so-called modulus of elasticity in shear.

To solve the null control problem associated to (3), we have to prove an observability inequality to its adjoint system with explicit dependence on k. To this end, we will proceed as in [3]. That is, we will write the solution to the adjoint system to (3) using the expansion in an appropriate basis of eigenfunctions that will include the exponential in time with the corresponding eigenvalues. The presence of a sequence of imaginary eigenvalues and a sequence of real eigenvalues makes the problem much more complicated than the problem studied in [3]. In fact, even if the existence of a biorthogonal basis to the two families of exponentials can be deduced from the results in [9], this construction is not useful in our problem since an explicit dependence on k is missing. In this paper, we apply the recent work by Komornik and Tenenbaum [14] to prove the needed estimates.

As mentioned before, to prove Theorem 1.1, we will adopt the strategy of obtaining a pair of controls $(m, δ)$ as a limit $\begin{matrix} (m_{k}, δ_{k}) \to (m, δ) weakly in H^{- 1} (0, T) \times L^{2} (0, T), as k \to \infty, \end{matrix}$ where $(m_{k}, δ_{k}) \in H^{- 1} (0, T) \times L^{2} (0, T)$ is a pair of controls which drives the system (3) zero, by considering initial datum $(u_{0}, u_{1}, v_{0}, v_{1}, θ_{0}) = (- v_{0 x}, - v_{1 x}, v_{0}, v_{1}, θ_{0})$ , with $(v_{0}, v_{1}, θ_{0}) \in H^{2} (0, π) \times H^{1} (0, π) \times L^{2} (0, π)$ .

The paper is organized as follows. In Section 2 we study the well-posedness for the above mentioned systems. In Section 3 we make a spectral analysis for the adjoint operator of system (3). By finding explicit eigenvalues and eigenfunctions of this operator, we describe the “good” eigenspace which allows us to construct convergent controls when the parameter k tends to infinity. Section 4 is devoted to prove an observability inequality in the “good” subspace. The proof of Theorem 1.1 is given in Section 5. Finally, in Section 6 we discuss further results and open problems. For the sake of completeness, in an Appendix at the end of the paper, we include a proof of a corollary of Theorem 1.1 in [14] which is used in Section 4.
2. Existence and characterization of the solutions

In order to study the well-posedness for (1), we will start by studying the solutions by transposition to (3) (see e.g. [17] for the general definition of solution by transposition). The solutions of (1) will be obtained as a limit, as $k \to \infty$ , of a sequence of solutions to (3).

Firstly, let us consider the non-homogeneous adjoint system: $\begin{matrix} (4) & \{\begin{matrix} a p_{t t} - p_{x x} + k (p + q_{x}) = f, & x \in (0, π), t \in (0, T), \\ b q_{t t} - k {(p + q_{x})}_{x} = g, & x \in (0, π), t \in (0, T), \\ - y_{t} - y_{x x} - ξ p_{x} + μ q_{x x} = h, & x \in (0, π), t \in (0, T), \\ p (0, t) = p (π, t) = q_{x} (0, t) = q_{x} (π, t) = 0, & t \in (0, T), \\ y_{x} (0, t) = y_{x} (π, t) = 0, & t \in (0, T), \\ p (x, T) = p_{0} (x), p_{t} (x, T) = p_{1} (x), & x \in (0, π), \\ q (x, T) = q_{0} (x), q_{t} (x, T) = q_{1} (x), & x \in (0, π), \\ y (x, T) = y_{0} (x), & x \in (0, π) . \end{matrix} \end{matrix}$

Let us introduce the finite energy space $\begin{matrix} (5) & H : = H_{0}^{1} (0, π) \times L^{2} (0, π) \times H^{1} (0, π) \times {[L^{2} (0, π)]}^{2} \end{matrix}$ equipped with the norm $\begin{matrix} (6) & {‖ (p_{0}, p_{1}, q_{0}, q_{1}, y_{0}) ‖}_{k} = {(\int_{0}^{π} (a | p_{1} |^{2} + b | q_{1} |^{2} + | p_{0 x} |^{2} + k | p_{0} + q_{0 x} |^{2} + | y_{0} |^{2}) d x)}^{1 / 2} . \end{matrix}$

For any $(p_{0}, p_{1}, q_{0}, q_{1}, y_{0}) \in H$ and $(f, g, h) \in V : = L^{1} (0, T; H_{0}^{1} (0, π) \times {[L^{2} (0, π)]}^{2})$ , system (4) possess a unique solution in the class $\begin{array}{l} (7) & \begin{matrix} (p, q) \in C^{0} ([0, T]; H_{0}^{1} (0, π) \times H^{1} (0, π)) \cap C^{1} ([0, T]; {[L^{2} (0, π)]}^{2}), \\ y \in L^{2} (0, T; H^{1} (0, π)) \cap C^{0} ([0, T]; L^{2} (0, π)) . \end{matrix} \end{array}$ Moreover, there exists a constant $C = C (k, T) > 0$ such that the following inequality holds: $\begin{matrix} (8) & {‖ (p (t), p_{t} (t), q (t), q_{t} (t), y (t)) ‖}_{k} ⩽ C ({‖ (p_{0}, p_{1}, q_{0}, q_{1}, y_{0}) ‖}_{k} + {‖ (f, g, h) ‖}_{V}), \forall t \in [0, T], \end{matrix}$ where $‖ \cdot ‖_{V}$ denotes the usual norm in $V$ .

We recall that the results in [3] imply a “hidden regularity” for the first two components of the system at $x = 0$ . We have then the following:

Theorem 2.1.
For any $T > 0$ , there exists a constant $C = C (T) > 0$ , independent of k, such that the solution $(p, q)$ of ( 4 ) satisfies the inequality $\begin{matrix} (9) & {‖ (p_{x} (0, \cdot), q (0, \cdot)) ‖}_{L^{2} (0, T) \times H^{1} (0, T)}^{2} ⩽ C ({‖ (p_{0}, p_{1}, q_{0}, q_{1}, y_{0}) ‖}_{k}^{2} + {‖ (f, g, h) ‖}_{V}^{2}), \end{matrix}$ for any $(p_{0}, p_{1}, q_{0}, q_{1}, y_{0}) \in H$ and $(f, g, h) \in V$ .

For the proof see [3, Prop. 2.2].

Concerning the asymptotic behavior of the solutions of (4), as $k \to \infty$ , the following result holds.
Theorem 2.2.
Let $(f, g, h) \in V$ be fixed. Suppose that the initial data $(p_{0 k}, p_{1 k}, q_{0 k}, q_{1 k}, y_{0 k}) \in H$ satisfy $\begin{matrix} (10) & \sqrt{k} ‖ p_{0 k} + q_{0 k, x} ‖_{L^{2} (0, π)} ⩽ C, \forall k ⩾ 1, \end{matrix}$ with C being a positive constant independent of k and are such that, as $k \to + \infty$ , $\begin{matrix} (11) & (p_{0 k}, p_{1 k}, q_{0 k}, q_{1 k}, y_{0 k}) \to (p_{0}, p_{1}, q_{0}, q_{1}, y_{0}) weakly in H . \end{matrix}$ Then, if $(p_{k}, q_{k}, y_{k})$ is the solution of ( 4 ), we have that $\begin{matrix} (12) & (p_{k}, p_{k t}, q_{k}, q_{k t}, y_{k}) \to (- q_{x}, - q_{x t}, q, q_{t}, y) weak- * in L^{\infty} (0, T; H), \end{matrix}$ where $(q, y)$ solves the system $\begin{matrix} (13) & \{\begin{matrix} b q_{t t} - a q_{x x t t} + q_{x x x x} = f_{x} + g, & x \in (0, π), t \in (0, T), \\ - y_{t} - y_{x x} + (ξ + μ) q_{x x} = h, & x \in (0, π), t \in (0, T), \\ q_{x} (o, t) = q_{x} (π, t) = q_{x x x} (0, t) = q_{x x x} (π, t) = 0, & t \in (0, T), \\ y_{x} (0, t) = y_{x} (π, t) = 0, & t \in (0, T), \\ q (x, T) = q_{0} (x), q_{t} (x, T) = q_{1} (x), y (x, T) = y_{0} (x), & x \in (0, π) . \end{matrix} \end{matrix}$
Proof.
Let $(p_{0 k}, p_{1 k}, q_{0 k}, q_{1 k}, y_{0 k}) \in H$ verify (10) and $(p_{k}, q_{k}, y_{k})$ be the corresponding solution to (4). Following the ideas in [3], we only need to prove that the energy $\begin{matrix} (14) & E_{k} (t) = \frac{1}{2} {‖ (p_{k} (t), p_{k t} (t), q_{k} (t), q_{k t} (t), y_{k} (t)) ‖}_{k}^{2} \end{matrix}$ is uniformly bounded in k.

In fact, multiplying the first three equations in (4) by $p_{k t}$ , $q_{k t}$ and $y_{k}$ , respectively, and integrating from 0 to π, we get $\begin{matrix} - \frac{d}{d t} E_{k} (t) + \int_{0}^{π} | y_{k x} |^{2} d x = \int_{0}^{π} (- ξ p_{k} + μ q_{k x}) y_{k x} d x + \int_{0}^{π} (f q_{k t} + g p_{k t} + h y_{k}) d x . \end{matrix}$ Using triangle, Young and Poincaré inequalities, it is not difficult to verify that, for some $C = C (ξ, μ) > 0$ , independent of $k ⩾ 1$ , $\begin{matrix} - \frac{d}{d t} E_{k} (t) + \frac{1}{2} \int_{0}^{π} | y_{k x} |^{2} d x ⩽ C \int_{0}^{π} (k | p_{k} + q_{k x} |^{2} + | p_{k x} |^{2} + | f |^{2} + | g |^{2} + | h |^{2}) d x . \end{matrix}$ So, Gronwall’s Lemma implies that $\begin{matrix} (15) & E_{k} (t) + \frac{1}{2} \int_{t}^{T} \int_{0}^{π} | y_{k x} |^{2} d x ⩽ C (E_{k} (T) + {‖ (f, g, h) ‖}_{V}^{2}) . \end{matrix}$

By (10) and (15), we have that the sequence $(p_{k}, p_{k t}, q_{k}, q_{k t}, y_{k})$ is uniformly (in k) bounded in $L^{\infty} (0, T; H)$ . The result is proved proceeding in a similar way than in [3, Th. 2.1]. □

Concerning system (3), we can argue as in [16] (see also [3]) to show that it is well-posed in the sense of transposition. More precisely, for any initial datum $(u_{0}, u_{1}, v_{0}, v_{1}, θ_{0}) \in H^{'}$ , the solution ${u, v, θ}$ belongs to the class $\begin{matrix} (u, v, θ) \in C^{0} ([0, T]; {[L^{2} (0, π)]}^{3}) \end{matrix}$ satisfying the identity $\begin{array}{l} \int_{Q} [f (x, t) u (x, t) + g (x, t) v (x, t) + h (x, t) θ (x, t)] d x d t \\ = a ⟨ u_{1}, p (\cdot, 0) ⟩ \\ - a \int_{0}^{π} u_{0} (x) p_{t} (x, 0) d x + b ⟨ v_{1}, q (\cdot, 0) ⟩ - b \int_{0}^{π} v_{0} (x) q_{t} (x, 0) d x - \int_{0}^{π} θ_{0} (x) y (x, 0) d x \\ (16) & + \int_{0}^{T} m_{1} q_{t} (0, t) d t + \int_{0}^{T} δ (t) y (0, t) d t, \end{array}$ where $(p, q, y)$ is solution of system (4), $⟨ \cdot, \cdot ⟩$ represents the duality between ${[H^{1} (0, π)]}^{'}$ and $H^{1} (0, π)$ or between $H^{- 1} (0, π)$ and $H_{0}^{1} (0, π)$ , and $m = m_{1 t}$ , with $m_{1} \in L^{2} (0, T)$ being of compact support in $(0, T)$ . Moreover, using Theorem 2.1, we can show the existence of a constant $C > 0$ , independent of k, such that $\begin{matrix} (17) & {‖ (u, v, θ) ‖}_{C^{0} ([0, T]; {[L^{2} (0, π)]}^{3})} ⩽ C ({‖ (u_{0}, u_{1}, v_{0}, v_{1}, θ_{0}) ‖}_{H^{'}} + {‖ (m_{1}, δ) ‖}_{{[L^{2} (0, T)]}^{2}}) . \end{matrix}$

The following result describes the asymptotic behavior, as $k \to \infty$ , for system (3). Theorem 2.3.
Let the initial datum $(u_{0}, u_{1}, v_{0}, v_{1}, θ_{0}) \in H^{'}$ be independent of k and satisfying $\begin{matrix} (18) & u_{0} + v_{0 x} = 0, u_{1} + v_{1 x} = 0 . \end{matrix}$ Suppose that the controls $(m_{k}, δ_{k})$ , with $m_{k} = m_{1 k t}$ are such that $\begin{matrix} (19) & (m_{1 k}, δ_{k}) \to (m_{1}, δ) weakly in {[L^{2} (0, T)]}^{2}, m_{1 k}, m_{1} being of compact support in (0, T) . \end{matrix}$ Let $(u_{k}, v_{k}, θ_{k})$ be the corresponding solution of ( 3 ). Then, as $k \to \infty$ , $\begin{matrix} (20) & (u_{k}, v_{k}, θ_{k}) \to (- v_{x}, v, θ) weak- * in L^{\infty} (0, T; {[L^{2} (0, π)]}^{3}), \end{matrix}$ where $(v, θ)$ is the unique solution of system ( 1 ) with $m = - m_{1 t}$ .
Proof.
For data $(u_{0}, u_{1}, v_{0}, v_{1}, m_{k}, δ_{k})$ satisfying the assumptions of Theorem 2.3, we consider, for each $k > 0$ , $(u_{k}, v_{k}, θ_{k})$ the unique solution of (3) in the sense of transposition (16).

By (17) and (19) it follows that $\begin{matrix} (u_{k}, v_{k}, θ_{k}) is bounded in L^{\infty} (0, T; {[L^{2} (0, π)]}^{3}) . \end{matrix}$ Then we can extract a subsequence, denoted in the same way, such that $\begin{matrix} (21) & (u_{k}, v_{k}, θ_{k}) \to (u, v, θ) weak- * in L^{\infty} (0, T; {[L^{2} (0, π)]}^{3}) . \end{matrix}$ Consider now $(p_{k}, q_{k}, y_{k})$ the solution to (4) corresponding to initial datum $(p_{0}, p_{1}, q_{0}, q_{1}, y_{0}) \in H$ satisfying $p_{0} + q_{0 x} = 0$ and $(f, g, h) \in V$ . We can pass to the limit, as $k \to \infty$ , in (16), obtaining, as consequence of Theorem 2.2 and (21), $\begin{array}{l} \iint_{Q} [v (x, t) (f_{x} + g) (x, t) + θ (x, t) h (x, t)] d x d t \\ = a \int_{0}^{π} v_{1 x} (x) q_{x} (x, 0) d x - a ⟨ v_{0 x}, q_{x t} (\cdot, 0) ⟩ \\ + b \int_{0}^{π} v_{1} (x) q (x, 0) d x - b \int_{0}^{π} v_{0} (x) q_{t} (x, 0) d x - \int_{0}^{π} θ_{0} (x) y (x, 0) d x \\ - \int_{0}^{T} m_{1} (t) q_{t} (0, t) d t + \int_{0}^{T} δ (t) y (0, t) d t \\ (22) & = 0, \end{array}$ where $(q, y)$ solves the system $\begin{matrix} \{\begin{matrix} b q_{t t} - a q_{x x t t} + q_{x x x x} = f_{x} + g, & x \in (0, π), t \in (0, T), \\ - y_{t} - y_{x x} + (ξ + μ) q_{x x} = h, & x \in (0, π), t \in (0, T), \\ q_{x} (0, t) = q_{x} (π, t) = q_{x x x} (0, t) = q_{x x x} (0, t) = 0, & t \in (0, T), \\ y_{x} (0, t) = y_{x} (π, t) = 0, & t \in (0, T), \\ q (x, T) = q_{t} (x, T) = y (x, T) = 0, & x \in (0, π) . \end{matrix} \end{matrix}$ Identity (22) tells us that $(v, θ)$ is the solution, by transposition, for (1). The uniqueness is proved in a standard way. □

3. Spectral decomposition

We will now study the eigenvalues and eigenfunctions for the operator involved in (4). More precisely, let us consider the following homogeneous system: $\begin{matrix} (23) & \{\begin{matrix} a ϕ_{t t} - ϕ_{x x} + k (ϕ + ψ_{x}) = 0, & x \in (0, π), t \in (0, T), \\ b ψ_{t t} - k {(ϕ + ψ_{x})}_{x} = 0, & x \in (0, π), t \in (0, T), \\ - ρ_{t} - ρ_{x x} - ξ ϕ_{x} + μ ψ_{x x} = 0, & x \in (0, π), t \in (0, T), \\ ϕ (0, t) = ϕ (π, t) = ψ_{x} (0, t) = ψ_{x} (π, t) = 0, & t \in (0, T), \\ ρ_{x} (0, t) = ρ_{x} (π, t) = 0, & t \in (0, T), \\ ϕ (x, T) = ϕ_{0} (x), ϕ_{t} (x, T) = ϕ_{1} (x), & x \in (0, π), \\ ψ (x, T) = ψ_{0} (x), ψ_{t} (x, T) = ψ_{1} (x), & x \in (0, π), \\ ρ (x, T) = ρ_{0} (x), & x \in (0, π), \end{matrix} \end{matrix}$ which can be rewritten as $\begin{matrix} \frac{d}{d t} Φ = A_{k} Φ, \end{matrix}$ where $Φ = {(ϕ, ϕ_{t}, ψ, ψ_{t}, ρ)}^{T}$ and $\begin{matrix} A_{k} = [\begin{matrix} 0 & 1 & 0 & 0 & 0 \\ \frac{1}{a} \frac{\partial^{2}}{\partial x^{2}} - \frac{k}{a} & 0 & \frac{- k}{a} \frac{\partial}{\partial x} & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \frac{k}{b} \frac{\partial}{\partial x} & 0 & \frac{k}{b} \frac{\partial^{2}}{\partial x^{2}} & 0 & 0 \\ ξ \frac{\partial}{\partial x} & 0 & - μ \frac{\partial^{2}}{\partial x^{2}} & 0 & \frac{\partial^{2}}{\partial x^{2}} \end{matrix}] . \end{matrix}$

We will now work in an appropriate subspace with the “good” properties in order to obtain a null control of (3) and then pass to the limit to prove our main result. We will show explicitly a hidden regularity associated to this subspace when the initial data in (23) are less regular.

To perform a spectral analysis, we consider $\begin{matrix} A_{k} Φ = λ Φ . \end{matrix}$ We obtain five families of eigenvalues $i λ_{k n}$ , $- i λ_{k n}$ , $i {\tilde{λ}}_{k n}$ , $- i {\tilde{λ}}_{k n}$ , $- n^{2}$ , and their respective families of eigenfunctions $\begin{array}{l} Φ_{k n}^{1} = (\begin{matrix} sin n x \\ i λ_{k n} sin n x \\ β_{k n} cos n x \\ i λ_{k n} β_{k n} cos n x \\ γ_{k n} cos n x \end{matrix}), Φ_{k n}^{2} = (\begin{matrix} sin n x \\ - i λ_{k n} sin n x \\ β_{k n} cos n x \\ - i λ_{k n} β_{k n} cos n x \\ γ_{k n} cos n x \end{matrix}), \\ Φ_{k n}^{3} = (\begin{matrix} sin n x \\ i {\tilde{λ}}_{k n} sin n x \\ {\tilde{β}}_{k n} cos n x \\ i {\tilde{λ}}_{k n} {\tilde{β}}_{k n} cos n x \\ {\tilde{γ}}_{k n} cos n x \end{matrix}), Φ_{k n}^{4} = (\begin{matrix} sin n x \\ - i {\tilde{λ}}_{k n} sin n x \\ {\tilde{β}}_{k n} cos n x \\ - i {\tilde{λ}}_{k n} {\tilde{β}}_{k n} cos n x \\ {\tilde{γ}}_{k n} cos n x \end{matrix}), Φ_{k n}^{5} = (\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ cos n x \end{matrix}), \end{array}$ where $\begin{matrix} (24) & λ_{k n} = {(\frac{n^{2}}{2 a} + \frac{k n^{2}}{2 b} + \frac{k}{2 a} - \frac{1}{2} \sqrt{r_{k n}})}^{\frac{1}{2}}, {\tilde{λ}}_{k n} = {(\frac{n^{2}}{2 a} + \frac{k n^{2}}{2 b} + \frac{k}{2 a} + \frac{1}{2} \sqrt{r_{k n}})}^{\frac{1}{2}}, \end{matrix}$ with $r_{k n} = \frac{k^{2}}{a^{2}} + \frac{2 k n^{2}}{a^{2}} + \frac{2 k^{2} n^{2}}{a b} + {(\frac{n^{2}}{a} - \frac{k n^{2}}{b})}^{2}$ , $\begin{array}{l} (25) & β_{k n} = \frac{- a λ_{k n}^{2} + n^{2} + k}{n k}, γ_{k n} = \frac{- ξ n + μ n^{2} β_{k n}}{λ_{k n}^{2} + n^{4}}, \\ (26) & {\tilde{β}}_{k n} = \frac{- a {\tilde{λ}}_{k n}^{2} + n^{2} + k}{n k} and {\tilde{γ}}_{k n} = \frac{- ξ n + μ n^{2} {\tilde{β}}_{k n}}{{\tilde{λ}}_{k n}^{2} + n^{4}}, \end{array}$ for $n \in N$ . Additionally, for $n = 0$ , the value $λ_{k 0} = 0$ has a two-dimensional eigenspace generated by $Φ_{k 0}^{1} = Φ_{k 0}^{2}$ and $Φ_{k 0}^{5}$ by setting $β_{k 0} = γ_{k 0} = 1$ . Note also that ${\tilde{λ}}_{k 0} = \sqrt{k / a}$ is not an eigenvalue for $A_{k}$ . For this reason and to simplify the series that follow, we define $Φ_{k 0}^{3} = Φ_{k 0}^{4} = 0$ .

Denoting $λ_{k n}^{\pm} = \pm λ_{k n}$ , we have the following estimates for these parameters when k goes to infinity:

Lemma 3.1.
For each fixed $n \in N$ , we have that $\begin{matrix} (27) & λ_{n}^{\pm} : = lim_{k \to \infty} λ_{k n}^{\pm} = \pm \sqrt{\frac{n^{4}}{a n^{2} + b}}, \end{matrix}$ and ${i λ_{n}^{\pm}, - n^{2}}$ is the complete set of eigenvalues of the limit system (13). Also, there exist $N \in N$ , $C > 0$ such that, for each $n, k ⩾ N$ , we have
$| λ_{k n}^{\pm} - λ_{n}^{\pm} | ⩽ C / k$ ;

$| λ_{k n} | + | {\tilde{λ}}_{k n} | ⩽ C n$ ;

$| γ_{k n} | + | {\tilde{γ}}_{k n} | ⩽ C / n$ ;

$| {\tilde{β}}_{k n} | ⩽ C n$ ;

$| β_{k n} | ⩽ C (1 / n + 1 / k)$ .

Proof.
For the proof of (27) see [3, Prop. 3.1]. It is straightforward to verify that the eigenvalues of system (13) are given by ${i λ_{n}^{\pm}, - n^{2}}$ .

The estimates from 1 to 4 can be obtained by direct computations. From these inequalities and the definition of $β_{k, n}$ , we have that $\begin{matrix} | β_{k n} | & = \frac{1}{k n} | - a {(λ_{k n}^{\pm} - λ_{n}^{\pm})}^{2} + 2 a λ_{n}^{\pm} (λ_{n}^{\pm} - λ_{k n}^{\pm}) + n^{2} - a {λ_{n}^{\pm}}^{2} + k | \\ ⩽ \frac{C}{k n} ({| λ_{k n}^{\pm} - λ_{n}^{\pm} |}^{2} + | λ_{n}^{\pm} | | λ_{n}^{\pm} - λ_{k n}^{\pm} | + | n^{2} - a {λ_{n}^{\pm}}^{2} | + k), \end{matrix}$ which proves 5. □

Let us define, for $m \in N$ , the spaces $\begin{array}{l} V^{m} : = H^{m} (0, π) \cap H_{0}^{1} (0, π), V^{- m} : = H^{- m} (0, π), \\ Z^{m} : = H^{m} (0, π), Z^{- m} : = {[H^{m} (0, π)]}^{'} \end{array}$ and $\begin{matrix} V^{0} = Z^{0} : = L^{2} (0, π) . \end{matrix}$ Now, for $m \in Z$ , let us introduce $H_{k}^{m}$ and ${\tilde{H}}_{k}^{m}$ the closed subspaces of $\begin{matrix} H^{m} : = V^{m} \times V^{m - 1} \times Z^{m} \times Z^{m - 1} \times Z^{0} \end{matrix}$ defined by $\begin{matrix} (28) & H_{k}^{m} = {\sum_{n = 0}^{\infty} (a_{n} Φ_{k n}^{1} + \overline{a_{n}} Φ_{k n}^{2}) + \sum_{n = 0}^{\infty} b_{n} Φ_{k n}^{5} : a_{n} \in C, b_{n} \in R, \sum_{n = 1}^{\infty} (n^{2 m} | a_{n} |^{2} + | b_{n} |^{2}) < \infty} \end{matrix}$ and $\begin{matrix} (29) & {\tilde{H}}_{k}^{m} = {\sum_{n = 0}^{\infty} (c_{n} Φ_{k n}^{3} + \overline{c_{n}} Φ_{k n}^{4}) : c_{n} \in C, \sum_{n = 1}^{\infty} n^{2 (m + 1)} | c_{n} |^{2} < \infty} . \end{matrix}$ Observe that $\begin{matrix} (30) & H_{k}^{m} \subset V^{m} \times V^{m - 1} \times Z^{m + 1} \times Z^{m} \times Z^{0} and {\tilde{H}}_{k}^{m} \subset V^{m + 1} \times V^{m} \times Z^{m} \times Z^{m - 1} \times Z^{0} . \end{matrix}$ Remark 3.2.
Taking into account the boundary conditions satisfied by the eigenfunctions which define the spaces $H^{m}$ , we can identify $H^{1}$ as being the space $H$ defined in (5).
Proposition 3.3.
With the previous definitions, we have $H^{m} = H_{k}^{m} \oplus {\tilde{H}}_{k}^{m}$ .
Proof.
We denote $\begin{array}{l} Ψ_{k n}^{1} = (\begin{matrix} 1 \\ i λ_{k n} \\ β_{k n} \\ i λ_{k n} β_{k n} \\ γ_{k n} \end{matrix}), Ψ_{k n}^{2} = (\begin{matrix} 1 \\ - i λ_{k n} \\ β_{k n} \\ - i λ_{k n} β_{k n} \\ γ_{k n} \end{matrix}), Ψ_{k n}^{3} = (\begin{matrix} 1 \\ i {\tilde{λ}}_{k n} \\ {\tilde{β}}_{k n} \\ i {\tilde{λ}}_{k n} {\tilde{β}}_{k n} \\ {\tilde{γ}}_{k n} \end{matrix}), \\ Ψ_{k n}^{4} = (\begin{matrix} 1 \\ - i {\tilde{λ}}_{k n} \\ {\tilde{β}}_{k n} \\ - i {\tilde{λ}}_{k n} {\tilde{β}}_{k n} \\ {\tilde{γ}}_{k n} \end{matrix}), Ψ_{k n}^{5} = (\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{matrix}), \end{array}$ in such a way that $Φ_{k n}^{j} = X_{n} Ψ_{k n}^{j}$ , where $\begin{matrix} X_{n} = (\begin{matrix} sin n x & 0 & 0 & 0 & 0 \\ 0 & sin n x & 0 & 0 & 0 \\ 0 & 0 & cos n x & 0 & 0 \\ 0 & 0 & 0 & cos n x & 0 \\ 0 & 0 & 0 & 0 & cos n x \end{matrix}) . \end{matrix}$

Considering $e_{i}$ , $i = 1, \dots, 5$ , the vectors of the canonical basis of $R^{5}$ , it is not difficult to verify that, for $n \in N$ , $\begin{matrix} e_{1}^{T} = a_{k n}^{1} Ψ_{k n}^{1} + \overline{a_{k n}^{1}} Ψ_{k n}^{2} + c_{k n}^{1} Ψ_{k n}^{3} + \overline{c_{k n}^{1}} Ψ_{k n}^{4} + b_{k n}^{1} Ψ_{k n}^{5}, \end{matrix}$ with $\begin{array}{l} a_{k n}^{1} = \frac{- {\tilde{β}}_{k n}}{2 (β_{k n} - {\tilde{β}}_{k n})}, c_{k n}^{1} = \frac{β_{k n}}{2 (β_{k n} - {\tilde{β}}_{k n})} and b_{k n}^{1} = \frac{- (β_{k n} {\tilde{γ}}_{k n} - {\tilde{β}}_{k n} γ_{k n})}{(β_{k n} - {\tilde{β}}_{k n})}, \\ e_{2}^{T} = a_{k n}^{2} Ψ_{k n}^{1} + \overline{a_{k n}^{2}} Ψ_{k n}^{2} + c_{k n}^{2} Ψ_{k n}^{3} + \overline{c_{k n}^{2}} Ψ_{k n}^{4}, \end{array}$ with $\begin{array}{l} a_{k n}^{2} = \frac{i {\tilde{β}}_{k n}}{2 λ_{k n} (β_{k n} - {\tilde{β}}_{k n})} and c_{k n}^{2} = \frac{- i β_{k n}}{2 {\tilde{λ}}_{k n} (β_{k n} - {\tilde{β}}_{k n})}, \\ e_{3}^{T} = a_{k n}^{3} Ψ_{k n}^{1} + \overline{a_{k n}^{3}} Ψ_{k n}^{2} + c_{k n}^{3} Ψ_{k n}^{3} + \overline{c_{k n}^{3}} Ψ_{k n}^{4} + b_{k n}^{3} Ψ_{k n}^{5}, \end{array}$ with $\begin{array}{l} a_{k n}^{3} = \frac{1}{2 (β_{k n} - {\tilde{β}}_{k n})}, c_{k n}^{3} = \frac{- 1}{2 (β_{k n} - {\tilde{β}}_{k n})} and b_{k n}^{3} = \frac{{\tilde{γ}}_{k n} - γ_{k n}}{β_{k n} - {\tilde{β}}_{k n}}, \\ e_{4}^{T} = a_{k n}^{4} Ψ_{k n}^{1} + \overline{a_{k n}^{4}} Ψ_{k n}^{2} + c_{k n}^{4} Ψ_{k n}^{3} + \overline{c_{k n}^{4}} Ψ_{k n}^{4}, \end{array}$ with $\begin{matrix} a_{k n}^{4} = \frac{- i}{2 λ_{k n} (β_{k n} - {\tilde{β}}_{k n})} and c_{k n}^{4} = \frac{i}{2 {\tilde{λ}}_{k n} (β_{k n} - {\tilde{β}}_{k n})} . \end{matrix}$ Finally, $\begin{matrix} e_{5}^{T} = Ψ_{k n}^{5} . \end{matrix}$

Next, using the well known fact that each of the families ${sin n x}_{n \in N}$ and ${cos n x}_{n \in N}$ is a Riesz basis of $L^{2} (0, π)$ , it follows that each element of $H^{m}$ is included in $H_{k}^{m} + {\tilde{H}}_{k}^{m}$ . Indeed, given $(ϕ_{0}, ϕ_{1}, ψ_{0}, ψ_{1}, ρ_{0}) \in H^{m}$ , we write $\begin{matrix} {(ϕ_{0}, ϕ_{1}, ψ_{0}, ψ_{1}, ρ_{0})}^{T} = \sum_{n = 0}^{\infty} X_{n} (ϕ_{0 n} e_{1}^{T} + ϕ_{1 n} e_{2}^{T} + ψ_{0 n} e_{3}^{T} + ψ_{1 n} e_{4}^{T} + ρ_{0 n} e_{5}^{T}), \end{matrix}$ and using the previous decomposition we get $\begin{matrix} (31) & \begin{matrix} {(ϕ_{0}, ϕ_{1}, ψ_{0}, ψ_{1}, ρ_{0})}^{T} & = \sum_{n = 0}^{\infty} [ϕ_{0 n} (a_{k n}^{1} Φ_{k n}^{1} + \overline{a_{k n}^{1}} Φ_{k n}^{2} + c_{k n}^{1} Φ_{k n}^{3} + \overline{c_{k n}^{1}} Φ_{k n}^{4} + b_{k n}^{1} Φ_{k n}^{5}) \\ + ϕ_{1 n} (a_{k n}^{2} Φ_{k n}^{1} + \overline{a_{k n}^{2}} Φ_{k n}^{2} + c_{k n}^{2} Φ_{k n}^{3} + \overline{c_{k n}^{2}} Φ_{k n}^{4}) \\ + ψ_{0 n} (a_{k n}^{3} Φ_{k n}^{1} + \overline{a_{k n}^{3}} Φ_{k n}^{2} + c_{k n}^{3} Φ_{k n}^{3} + \overline{c_{k n}^{3}} Φ_{k n}^{4} + b_{k n}^{3} Φ_{k n}^{5}) \\ + ψ_{1 n} (a_{k n}^{4} Φ_{k n}^{1} + \overline{a_{k n}^{4}} Φ_{k n}^{2} + c_{k n}^{4} Φ_{k n}^{3} + \overline{c_{k n}^{4}} Φ_{k n}^{4}) \\ + ρ_{0 n} Φ_{k n}^{5}] \\ = \sum_{n = 0}^{\infty} (A_{k n} Φ_{k n}^{1} + \overline{A_{k n}} Φ_{k n}^{2} + C_{k n} Φ_{k n}^{3} + \overline{C_{k n}} Φ_{k n}^{4} + B_{k n} Φ_{k n}^{5}), \end{matrix} \end{matrix}$ where $\begin{array}{l} A_{k n} = a_{k n}^{1} ϕ_{0 n} + a_{k n}^{2} ϕ_{1 n} + a_{k n}^{3} ψ_{0 n} + a_{k n}^{4} ψ_{1 n}, \\ B_{k n} = b_{k n}^{1} ϕ_{0 n} + b_{k n}^{3} ψ_{0 n} + ρ_{0 n} \end{array}$ and $\begin{matrix} C_{k n} = c_{k n}^{1} ϕ_{0 n} + c_{k n}^{2} ϕ_{1 n} + c_{k n}^{3} ψ_{0 n} + c_{k n}^{4} ψ_{1 n} . \end{matrix}$

Writing explicitely $β_{k n} - {\tilde{β}}_{k n}$ one can see that for appropriate positive constants $\tilde{C}$ and C $\begin{matrix} \tilde{C} n ⩽ | β_{k n} - {\tilde{β}}_{k n} | ⩽ C n . \end{matrix}$ This result with the estimates given in Lemma 3.1 implies that $\begin{matrix} (32) & \sum_{n = 1}^{\infty} (n^{2 m} A_{k n}^{2} + B_{k n}^{2} + n^{2 (m + 1)} C_{k n}^{2}) < \infty \end{matrix}$ and, then, we conclude that $H^{m} \subset H_{k}^{m} + {\tilde{H}}_{k}^{m}$ .

Finally, in order to prove that $H_{k}^{m} \cap {\tilde{H}}_{k}^{m} = {0}$ , we will construct a biorthonormal family of ${Φ_{k n}^{j}}$ . That is, we look for a family ${Φ_{k n}^{j }}$ such that ${⟨ Φ_{k n}^{j }, Φ_{k ν}^{l} ⟩}_{{[L^{2} (0, π)]}^{5}} = δ_{n ν}^{j l}$ with $δ_{n ν}^{j l} = 1$ if $j = l$ and $n = ν$ , and $δ_{n ν}^{j l} = 0$ otherwise. To this end, we denote by $Ψ_{k n}$ the $5 \times 5$ matrix whose columns are $Ψ_{k n}^{j}$ , and $\begin{matrix} Λ_{k n} = (\begin{matrix} a_{k n}^{1} & a_{k n}^{2} & a_{k n}^{3} & a_{k n}^{4} & 0 \\ \overline{a_{k n}^{1}} & \overline{a_{k n}^{2}} & \overline{a_{k n}^{3}} & \overline{a_{k n}^{4}} & 0 \\ c_{k n}^{1} & c_{k n}^{2} & c_{k n}^{3} & c_{k n}^{4} & 0 \\ \overline{c_{k n}^{1}} & \overline{c_{k n}^{2}} & \overline{c_{k n}^{3}} & \overline{c_{k n}^{4}} & 0 \\ b_{k n}^{1} & 0 & b_{k n}^{3} & 0 & 1 \end{matrix}), \end{matrix}$ then we have $I_{5} = Ψ_{k n} Λ_{k n}$ and hence $\begin{matrix} (33) & I_{5} = Λ_{k n} Ψ_{k n} . \end{matrix}$

On the other hand, from the properties of the trigonometric functions $sin n x$ and $cos n x$ in $L^{2} (0, π)$ , we have $\begin{matrix} (34) & {⟨ X_{n}, X_{ν} ⟩}_{{[L^{2} (0, π)]}^{5}} = δ_{n ν} \frac{π}{2} I_{5} . \end{matrix}$

Finally, if $Λ_{k n}^{j}$ denotes the j-line of $Λ_{k n}$ , we define, for $j = 1, \dots, 5$ , the vector $\begin{matrix} Φ_{k n}^{j } = \frac{2}{π} X_{n} {Λ_{k n}^{j}}^{T} . \end{matrix}$ Taking into account (33) and (34), it follows that ${Φ_{k n}^{j } : j \in 1, \dots, 5, n \in N}$ is a biortogonal family of ${Φ_{k n}^{j} : j \in 1, \dots, 5, n \in N}$ .

We have now to show that $H_{k}^{m} \cap {\tilde{H}}_{k}^{m} = {\vec{0}}$ . In fact, let us take $z \in H_{k}^{m} \cap {\tilde{H}}_{k}^{m}$ . This means that there exists a sequence ${z^{ν}}_{ν \in N}$ , with $z^{ν} \in span {Φ_{k n}^{1}, Φ_{k n}^{2}, Φ_{k n}^{5}}$ , such that $z^{ν} \to z$ in $H^{m}$ . Also, there exists a sequence ${w^{ν}}_{ν \in N}$ , with $w^{ν} \in span {Φ_{k n}^{3}, Φ_{k n}^{4}}$ , such that $w^{ν} \to z$ in $H^{m}$ . From Cauchy–Schwarz inequality, it is not difficult to see that for every $j = 1, \dots, 5$ and any $n \in N$ , the following convergence holds: $\begin{matrix} (z^{ν}, Φ_{k n}^{j }) \to (z, Φ_{k n}^{j }), as ν \to \infty . \end{matrix}$ That means in particular that $\begin{matrix} (z, Φ_{k n}^{3 }) = 0 and (z, Φ_{k n}^{4 }) = 0 . \end{matrix}$ The same argument with the sequence ${w^{ν}}_{ν \in N}$ proves that $z = \vec{0}$ . □

4. An observability inequality

The aim of this section is to prove an observability inequality for system (23). For this, we state the following result:

Theorem 4.1.
Let $T > 2 π \sqrt{a}$ , then, there exists a constant $C > 0$ , independent of k, such that, for any data $(ϕ_{0}, ϕ_{1}, ψ_{0}, ψ_{1}, ρ_{0}) \in H_{k}^{0}$ , the corresponding solution $(ϕ, ψ)$ to ( 23 ) satisfies the following observability inequality $\begin{matrix} (35) & {‖ (ϕ (\cdot, 0), ϕ_{t} (\cdot, 0), ψ (\cdot, 0), ψ_{t} (\cdot, 0), ρ (\cdot, 0)) ‖}_{H_{k}^{0}}^{2} ⩽ C (\int_{0}^{T} {| ψ_{t} (0, t) |}^{2} + {| ρ (0, t) |}^{2} d t) . \end{matrix}$
Remark 4.2.
By the standard controllability-observability duality, Theorem 4.1 implies a partial null controllability result for system (3) in a proper subspace of solutions, which is generated by the elements of the spectrum that converge, as $k \to \infty$ , towards the solutions of the thermoelastic Rayleigh system.

The proof of Theorem 4.1 is based in the following result which is, in fact, a corollary of Theorem 1.1 in [14]. For completeness and following the ideas in [13], we will give a proof in an appendix at the end of the paper. Theorem 4.3.
Consider ${λ_{n}}_{n \in Z}$ and ${μ_{k}}_{k \in N}$ sequences of the real numbers such that there exist $C > 0$ , $N \in N$ , $γ > 0$ and $γ_{\infty} > 0$ such that $\begin{array}{l} (36) & γ_{\infty} : = inf_{| n | > N} {λ_{n + 1} - λ_{n}} > 0, γ : = inf_{n \in Z} {λ_{n + 1} - λ_{n}} > 0, \\ (37) & inf_{k \in N} {μ_{k + 1} - μ_{k}} > 0, \sum_{| μ_{k} - μ | ⩽ t} 1 ⩽ C t^{1 / α} (μ > 0, α > 1, t ⩾ 0), \\ (38) & inf_{n \in Z, k \in N} | i λ_{n} \pm μ_{k} | > 0 . \end{array}$ Let $T > 0$ be such that $T > 2 π / γ_{\infty}$ . Then, there exists $C_{1} > 0$ such that $\begin{matrix} (39) & \sum_{n \in Z} | a_{n} |^{2} + \sum_{k \in N} | b_{k} |^{2} e^{- μ_{k} T} ⩽ C_{1} \int_{0}^{T} {| \sum_{n \in Z} a_{n} e^{i λ_{n} t} + \sum_{k \in N} b_{k} e^{- μ_{k} t} |}^{2} d t, \end{matrix}$ for all sequences ${a_{n}}_{n \in Z}$ , ${b_{k}}_{k \in N}$ in $l^{2}$ . Here, constant $C_{1}$ depends at most on α, γ, $γ_{\infty}$ , T and on the implicit constants in the assumptions.
Proof of Theorem 4.1.
We define, for each $k ⩾ 1$ and $n \in N$ , $λ_{n}^{k} = λ_{k n}$ and $λ_{- n}^{k} = - λ_{k n}$ . Note that $λ_{0}^{k} = λ_{k 0} = 0$ . It is not difficult to see that estimate 1 in Lemma 3.1 implies the existence of positive real numbers $k_{0}$ and M such that, for each $k > k_{0}$ , the sequence ${λ_{n}^{k}}_{n}$ is strictly increasing for $| n | > M$ . Hence, we can reorder the whole sequence $λ_{n}^{k}$ in order to be strictly increasing. We still denote the new sequence by ${λ_{n}^{k}}_{n \in Z}$ .

We recall that Proposition 6.1 in [3] establishes the following gap property: It exists $ν_{0} \in N$ independent of k such that $\begin{matrix} (40) & inf_{| n | ⩾ m} {λ_{n + 1}^{k} - λ_{n}^{k}} = γ_{k}, with γ_{k} \to 1 / \sqrt{a}, as k \to \infty . \end{matrix}$ Taking $N = max {M, ν_{0}}$ , we are in the conditions of Theorem 4.3.

Given $(ϕ_{0}, ϕ_{1}, ψ_{0}, ψ_{1}, ρ_{0}) \in H_{k}^{0}$ , we have $\begin{array}{l} {(ϕ_{0}, ϕ_{1}, ψ_{0}, ψ_{1}, ρ_{0})}^{T} & = \sum_{n = 0}^{\infty} {(ϕ_{0 n} sin (n x), ϕ_{1 n} sin (n x), ψ_{0 n} cos (n x), ψ_{1 n} cos (n x), ρ_{0 n} cos (n x))}^{T} \\ = \sum_{n = 0}^{\infty} a_{k n} Φ_{k n}^{1} + \overline{a_{k n}} Φ_{k n}^{2} + b_{k n} Φ_{k n}^{5}, \end{array}$ for some $a_{k n} \in C$ and $b_{k n} \in R$ . Defining $a_{k (- n)} = \overline{a_{k n}}$ , we get $\begin{matrix} (41) & \begin{array}{l} ϕ_{0 n} = a_{k n} + a_{k (- n)}, ϕ_{1 n} = i (a_{k n} λ_{n}^{k} + a_{k (- n)} λ_{- n}^{k}) \\ ψ_{0 n} = ϕ_{0 n} β_{k n}, ψ_{1 n} = ϕ_{1 n} β_{k n}, \\ ρ_{0 n} = γ_{k n} ϕ_{0 n} + b_{k n}, \end{array} \end{matrix}$ where $β_{k n}$ and $γ_{k n}$ are given in (25).

Since $\begin{matrix} (42) & ψ_{t} (x, t) = i \sum_{n \in Z} λ_{n}^{k} β_{k n} a_{k n} e^{i λ_{n}^{k} t} cos (n x) \end{matrix}$ and the gap conditions (36) hold, we can apply Haraux’s version [11] of classical Ingham Theorem [12], to deduce that, for every $T > 2 π \sqrt{a}$ , there exist positive constants $c_{0}$ and $c_{1}$ , depending on T, γ and $γ_{\infty}$ , such that $\begin{matrix} (43) & c_{0} \sum_{n = 1}^{\infty} λ_{k n}^{2} β_{k n}^{2} | a_{k n} |^{2} ⩽ \int_{0}^{T} {| ψ_{t} (0, t) |}^{2} d t ⩽ c_{1} \sum_{n = 1}^{\infty} λ_{k n}^{2} β_{k n}^{2} | a_{k n} |^{2} . \end{matrix}$

Taking into account (41) and estimates in Lemma 3.1, we get $\begin{matrix} (44) & \sum_{n = 1}^{\infty} λ_{k n}^{2} β_{k n}^{2} | a_{k n} |^{2} = \frac{1}{4} \sum_{n = 1}^{\infty} (| ϕ_{0 n} |^{2} λ_{k n}^{2} + | ϕ_{1 n} |^{2}) β_{k n}^{2} \end{matrix}$ and $\begin{matrix} (45) & \sum_{n = 1}^{\infty} (| ϕ_{0 n} |^{2} λ_{k n}^{2} + | ϕ_{1 n} |^{2}) β_{k n}^{2} \sim \sum_{n = 1}^{\infty} (| ϕ_{0 n} |^{2} + \frac{1}{n^{2}} | ϕ_{1 n} |^{2} + (n^{2} + 1) | ψ_{0 n} |^{2} + | ψ_{1 n} |^{2}) . \end{matrix}$ Combining (43)–(45), we can guarantee the existence of positive constants $C_{0}$ and $C_{1}$ , depending on T, γ and $γ_{\infty}$ , such that $\begin{matrix} (46) & C_{0} {‖ (ϕ_{0}, ϕ_{1}, ψ_{0}, ψ_{1}, 0) ‖}_{H_{k}^{0}}^{2} ⩽ \int_{0}^{T} {| ψ_{t} (0, t) |}^{2} d t ⩽ C_{1} {‖ (ϕ_{0}, ϕ_{1}, ψ_{0}, ψ_{1}, 0) ‖}_{H_{k}^{0}}^{2} . \end{matrix}$

On the other hand, we have that $\begin{matrix} (47) & ρ (x, t) = \sum_{n \in Z} γ_{k n} \hat{a_{k n}} e^{i λ_{n}^{k} (T - t)} cos (n x) + \sum_{n = 1}^{\infty} b_{k n} e^{- n^{2} (T - t)} cos (n x), \end{matrix}$ where $\hat{a_{k n}} = a_{k n}$ , for $n \neq 0$ , and $\hat{a_{k 0}} = a_{k 0} + b_{k 0}$ . Thus $\begin{matrix} (48) & {‖ ρ (\cdot, 0) ‖}_{L^{2} (0, π)}^{2} ⩽ C \sum_{n = 0}^{\infty} (γ_{k n}^{2} | a_{k n} |^{2} + | b_{k n} |^{2} e^{- n^{2} T}) . \end{matrix}$ Also, from (40) and using Theorem 4.3, for any $T > 2 π \sqrt{a}$ , it follows that $\begin{array}{l} \int_{0}^{T} {| ρ (0, t) |}^{2} d t & = \int_{0}^{T} {| \sum_{n \in Z} γ_{k n} a_{k n} e^{i λ_{n}^{k} (T - t)} + \sum_{n = 0}^{\infty} b_{k n} e^{- n^{2} (T - t)} |}^{2} d t \\ ⩾ C \sum_{n = 0}^{\infty} (γ_{k n}^{2} | a_{k n} |^{2} + | b_{k n} |^{2} e^{- n^{2} T}) . \end{array}$ Therefore, using (48), we obtain $\begin{matrix} (49) & {‖ ρ (\cdot, 0) ‖}_{L^{2} (0, π)}^{2} ⩽ C \int_{0}^{T} {| ρ (0, t) |}^{2} d t . \end{matrix}$

From (46), (49) and taking into account the reversibility of two first equations in (23), we conclude the proof. □

5. Proof of Theorem 1.1

Let us fix the initial datum $(v_{0}, v_{1}, θ_{0}) \in H^{2} (0, π) \times H^{1} (0, π) \times L^{2} (0, π)$ .

The proof of Theorem 1.1 is done in three steps. They are:

Prove the existence of a pair of controls $(m_{k}, δ_{k})$ controlling to zero system (3) for initial datum $(u_{0}, u_{1}, v_{0}, v_{1}, θ_{0}) = (- v_{0 x}, - v_{1 x}, v_{0}, v_{1}, θ_{0}) \in H^{1}$ .

Prove that the pair $(m_{k}, δ_{k})$ is uniformly bounded (with respect to k), and then it has a weakly convergent subsequence;

Prove that the pair $(m, δ)$ , limit of the subsequence, is the control of the limit system.

Step 1. Observe that since we are taking $T > 2 π \sqrt{a}$ the proof of (35) remains valid if we introduce a cut-off function $β (t) \in C^{2} [0, T]$ , $0 ⩽ β ⩽ 1$ , $β = 0$ for $t \in [0, δ / 4] \cup [T - δ / 4, T]$ and $β = 1$ for $t \in [δ / 2, T - δ / 2]$ , with $δ > 0$ such that $T - δ > 2 π \sqrt{a}$ . In this way, we have $\begin{matrix} (50) & {‖ (ϕ (\cdot, 0), ϕ_{t} (\cdot, 0), ψ (\cdot, 0), ψ_{t} (\cdot, 0), ρ (\cdot, 0)) ‖}_{H_{k}^{0}}^{2} ⩽ C \int_{0}^{T} (β (t) {| ψ_{t} (0, t) |}^{2} + {| ρ (0, t) |}^{2}) d t . \end{matrix}$

For each $(ϕ_{k 0}, ϕ_{k 1}, ψ_{k 0}, ψ_{k 1}, ρ_{k 0}) \in H_{k}^{0}$ , we denote by $(ϕ_{k}, ψ_{k}, ρ_{k})$ the corresponding solution to the system (23), and we consider the functional $\begin{array}{l} J_{k} (ϕ_{k 0}, ϕ_{k 1}, ψ_{k 0}, ψ_{k 1}, ρ_{k 0}) \\ = \frac{b}{2} \int_{0}^{T} β (t) {| ψ_{k t} (0, t) |}^{2} d t + \frac{1}{2} \int_{0}^{T} {| ρ_{k} (0, t) |}^{2} d t + a \int_{0}^{π} v_{1 x} (x) ϕ_{k} (x, 0) d x \\ - a ⟨ ϕ_{k t} (\cdot, 0), v_{0 x} ⟩ - b \int_{0}^{π} v_{1} (x) ψ_{k} (x, 0) d x + b \int_{0}^{π} v_{0} (x) ψ_{k t} (x, 0) d x \\ + \int_{0}^{π} θ_{0} (x) ρ_{k} (x, 0) d x . \end{array}$ Notice by (46) that $J_{k}$ is well-defined. It is also not difficult to see that this functional is continuous and strictly convex. We still have from (50) that $J_{k}$ is also coercive. Hence, it has a unique minimum $({\hat{ϕ}}_{k 0}, {\hat{ϕ}}_{k 1}, {\hat{ψ}}_{k 0}, {\hat{ψ}}_{k 1}, {\hat{ρ}}_{k 0}) \in H_{k}^{0}$ . We consider $({\hat{ϕ}}_{k}, {\hat{ψ}}_{k}, {\hat{ρ}}_{k})$ the solution to (23) associated to the minimum. Critical point properties implies that the Gâteaux derivative of $J_{k}$ in any direction $(ϕ_{k 0}, ϕ_{k 1}, ψ_{k 0}, ψ_{k 1}, ρ_{k 0}) \in H_{k}^{0}$ will vanish, that is, $\begin{array}{l} b \int_{0}^{T} β (t) ψ_{k t} (0, t) {\hat{ψ}}_{k t} (0, t) d t + \int_{0}^{T} ρ_{k} (0, t) {\hat{ρ}}_{k} (0, t) d t \\ + a \int_{0}^{π} v_{1 x} (x) ϕ_{k} (x, 0) d x - a ⟨ ϕ_{k t} (\cdot, 0), v_{0 x} ⟩ \\ (51) & - b \int_{0}^{π} v_{1} (x) ψ_{k} (x, 0) d x + b \int_{0}^{π} v_{0} (x) ψ_{k t} (x, 0) d x + \int_{0}^{π} θ_{0} (x) ρ_{k} (x, 0) d x = 0, \end{array}$ for all $(ϕ_{k 0}, ϕ_{k 1}, ψ_{k 0}, ψ_{k 1}, ρ_{k 0}) \in H_{k}^{0}$ , with $(ϕ_{k}, ψ_{k}, ρ_{k})$ being the associated solution to (23). In this way, the identity (51) implies that the solution $(u_{k}, v_{k}, θ_{k})$ of (3), with the pair of controls $(m_{k}, δ_{k})$ defined by $\begin{matrix} (52) & (m_{k}, δ_{k}) = ({(β (\cdot) {\hat{ψ}}_{k t} (0, \cdot))}_{t}, {\hat{ρ}}_{k} (0, \cdot)) \in H^{- 1} (0, T) \times L^{2} (0, T), \end{matrix}$ satisfies $(u_{k} (\cdot, T), u_{k t} (\cdot, T), v_{k} (\cdot, T), v_{k t} (\cdot, T), θ_{k} (\cdot, T)) = (0, 0, 0, 0, 0)$ in $(0, π)$ .

Step 2. We will prove that the sequence $(m_{k}, δ_{k})$ is uniformly bounded. Indeed, since $\begin{matrix} J_{k} ({\hat{ϕ}}_{k 0}, {\hat{ϕ}}_{k 1}, {\hat{ψ}}_{k 0}, {\hat{ψ}}_{k 1}, {\hat{ρ}}_{k 0}) ⩽ J_{k} (0, 0, 0, 0, 0) = 0, \end{matrix}$ we have $\begin{array}{l} \int_{0}^{T} (β (t) {| {\hat{ψ}}_{k t} (0, t) |}^{2} + {| {\hat{ρ}}_{k} (0, t) |}^{2}) d t \\ ⩽ C {‖ (- v_{0 x}, - v_{1 x}, v_{0}, v_{1}, θ_{0}) ‖}_{H^{1}} {‖ ({\hat{ϕ}}_{k} (\cdot, 0), {\hat{ϕ}}_{k t} (\cdot, 0), {\hat{ψ}}_{k} (\cdot, 0), {\hat{ψ}}_{k t} (\cdot, 0), {\hat{ρ}}_{k} (\cdot, 0)) ‖}_{H_{k}^{0}} \\ (53) & ⩽ C {‖ (- v_{0 x}, - v_{1 x}, v_{0}, v_{1}, θ_{0}) ‖}_{H^{1}} {(\int_{0}^{T} β (t) {| {\hat{ψ}}_{k t} (0, t) |}^{2} d t + \int_{0}^{T} {| {\hat{ρ}}_{k} (0, t) |}^{2} d t)}^{1 / 2} . \end{array}$ The last inequality came from the observability inequality (50). It follows by (53) that $\begin{matrix} (54) & {‖ (m_{1 k}, δ_{k}) ‖}_{{[L^{2} (0, T)]}^{2}} ⩽ C {‖ (v_{0 x}, - v_{1 x}, v_{0}, v_{1}, θ_{0}) ‖}_{H^{1}}, \end{matrix}$ where $m_{1 k} = β (\cdot) {\hat{ψ}}_{k t} (0, \cdot)$ . Notice that $m_{k} = {(m_{1 k})}_{t}$ .

Step 3. As a consequence of (54), there exists a subsequence of ${(m_{1 k}, δ_{k})}_{k}$ , still denoted by the index k, such that $\begin{matrix} (55) & (m_{1 k}, δ_{k}) \to (m_{1}, δ) weakly in {[L^{2} (0, T)]}^{2} . \end{matrix}$ With this last convergence, we are in the conditions of Theorem 2.3 and we can guarantee that, as $k \to \infty$ , the following convergence holds: $\begin{matrix} (u_{k}, v_{k}, θ_{k}) \to (- v_{x}, v, θ) weak- * in L^{\infty} (0, T; {[L^{2} (0, π)]}^{3}), \end{matrix}$ where $(v, θ)$ solves (1). We claim that $m = - m_{1 t}$ and δ are the controls such that the solution $(v, θ)$ of (1) verifies (2). To see this, it is enough to pass (51) to the limit taking as test functions the solutions of (23) in separated variables. In fact, considering, for $n \in N$ , the solution $(ϕ_{k}, ψ_{k}, ρ_{k}) = (e^{- i λ_{k n} t} sin n x, e^{- i λ_{k n} t} β_{k n} cos n x, (e^{- i λ_{k n} t} γ_{k n} + e^{- n^{2} t}) cos n x)$ , we get, after passing to the limit, as $k \to \infty$ , and using the convergences (27) and (55), that $\begin{array}{l} \int_{0}^{T} m_{1} (t) ψ_{t} (0, t) d t + \int_{0}^{T} δ (t) ρ (0, t) d t - a \int_{0}^{π} v_{1 x} (x) ψ_{x} (x, 0) + a \int_{0}^{π} v_{0 x} (x) ψ_{x t} (x, 0) d x \\ (56) & - b \int_{0}^{π} v_{1} (x) ψ (x, 0) d x + b \int_{0}^{π} v_{0} (x) ψ_{t} (x, 0) d x + \int_{0}^{π} θ_{0} (x) ρ (x, 0) d x = 0 . \end{array}$ This last identity and the formulation by transposition of (1), gives us $\begin{array}{l} a \int_{0}^{π} v_{x t} (x, T) ψ_{x} (x, T) d x - a \int_{0}^{π} v_{x} (x, T) ψ_{x t} (x, T) d x \\ (57) & + b \int_{0}^{π} v_{t} (x, T) ψ (x, T) d x - b \int_{0}^{π} v (x, T) ψ_{t} (x, T) d x + \int_{0}^{π} θ (x, T) ρ (x, T) d x = 0, \end{array}$ where $(ψ, ρ) = (\frac{1}{n} e^{- i λ_{n} t} cos n x, (e^{- i λ_{n} t} γ_{n} + e^{- n^{2} t}) cos n x)$ , with $γ_{n} = (μ - ξ) n / (λ_{n}^{2} + n^{4})$ . By (57) we find that $\begin{array}{l} e^{- i λ_{n} T} [(a n + \frac{b}{n}) \int_{0}^{π} v_{t} (x, T) cos n x d x + i λ_{n} (a n + \frac{b}{n}) \int_{0}^{π} v (x, T) cos n x d x \\ (58) & + γ_{n} \int_{0}^{π} θ (x, T) cos n x d x] + e^{- n^{2} T} \int_{0}^{π} θ (x, T) cos n x d x = 0, \end{array}$ which implies (2).

6. Further comments and open problems

In this section we briefly mention some direct corollaries from the results in the paper, and we also indicate open problems on the subject.

Similar results are expected for other boundary conditions, but a further study of this issue is needed. This is because, at least to apply the strategy adopted here, an explicit computation of the spectrum of the problem is necessary.

The cascade coupling structure of system (1) allows us to apply spectral methods performed in [3]. It remains an open problem to consider the asymptotic null control problem of a strongly coupled system like $\begin{matrix} (59) & \{\begin{matrix} b v_{t t} + v_{x x x x} - a v_{x x t t} + (ξ + μ) θ_{x x} = 0, & x \in (0, π), t \in (0, T), \\ θ_{t} - θ_{x x} + α v_{t x x} = 0, & x \in (0, π), t \in (0, T), \\ v_{x} (0, t) = v_{x} (π, 0) = 0, & t \in (0, T), \\ v_{x x x} (0, t) = m (t), v_{x x x} (π, t) = 0, & t \in (0, T), \\ θ_{x} (0, t) = δ (t), θ_{x} (π, t) = 0, & t \in (0, T), \\ v (x, 0) = v_{0} (x), v_{t} (x, 0) = v_{1} (x), & x \in (0, π), \\ θ (x, 0) = θ_{0} (x), & x \in (0, π), \end{matrix} \end{matrix}$ as is the kind of coupling studied in [10].

Another interesting and challenging open problem is to control system (1) (or (59)) with a control acting only on one equation, that is, with $m (t) \equiv 0$ or $δ (t) \equiv 0$ .

Footnotes

Appendix

In this appendix, we prove Theorem 4.3. For this, we follow the ideas of Komornik [13].

Theorem 1.1 in [14] states that, if $\tilde{γ} > 0$ and $p \in N$ are such that $\begin{matrix} (60) & inf_{n \in Z} \frac{λ_{n + p} - λ_{n}}{p} ⩾ \tilde{γ}, \end{matrix}$ then (39) holds true for any $T > 2 π / \tilde{γ}$ .

We assume that $N \in N$ and $T > 0$ are such that $T > 2 π / γ_{\infty}$ with $\begin{matrix} (61) & γ_{\infty} : = inf_{| n | > N} {λ_{n + 1} - λ_{n}} > 0 . \end{matrix}$ We define, for a natural number $p > 2 N$ , $\begin{matrix} {\tilde{γ}}_{p} = \frac{(p - 2 N) γ_{\infty}}{p} . \end{matrix}$ Observe that ${\tilde{γ}}_{p} \to γ_{\infty}$ , as $p \to \infty$ . Then, we can take p large enough such that $T > 2 π / {\tilde{γ}}_{p}$ . It is not difficult to see that (60) is satisfied for $\tilde{γ} = {\tilde{γ}}_{p}$ . Indeed, we have $\begin{matrix} λ_{n + p} - λ_{n} ⩾ \{\begin{matrix} p γ_{\infty} & if n > N, \\ λ_{p - N} - λ_{N} ⩾ (p - 2 N) γ_{\infty} & if - N ⩽ n ⩽ N, \\ λ_{- N} - λ_{N - p} ⩾ (p - 2 N) γ_{\infty} & if - N ⩽ n + p ⩽ N, \\ p γ_{\infty} & if n + p < - N, \end{matrix} \end{matrix}$ and, therefore, $\begin{matrix} \frac{λ_{n + p} - λ_{n}}{p} ⩾ \frac{(p - 2 N) γ_{\infty}}{p} = {\tilde{γ}}_{p}, \end{matrix}$ for all $n \in Z$ . Then Theorem 1.1 in [14] can be applied and the proof is complete.

Acknowledgements

We thank Vilmos Komornik for fruitful discussions about the proof of Theorem . The research was done partially supported by Grant 2019/0014 Paraíba State Research Foundation (FAPESQ), CNPq, CAPES, MathAmSud ACIPDE, FONDECYT 1211292, Basal CMM U. de Chile, ANID – Millennium Science Initiative Program NCN19-161, and by program UNAM-DGAPA-PAPIIT-IN109522 and Conacyt-A1-S-17475. The last version of this paper was written as the third author was at Universidade Federal da Paraíba as visiting professor.

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