Boundary Controllability of a Pseudoparabolic Equation

Abstract

We deal with the controllability problem for the pseudoparabolic equation by means of boundary controls. Due to the unusual spectrum of this kind of equations, we prove that the null controllability property is false. Furthermore, by the explicit solution, we show that the approximate controllability holds.

1. Introduction

This paper is devoted to the study of controllability properties of the 1D pseudoparabolic equation: 𝑦_𝑡−𝑦_𝑥𝑥−𝑘𝑦_𝑥𝑥𝑡()(),𝑦()()(𝑡)(),𝑦()=0,𝑥∈0,1,𝑡∈0,𝑇0,𝑡=0,𝑦1,𝑡=𝑢,𝑡∈0,𝑇𝑥,0=𝑦₀(𝑥),𝑥∈(0,1), (1) where 𝑦=𝑦(𝑥,𝑡) is the state, 𝑢=𝑢(𝑡) is a control function acting on the boundary {1}×(0,𝑇), and 𝑘>0 is a constant.

The pseudoparabolic equations are a kind of Sobolev-Galpern type equations. They have occurred in numerous physical applications among which include problems involving seepage of fluids through fissured rocks [1], unsteady flows of second-order fluids [2, 3], and the theory of thermodynamics involving two temperatures [4]. They can also be used as a regularization of ill-posed transport problems, especially as a quasicontinuous approximation to discrete models for population dynamics [5]. Furthermore, pseudoparabolic equations are closely related to the well-know BBM equations [6] which are advocated as a refinement of KdV equations.

In the last two decades, important progress has been made in the controllability analysis of parabolic equations. We refer to the works [7–13] and the references therein. It is well known that the null and approximate controllability hold for the classical parabolic equations. However, for some special models, there arise some new results. For example, in [14], the authors considered the heat equation with memory: 𝑦_𝑡−Δ𝑦+^𝑡₀Δ𝑦(𝑥,𝑠)𝑑𝑠=0. (2) By establishing that the observability inequality for the heat equation with memory is not true, they proved that there exists a set of initial conditions such that the null controllability property fails by means of boundary controls. Recently, Doubova and Fernández-Cara [15] studied the approximate distributed and boundary controllability of viscoelastic fluids of the Jeffreys kind, which can be equivalently rewritten as a parabolic equation with memory 𝑦_𝑡−𝜈Δ𝑦−𝑏^𝑡₀𝑒^{−𝑎(𝑡−𝑠)}Δ𝑦(𝑠)𝑑𝑠+∇𝜋=𝑒^−𝑎𝑡∇⋅𝜏₀. (3) The main tool for proving the approximate controllability result is a unique continuation property for its adjoint system. The authors pointed out that this parabolic equation with memory transforms from a damped wave equation (see [15]): 𝑧_𝑡𝑡−𝑎𝑧_𝑡−𝜈Δ𝑧_𝑡−𝑏Δ𝑧+∇𝑒^𝑎𝑡𝜋=∇⋅𝜏₀. (4) In recent years, there are more and more works addressing the controllability problems of damped wave equations (see e.g., [16–19] and the references cited therein).

In this paper, we focus on another kind of parabolic equation with damped term 𝑦_𝑡𝑥𝑥, that is, so-called the pseudoparabolic equation. Inspired by the above works, whether the pseudoparabolic equation is controllable or not seems very interesting. Indeed, as the third-order term arises, some properties of (1) are quite different from those of the parabolic equations. One of the most essential differences is that, comparing with the fact that the eigenvalues of heat equations accumulate at −∞, the eigenvalues of (1) have an accumulation real point −1/𝑘 (it will be shown in Section 2). A similar property was pointed in [20]. This difference causes that the controllability properties of (1) become deeply different from the ones for the parabolic equations. We will show that system (1) is not null controllable under the influence of such unusual spectrum. To this end, we turn the control problem into a moment problem. Thanks to the Paley-Wiener theorem, the result is got, and our approach avoids the proof of the observability inequalities which was used in [14]. On the other hand, we establish the approximate controllability of system (1) in some Fourier definition of Sobolev spaces 𝐻_𝛼. The proof is based on a duality method and the explicit solution of the adjoint system of (1), which we obtain by variable separations instead of the Laplace transform in [15]. The techniques we use to prove the approximate controllability contain some ideas of the unique continuation properties.

Throughout this paper, we will use the following notations. The 𝑛th Fourier coefficient (with respect to the orthonormal basis (sin(𝑛𝜋𝑥))_𝑛≥1 of 𝐿²(0,1)) of any integrable function 𝑦∶(0,1)→ℝ is defined as 𝑦_𝑛=¹₀𝑦(𝑥)sin(𝑛𝜋𝑥)𝑑𝑥. (5) So, 𝑦 can be written in the form 𝑦(𝑥)=_𝑛≥1𝑦_𝑛sin(𝑛𝜋𝑥). (6) For any 𝛼∈ℝ, let 𝐻_𝛼∶=𝑦∶(0,1)⟶ℝ;_𝑛≥1𝑛^2𝛼𝑦_𝑛²<∞. (7) Endowed with the scalar product 𝑦,𝑧_𝛼=_𝑛≥1𝑛^2𝛼𝑦_𝑛𝑧_𝑛. (8)𝐻_𝛼 is a Hilbert space. Moreover, 𝐻₁=𝐻¹₀(0,1), 𝐻₂=𝐻²(0,1)⋂𝐻¹₀(0,1), and 𝐻_−𝛼=𝐻^′_𝛼 (the dual space of 𝐻_𝛼 with respect to the central space 𝐻₀=𝐿²(0,1)) for any 𝛼≥0. Finally, for any 𝑓=∑_𝑛≥1𝑐_𝑛sin(𝑛𝜋𝑥)∈𝐻_−𝛼 and any 𝑔=∑_𝑛≥1𝑑_𝑛sin(𝑛𝜋𝑥)∈𝐻_𝛼, we have that 𝑓,𝑔_{𝐻−𝛼,𝐻𝛼}=_𝑛≥1𝑐_𝑛𝑑_𝑛, (9) where ⟨⋅,⋅⟩_{𝐻−𝛼,𝐻𝛼} stands for the duality pairing between 𝐻_−𝛼 and 𝐻_𝛼.

The main aim of this paper is to analyze the controllability properties of (1). It will be said that (1) is approximate controllable by boundary control at time 𝑇; if for any 𝑦₀∈𝐻¹₀(0,1), the set of reachable states 𝑅(𝑇)=𝑦(𝑇)()∶(1)=𝑦𝑇;𝑢𝑦isthesolutionofwith𝑢∈𝐻¹₀(0,𝑇) (10) is dense in 𝐻_𝛼. And it will be said that (1) is null controllable at time 𝑇; if for any given 𝑦₀∈𝐻¹₀(0,1), there exist controls 𝑢∈𝐻¹₀(0,𝑇) such that the associated solutions to (1) satisfy 𝑦(𝑥,𝑇)=0inΩ. (11)

For the sake of simplicity, we will take 𝑘=1 throughout this paper. All the results can be extended without difficulty to 𝑘>0 arbitrary.

The rest of this paper is organized as follows. In Section 2, we will show some elementary properties for (1) and their adjoint equation. Section 3 is devoted to studying the null and approximate controllability of (1), respectively. In Section 4, some open questions related to this work are provided.

2. Preliminaries

In this section, we first consider the existence and uniqueness of the solution to problem (1).

Let 𝑦(𝑥,𝑡)=𝑧(𝑥,𝑡)+𝑥𝑢(𝑡). We readily obtain that 𝑧 is a solution of the system 𝑧_𝑡−𝑧_𝑥𝑥−𝑧_𝑥𝑥𝑡=−𝑥𝑢^′()(),𝑧()()(),𝑧(),𝑥∈0,1,𝑡∈0,𝑇0,𝑡=𝑧1,𝑡=0,𝑡∈0,𝑇𝑥,0=𝑦₀(𝑥)−𝑥𝑢(0),𝑥∈(0,1). (12) The existence and uniqueness of the solution to the problem (12) are well known (see [21, 22]); that is, for every 𝑦₀∈𝐻¹₀(0,1) and 𝑢∈𝐻¹₀(0,𝑇), system (12) admits a unique solution 𝑧∈𝐶([0,𝑇];𝐿²(0,1)). In turn, we see that if 𝑦₀∈𝐻¹₀(0,1) and 𝑢∈𝐻¹₀(0,𝑇), system (1) admits a unique solution 𝑦∈𝐶([0,𝑇];𝐿²(0,1)).

In order to prove the controllability of system (1), let us consider the following homogeneous initial boundary value problem: 𝑦_𝑡−𝑦_𝑥𝑥−𝑦_𝑥𝑥𝑡()(),𝑦()()(),𝑦()=0,𝑥∈0,1,𝑡∈0,𝑇0,𝑡=𝑦1,𝑡=0,𝑡∈0,𝑇𝑥,0=𝑦₀(𝑥),𝑥∈(0,1). (13) We will give an explicit solution of the problem (13) by the method of separation of variables.

Proposition 1. If the initial condition 𝑦₀ is given by 𝑦₀(𝑥)=∑_𝑛≥1𝑎_𝑛sin(𝑛𝜋𝑥), then the solution to (13) is 𝑦(𝑥,𝑡)=_𝑛≥1𝑎_𝑛𝑒^𝜆𝑛𝑡sin(𝑛𝜋𝑥), (14) where 𝜆_𝑛=𝜇_𝑛/(1−𝜇_𝑛) and 𝜇_𝑛=−(𝑛𝜋)². Moreover, 𝜆_𝑛<0, {𝜆_𝑛}_𝑛≥1 is a monotone decreasing sequence, and 𝜆_𝑛→−1 as 𝑛→∞.

Proof. Let 𝑦(𝑥,𝑡)=𝑇(𝑡)𝑋(𝑥)≠0. From the first equality of (13), we have 𝑇^′(𝑡)𝑇(𝑡)+𝑇^′(𝑡)≡𝑋^′′(𝑥)𝑋(𝑥). (15) We can see that the identity is true if and only if both sides of it are equal to one constant. Let the constant be 𝜇. Then we get 𝑋^′′(𝑥)(𝑥)𝑇=𝜇𝑋,0<𝑥<1,_′(𝑡)−𝜇𝑇(𝑡)−𝜇𝑇^′(𝑡)=0,𝑡>0. (16) Since the solution 𝑦(𝑥,𝑡) satisfies the boundary condition, 𝑋(0)=𝑋(1)=0 is necessary. Thus, we obtain an eigenvalue problem 𝑋^′′(𝑥)(𝑥)𝑋(0)(1)=𝜇𝑋,0<𝑥<1,=𝑋=0. (17) By using a simple calculation, we have that 𝜇=𝜇_𝑛=−(𝑛𝜋)² and 𝑋_𝑛(𝑥)=𝐵₀sin(𝑛𝜋𝑥), (18) where 𝐵₀ is an arbitrary constant. Now, let us turn to the second equation of (16) 𝑇^′(𝑡)−𝜇_𝑛𝑇^′(𝑡)−𝜇_𝑛𝑇(𝑡)=0,𝑡>0. (19) It is easy to see that 𝑇_𝑛(𝑡)=𝐵₁𝑒^𝜆𝑛𝑡 (20) is the solution to (19), where 𝜆_𝑛=𝜇_𝑛/(1−𝜇_𝑛) and 𝐵₁ is an arbitrary constant. Combining the initial condition 𝑦₀, we can write the solution of (13) as 𝑦(𝑥,𝑡)=_𝑛≥1𝑎_𝑛𝑒^𝜆𝑛𝑡sin(𝑛𝜋𝑥). (21) The expressions of 𝜆_𝑛 imply that 𝜆_𝑛<0, {𝜆_𝑛}_𝑛≥1 is a monotone decreasing sequence, and 𝜆_𝑛→−1 as 𝑛→∞.

Remark 2. It is important to observe that the spectrum of (1) is quite different from that of heat equation. This will be essential when dealing with the controllability problem of (1).

As an easy consequence of the above representation formula, we have the following result.

Proposition 3. Let 𝛼∈ℝ. If 𝑦₀∈𝐻_𝛼, then 𝑦∈𝐶(ℝ⁺;𝐻_𝛼). In addition, if 𝛼>3/2, then ∑_𝑛≥1𝑛|𝑎_𝑛|<+∞ and 𝑦_𝑥(1,⋅)∈𝐶(ℝ⁺).

Proof. If 𝑦₀∈𝐻_𝛼, we have ∑_𝑛≥1𝑛^2𝛼|𝑎_𝑛|²<+∞. It holds that 𝑛^2𝛼𝑎_𝑛𝑒^{𝜆𝑛𝑡2}≤𝑛^2𝛼𝑎_𝑛²𝑒^{2𝜆𝑛𝑡}≤𝑛^2𝛼𝑎_𝑛², (22) and hence, 𝑦∈𝐶(ℝ⁺;𝐻_𝛼).
On the other hand, taking the derivative with respect to 𝑥 in (14), we obtain that 𝑦_𝑥(1,𝑡)=_𝑛≥1𝑎_𝑛𝑒^𝜆𝑛𝑡(𝑛𝜋)(−1)^𝑛. (23) Since _𝑛≥1𝑛𝑎_𝑛𝑒^𝜆𝑛𝑡≤_𝑛≥1𝑛𝑎_𝑛≤_𝑛≥1𝑛^2𝛼𝑎_𝑛^21/2⋅_𝑛≥1𝑛^{−2(𝛼−1)1/2}, (24) we see that ∑_𝑛≥1𝑛|𝑎_𝑛|<+∞ and 𝑦_𝑥(1,⋅)∈𝐶(ℝ⁺), provided that 𝛼>3/2.

Now, let us turn to the adjoint system to system (1) as follows: 𝜑_𝑡+𝜑_𝑥𝑥−𝜑_𝑥𝑥𝑡()(),𝜑()()(),𝜑()=0,𝑥∈0,1,𝑡∈0,𝑇0,𝑡=𝜑1,𝑡=0,𝑡∈0,𝑇𝑥,𝑇=𝜑^𝑇(𝑥),𝑥∈(0,1). (25) Based on the method in Proposition 1, we have that if 𝜑^𝑇 is decomposed as 𝜑^𝑇(𝑥)=_𝑛≥1𝑏_𝑛sin(𝑛𝜋𝑥), (26) the solution of (25) is given by 𝜑(𝑥,𝑡)=_𝑛≥1𝑏_𝑛𝑒^{𝜆𝑛(𝑇−𝑡)}sin(𝑛𝜋𝑥), (27) which yields 𝜑_𝑥(1,𝑡)=_𝑛≥1𝑏_𝑛𝑒^{𝜆𝑛(𝑇−𝑡)}(𝑛𝜋)(−1)^𝑛. (28) Corresponding to Proposition 3, we have the following result.

Proposition 4. Let 𝛼∈ℝ. If 𝜑^𝑇∈𝐻_−𝛼, then 𝜑∈𝐶(ℝ⁺;𝐻_−𝛼). In addition, if 𝛼<−3/2, then ∑_𝑛≥1𝑛|𝑏_𝑛|<+∞ and 𝜑_𝑥(1,⋅)∈𝐶(ℝ⁺).

Proof. If 𝜑^𝑇∈𝐻_−𝛼, we have ∑_𝑛≥1𝑛^−2𝛼|𝑏_𝑛|²<∞. It follows that 𝑛^−2𝛼𝑏_𝑛𝑒^{𝜆𝑛(𝑇−𝑡)2}≤𝑛^−2𝛼𝑏_𝑛². (29) Hence 𝜑∈𝐶(ℝ⁺;𝐻_−𝛼). On the other hand, we have by the Cauchy-Schwarz inequality that _𝑛≥1𝑛𝑏_𝑛≤_𝑛≥1𝑛^2𝛼+21/2⋅_𝑛≥1𝑛^−2𝛼𝑏_𝑛^21/2. (30) It is clear that ∑_𝑛≥1𝑛|𝑏_𝑛|<+∞, provided that 𝛼<−3/2.

3. The Main Results

This section is devoted to the study of the controllability of system (1). We first do some transformation by the duality principle.

Multiplying (formally) the first equation in (1) by 𝜑 which is the solution of (25) and integrating on (0,1)×(0,𝑇), we have ^𝑇₀¹₀𝑦_𝑡𝜑𝑑𝑥𝑑𝑡−^𝑇₀¹₀𝑦_𝑥𝑥𝜑𝑑𝑥𝑑𝑡−^𝑇₀¹₀𝑦_𝑥𝑥𝑡𝜑𝑑𝑥𝑑𝑡=0. (31) Integrating by parts, we get ¹₀𝑦(𝑥,𝑇)𝜑(𝑥,𝑇)𝑑𝑥−^𝑇₀¹₀𝑦𝜑_𝑡+𝑑𝑥𝑑𝑡_𝑇0𝑦(1,𝑡)𝜑_𝑥(1,𝑡)𝑑𝑡−^𝑇₀¹₀𝑦𝜑_𝑥𝑥+𝑑𝑥𝑑𝑡_𝑇0𝑦_𝑡(1,𝑡)𝜑_𝑥()−1,𝑡𝑑𝑡₁₀𝑦(𝑥,𝑇)𝜑_𝑥𝑥()+𝑥,𝑇𝑑𝑥_𝑇010𝑦𝜑_𝑥𝑥𝑡+𝑑𝑥𝑑𝑡₁₀𝑦₀(𝑥)𝜑_𝑥𝑥(𝑥,0)−𝜑(𝑥,0)𝑑𝑥=0. (32) Noticing the first equation in (25), we obtain ¹₀𝑦(𝑥,𝑇)𝜑_𝑥𝑥()()−𝑥,𝑇−𝜑𝑥,𝑇𝑑𝑥₁₀𝑦₀(𝑥)𝜑_𝑥𝑥()(=𝑥,0−𝜑𝑥,0)𝑑𝑥_𝑇0𝑦_𝑡(1,𝑡)𝜑_𝑥(1,𝑡)𝑑𝑡+^𝑇₀𝑦(1,𝑡)𝜑_𝑥(1,𝑡)𝑑𝑡. (33) It is clear that (33) can be rewritten as −𝜑−𝜑_𝑥𝑥,𝑦_{𝐻−𝛼,𝐻𝛼}^𝑇₀=^𝑇₀𝑢(𝑡)+𝑢^′(𝑡)𝜑_𝑥(1,𝑡)𝑑𝑡. (34) Now, we are ready to show the controllability of system (1).

3.1. Null Controllability

Following some of the key ideas developed by Micu [23], we are able to show that the null controllability property fails.

Theorem 5. For 𝑇>0, there exist initial conditions 𝑦₀∈𝐻¹₀(0,1) such that for any control function 𝑢∈𝐻¹₀(0,𝑇), the associated solution 𝑦 to system (1) is not identically equal zero at time 𝑇.

Proof. It follows from (34) that the null controllability problem is equivalent to the existence of a control function 𝑢=𝑢(𝑡) such that 𝜑(𝑥,0)−𝜑_𝑥𝑥(𝑥,0),𝑦₀(𝑥)_{𝐻−𝛼,𝐻𝛼}=^𝑇₀𝑢(𝑡)+𝑢^′(𝑡)𝜑_𝑥(1,𝑡)𝑑𝑡. (35) Using (9), (27), and (28), we can transform the control problem into a moments problem. In other words, we need to find the control function 𝑢=𝑢(𝑡) such that ^𝑇₀𝑢(𝑡)+𝑢^′(𝑡)𝑒^{−𝜆𝑛𝑡}(𝑛𝜋)(−1)^𝑛𝑑𝑡=𝑎_𝑛1+𝑛²𝜋². (36) To this end, we consider that any initial data 𝑦₀ with the sequence {𝑎_𝑛}_𝑛≥1 satisfies 𝑎_𝑛=0 for 𝑛>𝑁. Suppose that for this kind of 𝑦₀, there exists a 𝑢 such that (36) holds. Let 𝐹(𝑧)∶=^𝑇₀𝑢(𝑡)+𝑢^′(𝑡)𝑒^𝑖𝑧𝑡𝑑𝑡. (37) By Paley-Wiener theorem, 𝐹 is an entire function and 𝐹𝑖𝜆_𝑛=^𝑇₀𝑢(𝑡)+𝑢^′(𝑡)𝑒^{−𝜆𝑛𝑡}𝑑𝑡=0,for𝑛>𝑁. (38) The fact that 𝑖𝜆_𝑛→−𝑖 as 𝑛→+∞ implies that 𝐹 is zero on a set with a finite accumulation point. Therefore, 𝐹≡0. It follows that 𝑎_𝑛=0 for each 𝑛≥1. Thus the proof of Theorem 5 is completed.

3.2. Approximate Controllability

Because of the lack of null controllability, the approximate controllability of system (1) becomes much more interesting. We will study the approximate controllability in this part. Without loss of generality, we assume that 𝑦₀=0 and by (34) we have −𝜑^𝑇(𝑥)−𝜑^𝑇_𝑥𝑥(𝑥),𝑦(𝑥,𝑇)_{𝐻−𝛼,𝐻𝛼}=^𝑇₀𝑢(𝑡)+𝑢^′(𝑡)𝜑_𝑥(1,𝑡)𝑑𝑡. (39)

In order to show the approximate controllability, two lemmas are needed later. The first one is an equivalent condition for the approximate controllability.

Lemma 6. For system (25), if we assume that 𝜑^𝑇=0 if and only if ∫^𝑇₀(𝑢(𝑡)+𝑢^′(𝑡))𝜑_𝑥(1,𝑡)𝑑𝑡=0 for any function 𝑢=𝑢(𝑡), then system (1) is approximatively controllable in 𝐻_𝛼.

Proof. Recall that the approximate controllability of system (1) in 𝐻_𝛼 is equivalent to that 𝑅(𝑇) that is dense in 𝐻_𝛼. Therefore, in order to conclude the proof, we only need to show that 𝑅(𝑇) is dense in 𝐻_𝛼. It follows from Hahn-Banach theorem that every continuous linear functional on 𝐻_𝛼 which vanishes on 𝑅(𝑇), must vanish everywhere on 𝐻_𝛼.
Now, suppose that 𝑅(𝑇) is not dense in 𝐻_𝛼. Then by Hahn-Banach theorem there exists 𝜑^𝑇∈𝐻_−𝛼 with 𝜑^𝑇−𝜑^𝑇_𝑥𝑥≠0, such that for any 𝑦(𝑥,𝑇)∈𝑅(𝑇), we have 𝜑^𝑇(𝑥)−𝜑^𝑇_𝑥𝑥(𝑥),𝑦(𝑥,𝑇)_{𝐻−𝛼,𝐻𝛼}=0. (40) By (39), we get that ^𝑇₀𝑢(𝑡)+𝑢^′(𝑡)𝜑_𝑥(1,𝑡)𝑑𝑡=0. (41) But from the condition 𝜑^𝑇=0 if and only if ∫^𝑇₀(𝑢(𝑡)+𝑢^′(𝑡))𝜑_𝑥(1,𝑡)𝑑𝑡=0 for any function 𝑢=𝑢(𝑡), we have that 0 is the only initial value 𝜑^𝑇∈𝐻_−𝛼 for which the solution 𝜑(𝑥,𝑡) of (25) fulfills ∫^𝑇₀(𝑢(𝑡)+𝑢^′(𝑡))𝜑_𝑥(1,𝑡)𝑑𝑡=0 for any function 𝑢=𝑢(𝑡). Therefore 𝜑^𝑇=0, and it is contrary to the choice of 𝜑^𝑇. This completes the proof of Lemma 6.

Lemma 6 implies that, in order to prove the approximate controllability of system (1) with boundary control, we only need to check the condition about the solution of the dual problem (25) in Lemma 6.

The following elementary lemma can be found in [24, 25].

Lemma 7. Let (𝛽_𝑛)_𝑛≥1 and (𝜆_𝑛)_𝑛≥1 be two sequences of complex numbers such that ∑_𝑛≥1|𝛽_𝑛|<∞ and Re𝜆_𝑛<Λ for each 𝑛≥1 and some number Λ∈ℝ. Assume that the 𝜆_𝑛s are pairwise distinct and that ∑_𝑛≥1𝛽_𝑛𝑒^𝜆𝑛𝑡=0 for a.e. 𝑡∈(0,𝑇). Then 𝛽_𝑛=0 for all 𝑛≥1.

We are now in a position to present the proof of the approximate controllability of (1) with boundary control.

Theorem 8. Let 𝑇>0. System (1) is approximately controllable in 𝐻_𝛼 with 𝛼<−3/2 at time 𝑇.

Proof. First, we prove that there is a control function 𝑢 such that 𝑅(𝑇) is dense in 𝐻_𝛼 with 𝛼<−3/2. Now, take 𝜑^𝑇∈𝐻_−𝛼 with 𝛼<−3/2 decomposed as in (26).
Set 𝜃=𝑢+𝑢^′ and denote 𝐺∶=𝑔∈𝐿²(0,𝑇);^𝑇₀𝑒^𝑡𝑔(𝑡)𝑑𝑡=0. (42) Then it follows that when 𝑢 ranges over 𝐻¹₀(0,𝑇) and 𝜃 ranges over 𝐺.
Assume that, for any 𝑢∈𝐻¹₀(0,𝑇), ^𝑇₀𝜃(𝑡)𝜑_𝑥(1,𝑡)𝑑𝑡=^𝑇₀𝑢(𝑡)+𝑢^′(𝑡)𝜑_𝑥(1,𝑡)𝑑𝑡=0. (43) By Lemma 6, we only need to prove that 𝜑^𝑇=0 or equivalently that 𝑏_𝑛=0 for each 𝑛 (see (26)). Noticing that for any 𝜃∈ Span (𝑒^𝑡)^⟂, 𝜃,𝜑_𝑥(1,⋅)_{𝐿2(0,𝑇)}=0. (44) We have 𝜑_𝑥(1,⋅)∈ Span (𝑒^𝑡)^⟂⟂= Span (𝑒^𝑡). Therefore, from (28), there exists a constant 𝐶∈ℝ such that _𝑛≥1𝑏_𝑛𝑒^{𝜆𝑛(𝑇−𝑡)}(𝑛𝜋)(−1)^𝑛=𝐶𝑒^𝑡, (45) for a.e. 𝑡∈(0,𝑇). If taking 𝜏=𝑇−𝑡, 𝑐_𝑛=𝑏_𝑛(𝑛𝜋)(−1)^𝑛, and 𝑐₀∶=−𝐶𝑒^𝑇, we obtain _𝑛≥1𝑐_𝑛𝑒^𝜆𝑛𝜏+𝑐₀𝑒^−𝜏=0, (46) for a.e. 𝜏∈(0,𝑇).
On the other hand, since 𝜑^𝑇∈𝐻_−𝛼 with 𝛼<−3/2 and by Proposition 4, we have that _𝑛≥1𝑐_𝑛=𝜋_𝑛≥1𝑛𝑏_𝑛<+∞. (47) Then by Lemma 7, we obtain that 𝑐_𝑛=0 for each 𝑛≥1. It implies that 𝜑^𝑇=0. The proof of Theorem 8 is completed.

4. Concluding Remark

In this paper, the controllability of the pseudoparabolic equation is studied. With a boundary control, it is proved that the system is not null controllable, but that an approximate controllability result is obtained in some appropriate functional spaces. Below is a list of unsolved and interesting questions related to our work.(1)It has been got that the approximate controllability holds in 𝐻_𝛼 with 𝛼<−3/2. The question whether the approximate controllability holds for 𝛼≥−3/2 remains open. The method in this paper does not work for that case.(2)It seems natural to expect that the controllability for multidimensional pseudoparabolic equations through a boundary controller or a locally distributed one. We will consider these problems in the forthcoming papers.(3)It would be quite interesting to study the controllability properties of (1) for the case variable coefficient (i.e., 𝑘=𝑘(𝑥,𝑡)). However, it seems very difficult, and many classical methods such as moment problem and strongly continuous semigroups may be false. Indeed, controllability of equation with variable coefficients always bring us much more difficult than that with constant coefficient. One needs a highly innovative way to obtain the observability inequalities or unique continuation properties. An example was presented in [26] to establish some controllability results for wave equations with variable coefficients by a Riemannian geometry method.

Acknowledgment

The authors would like to thank the anonymous reviewer for their valuable comments, which significantly contributed to improving the quality of the paper. This work was supported by the NSF of China under Grants 11171060, 11271381, 11301345, and 11071036, National Basic Research Program of China (973 Program) (no. 2011CB808002).