Approximate Controllability of a Semilinear Heat Equation

Abstract

We apply Rothe’s type fixed point theorem to prove the interior approximate controllability of the following semilinear heat equation: 𝑧_𝑡(𝑡,𝑥)=Δ𝑧(𝑡,𝑥)+1_𝜔𝑢(𝑡,𝑥)+𝑓(𝑡,𝑧(𝑡,𝑥),𝑢(𝑡,𝑥)) in (0,𝜏]×Ω,𝑧=0, on (0,𝜏)×𝜕Ω,𝑧(0,𝑥)=𝑧₀(𝑥),𝑥∈Ω, where Ω is a bounded domain in ℝ^𝑁(𝑁≥1), 𝑧₀∈𝐿²(Ω), 𝜔 is an open nonempty subset of Ω, 1_𝜔 denotes the characteristic function of the set 𝜔, the distributed control 𝑢 belongs to 𝐿²(0,𝜏;𝐿²(Ω)), and the nonlinear function 𝑓∶[0,𝜏]×ℝ×ℝ→ℝ is smooth enough, and there are 𝑎,𝑏,𝑐∈ℝ, 𝑅>0 and 1/2≤𝛽<1 such that |𝑓(𝑡,𝑧,𝑢)−𝑎𝑧|≤𝑐|𝑢|^𝛽+𝑏, for all 𝑢,𝑧∈ℝ,|𝑢|,|𝑧|≥𝑅. Under this condition, we prove the following statement: for all open nonempty subset 𝜔 of Ω, the system is approximately controllable on [0,𝜏]. Moreover, we could exhibit a sequence of controls steering the nonlinear system from an initial state 𝑧₀ to an 𝜖 neighborhood of the final state 𝑧₁ at time 𝜏>0.

1. Introduction

In this paper, we prove the interior approximate controllability of the following semilinear heat equation: 𝑧_𝑡(𝑡,𝑥)=Δ𝑧(𝑡,𝑥)+1_𝜔𝑢()((,(]()𝑧()𝑡,𝑥+𝑓𝑡,𝑧,𝑢𝑡,𝑥))in0,𝜏×Ω,𝑧=0,on0,𝜏×𝜕Ω,0,𝑥=𝑧₀(𝑥),𝑥∈Ω, (1) where Ω is a bounded domain in ℝ^𝑁(𝑁≥1), 𝑧₀∈𝐿²(Ω), 𝜔 is an open nonempty subset of Ω, 1_𝜔 denotes the characteristic function of the set 𝜔, the distributed control 𝑢 belong to 𝐿²(0,𝜏;𝐿²(Ω)), and the nonlinear function 𝑓∶[0,𝜏]×ℝ×ℝ→ℝ is smooth enough, and there are 𝑎,𝑏,𝑐∈ℝ, 𝑅>0 and 1/2≤𝛽<1 such that 𝑓(𝑡,𝑧,𝑢)−𝑎𝑧≤𝑐|𝑢|^𝛽+𝑏,∀𝑢,𝑧∈ℝ,|𝑢|,|𝑧|≥𝑅, (2) which implies that 𝑓(𝑡,𝑧,𝑢)≤𝑐|𝑢|^𝛽+𝑒|𝑧|+𝑏,∀𝑢,𝑧∈ℝ,|𝑢|,|𝑧|≥𝑅. (3)

Definition 1 (approximate controllability). The system (1) is said to be approximately controllable on [0,𝜏] if for every 𝑧₀, 𝑧₁∈𝑍=𝑈=𝐿²(Ω), 𝜀>0, there exists 𝑢∈𝐿²(0,𝜏;𝑈) such that the solution 𝑧(𝑡) of (1) corresponding to 𝑢 verifies 𝑧(0)=𝑧₀,𝑧(𝜏)−𝑧_1𝑧<𝜀; (4) see (Figure 1), where 𝑧(𝜏)−𝑧_1𝑍=_Ω𝑧(𝜏,𝑥)−𝑧₁(𝑥)²𝑑𝑥^1/2. (5)

Figure 1 

Remark 2. It is clear that exact controllability of the system (1) implies approximate controllability, null controllability, and controllability to trajectories of the system. But it is well known (see [1]) that due to the diffusion effect or the compactness of the semigroup generated by −Δ, the heat equation can never be exactly controllable. We observe also that in the linear case, controllability to trajectories and null controllability are equivalent. Nevertheless, the approximate controllability and the null controllability are in general independent. Therefore, in this paper, we shall concentrate only on the study of the approximate controllability of the system (1).

Now, before proceeding with the introduction of this paper, we should mention the work of other authors and show that ours is different in relation with new perturbation and the technique used. The approximate controllability of the heat equation under nonlinear perturbations 𝑓(𝑧) independents of 𝑡 and 𝑢 variable 𝑧_𝑡(𝑡,𝑥)=Δ𝑧(𝑡,𝑥)+1_𝜔𝑢()(𝑧)(]()𝑧()𝑡,𝑥+𝑓,in0,𝜏×Ω,𝑧=0,on0,𝜏×𝜕Ω,0,𝑥=𝑧₀(𝑥),𝑥∈Ω (6) has been studied by several authors, particularly in [2–4], depending on conditions that impose to the nonlinear term 𝑓(𝑧). For instance, in [3, 4], the approximate controllability of the system (6) is proved if 𝑓(𝑧) is sublinear at infinity, that is, 𝑓(𝑧)≤𝑑|𝑧|+𝑒. (7) Also, in the above reference, they mentioned that when 𝑓 is superlinear at the infinity, the approximate controllability of the system (6) fails.

Recently, the interior controllability of the semilinear heat equation (1) has been proved in [5–7] under the following condition: there are constants 𝑎,𝑐∈ℝ, with 𝑐≠−1, such that sup_{(𝑡,𝑧,𝑢)∈𝑄𝜏}𝑓(𝑡,𝑧,𝑢)−𝑎𝑧−𝑐𝑢<∞, (8) where 𝑄_𝜏=[0,𝜏]×ℝ×ℝ.

We note that the interior approximate controllability of the linear heat equation 𝑧_𝑡(𝑡,𝑥)=Δ𝑧(𝑡,𝑥)+1_𝜔𝑢()(]()𝑧()𝑡,𝑥,in0,𝜏×Ω,𝑧=0,on0,𝜏×𝜕Ω,0,𝑥=𝑧₀(𝑥),𝑥∈Ω, (9) has been studied by several authors, particularly by [8–10]; and in a general fashion in [11].

Now, we shall describe the strategy of this work.

First, we prove that the linear system 𝑧_𝑡(𝑡,𝑥)=Δ𝑧(𝑡,𝑥)+1_𝜔𝑢()()(]()𝑧()𝑡,𝑥+𝑎𝑧𝑡,𝑥,in0,𝜏×Ω,𝑧=0,on0,𝜏×𝜕Ω,0,𝑥=𝑧₀(𝑥),𝑥∈Ω (10) is approximately controllable.

After that, we write the system (1) as follows: 𝑧_𝑡(𝑡,𝑥)=Δ𝑧(𝑡,𝑥)+1_𝜔𝑢()(),(]()𝑧()𝑡,𝑥+𝑎𝑧+𝑔𝑡,𝑧,𝑢in0,𝜏×Ω,𝑧=0,on0,𝜏×𝜕Ω,0,𝑥=𝑧₀(𝑥),𝑥∈Ω, (11) where 𝑔(𝑡,𝑧,𝑢)=𝑓(𝑡,𝑧,𝑢)−𝑎𝑧 is a smooth enough functions satisfying the following conditions: there are 𝑎,𝑏,𝑐∈ℝ, 𝑅>0 and 1/2≤𝛽<1 such that 𝑔(𝑡,𝑧,𝑢)≤𝑐|𝑢|^𝛽+𝑏,∀𝑢,𝑧∈ℝ,|𝑢|,|𝑧|≥𝑅. (12) Finally, the approximate controllability of the system (11) follows from the controllability of (10), the compactness of the semigroup generated by the Laplacian operator −Δ, the condition (12) satisfied by the nonlinear term 𝑔, and the following results.

Proposition 3. Let (𝑋,Σ,𝜇) be a measure space with 𝜇(𝑋)<∞ and 1≤𝑞<𝑟<∞. Then, 𝐿_𝑟(𝜇)⊂𝐿_𝑞(𝜇) and 𝑓_𝑞≤𝜇(𝑋)^{(𝑟−𝑞)/𝑟𝑞}𝑓_𝑟,𝑓∈𝐿_𝑟𝜇. (13)

Proof. The proof of this proposition follows from Theorem I.V.6 from [12] by putting 𝑝=𝑟/𝑞>1 and considering the relation _𝑋𝑓^𝑞𝑝𝑑𝜇=_𝑋𝑓^𝑟𝑑𝜇,∀𝑓∈𝐿_𝑟𝜇. (14)

Theorem 4 (Rothe’s theorem [13, page 129]). Let 𝐸(𝜏) be a Hausdorff topological vector space. Let 𝐵⊂𝐸 be a closed convex subset such that the zero of 𝐸 is contained in the interior of 𝐵.
Let Φ∶𝐵→𝐸 be a continuous mapping with Φ(𝐵) relatively compact in 𝐸 and Φ(𝜕𝐵)⊂𝐵.
Then, there is a point 𝑥^∗∈𝐵 such that Φ(𝑥^∗)=𝑥^∗.

The technique we use here to prove the approximate controllability of the linear part of (10) is based on the classical unique continuation for elliptic equations (see [14]) and the following lemma.

Lemma 5 (see Lemma 3.14 from [15, page 62]). Let {𝛼_𝑗}_𝑗≥1 and {𝛽_𝑖,𝑗∶𝑖=1,2,…,𝑚}_𝑗≥1 be two sequences of real numbers such that 𝛼₁>𝛼₂>𝛼₃⋯. Then, ^∞_𝑗=1𝑒^𝛼𝑗𝑡𝛽_𝑖,𝑗=0,∀𝑡∈[0,𝜏],𝑖=1,2,…,𝑚. (15) If and only if 𝛽_𝑖,𝑗=0,𝑖=1,2,…,𝑚;𝑗=1,2,…,∞. (16)

2. Abstract Formulation of the Problem

In this section, we choose a Hilbert space where system (1) can be written as an abstract differential equation; to this end, we consider the following results appearing in [15, page 46], [16, page 335], and [17, page 147].

Let us consider the Hilbert space 𝑍=𝐿²(Ω) and 0<𝜆₁<𝜆₂<⋯<𝜆_𝑗→∞ the eigenvalues of −Δ with the Dirichlet homogeneous conditions, each one with finite multiplicity 𝛾_𝑗 equal to the dimension of the corresponding eigenspace. Then, we have the following well-known properties.(i) There exists a complete orthonormal set {𝜙_𝑗,𝑘} of eigenvectors of 𝐴=−Δ.(ii) For all 𝑧∈𝐷(𝐴), we have 𝐴𝑧=^∞_𝑗=1𝜆_𝑗^𝛾𝑗_𝑘=1𝜉,𝜙_𝑗,𝑘𝜙_𝑗,𝑘=^∞_𝑗=1𝜆_𝑗𝐸_𝑗𝑧, (17) where ⟨⋅,⋅⟩ is the inner product in 𝑍 and 𝐸_𝑗𝑧=^𝛾𝑗_𝑘=1𝑧,𝜙_𝑗,𝑘𝜙_𝑗,𝑘. (18) So, {𝐸_𝑗} is a family of complete orthogonal projections in 𝑍 and 𝑧=∑^∞_𝑗=1𝐸_𝑗𝑧, 𝑧∈𝑍.(iii)−𝐴 generates an analytic semigroup {𝑇(𝑡)} given by 𝑇(𝑡)𝑧=^∞_𝑗=1𝑒^{−𝜆𝑗𝑡}𝐸_𝑗𝑧,‖𝑇(𝑡)‖≤𝑒^−𝜆1𝑡,𝑡≥0. (19)

Consequently, systems (1), (10), and (11) can be written, respectively, as an abstract differential equations in 𝑍: 𝑧^′=−𝐴𝑧+𝐵_𝜔𝑢+𝑓^𝑒()𝑧𝑡,𝑧,𝑢,𝑧∈𝑍,𝑡≥0,_′=−𝐴𝑧+𝐵_𝜔𝑧𝑢+𝑎𝑧,𝑧∈𝑍,𝑡≥0,_′=−𝐴𝑧+𝐵_𝜔𝑢+𝑎𝑧+𝑔^𝑒(𝑡,𝑧,𝑢),𝑧∈𝑍,𝑡≥0, (20)(21)(22) where 𝑢∈𝐿²([0,𝜏];𝑈), 𝑈=𝑍, 𝐵_𝜔∶𝑈→𝑍, 𝐵_𝜔𝑢=1_𝜔𝑢 is a bounded linear operator, 𝑓^𝑒∶[0,𝜏]×𝑍×𝑈→𝑍 is defined by 𝑓^𝑒(𝑡,𝑧,𝑢)(𝑥)=𝑓(𝑡,𝑧(𝑥),𝑢(𝑥)), for all 𝑥∈Ω, and 𝑔^𝑒(𝑡,𝑧,𝑢)=𝑓^𝑒(𝑡,𝑧,𝑢)−𝑎𝑧. On the other hand, from condition (12), we get the following estimate.

Proposition 6. Under condition (12), the function 𝑔^𝑒∶[0,𝜏]×𝑍×𝑈→𝑍 defined by 𝑔^𝑒(𝑡,𝑧,𝑢)(𝑥)=𝑔(𝑡,𝑧(𝑥),𝑢(𝑥)), for all 𝑥∈Ω, satisfies for all 𝑢,𝑧∈𝑍=𝐿²(Ω): 𝑔^𝑒(𝑡,𝑧,𝑢)_𝑍≤2𝑐𝜇(Ω)^{𝛽/(2𝛽−1/2)}‖𝑢‖^𝛽_𝑍+2𝑏𝜇(Ω),‖𝑧‖,‖𝑢‖≥𝑅𝜇(Ω). (23)

Proof𝑔^𝑒(𝑡,𝑧,𝑢)_𝑍=_Ω𝑔(𝑡,𝑧(𝑥),𝑢(𝑥))²𝑑𝑥^1/2≤_Ω𝑐|𝑢(𝑥)|^𝛽+𝑏²𝑑𝑥^1/2≤_Ω4𝑐²|𝑢(𝑥)|^2𝛽+4𝑏²𝑑𝑥^1/2≤2𝑐_Ω|𝑢(𝑥)|^{2𝛽1/2𝛽𝛽}+2𝑏𝜇(Ω)=2𝑐‖𝑢‖^𝛽_{𝐿2𝛽(Ω)}+2𝑏𝜇(Ω). (24)
Now, since 1/2≤𝛽<1⇔1≤2𝛽<2, applying Proposition 3, we obtain that 𝑔^𝑒(𝑡,𝑧,𝑢)_𝑍≤2𝑐𝜇(Ω)^{(1−𝛽)/𝛽}‖𝑢‖^𝛽_𝑍+2𝑏𝜇(Ω). (25)

3. Interior Controllability of the Linear Equation

In this section, we shall prove the interior approximate controllability of the linear system (21). To this end, we note that, for all 𝑧₀∈𝑍 and 𝑢∈𝐿²(0,𝜏;𝑈), the initial value problem 𝑧^′=−𝐴𝑧+𝐵_𝜔𝑢(𝑡)(𝑡)𝑧(0)+𝑎𝑧,𝑧∈𝑍,=𝑧₀, (26)

where the control function 𝑢 belongs to 𝐿²(0,𝜏;𝑈), admits only one mild solution given by 𝑧(𝑡)=𝑒^𝑎𝑡𝑇(𝑡)𝑧₀+^𝑡₀𝑒^{𝑎(𝑡−𝑠)}𝑇(𝑡−𝑠)𝐵_𝜔𝑢(𝑠)𝑑𝑠,𝑡∈[0,𝜏]. (27)

Definition 7. For system (21), we define the following concept: the controllability map (for 𝜏>0) 𝐺_𝐚∶𝐿²(0,𝜏;𝑈)→𝑍 is given by 𝐺_𝐚𝑢=^𝜏₀𝑒^{𝑎(𝜏−𝑠)}𝑇(𝜏−𝑠)𝐵_𝜔𝑢(𝑠)𝑑𝑠, (28) whose adjoint operator 𝐺^∗_𝐚∶𝑍→𝐿²(0,𝜏;𝑍) is given by 𝐺^∗_𝐚𝑧(𝑠)=𝐵^∗_𝜔𝑒^{𝑎(𝜏−𝑠)}𝑇^∗(𝜏−𝑠)𝑧,∀𝑠∈[0,𝜏],∀𝑧∈𝑍. (29)

The following lemma holds in general for a linear bounded operator 𝐺∶𝑊→𝑍 between Hilbert spaces 𝑊 and 𝑍.

Lemma 8 (see [11, 15, 18]). Equation (21) is approximately controllable on [0,𝜏] if and only if one of the following statements holds.(a)Rang(𝐺_𝐚)=𝑍.(b)Ker(𝐺^∗_𝐚)={0}.(c)⟨𝐺_𝐚𝐺^∗_𝐚𝑧,𝑧⟩>0, 𝑧≠0  in 𝑍.(d)lim_𝛼→0⁺𝛼(𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚)⁻¹𝑧=0.(e)𝐵^∗_𝜔𝑒^𝑎𝑡𝑇^∗(𝑡)𝑧=0, for  all 𝑡∈[0,𝜏],⇒𝑧=0. (f)For all 𝑧∈𝑍 , we have 𝐺𝑢_𝛼=𝑧−𝛼(𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚)⁻¹𝑧, where𝑢_𝛼=𝐺^∗_𝐚𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝑧,𝛼∈(0,1]. (30)
So, lim_𝛼→0𝐺_𝐚𝑢_𝛼=𝑧 and the error 𝐸_𝛼𝑧 of this approximation is given by 𝐸_𝛼𝑧=𝛼𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝑧,𝛼∈(0,1]. (31)

Remark 9. Lemma 8 implies that the family of linear operators Γ_𝛼∶𝑍→𝐿²(0,𝜏;𝑈), defined for 0<𝛼≤1 by Γ_𝛼𝑧=𝐵^∗_𝜔𝑒^𝑎(⋅)𝑇^∗(⋅)𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝑧=𝐺^∗_𝐚𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝑧, (32)
is an approximate inverse for the right of the operator 𝐺_𝐚 in the sense that lim_𝛼→0𝐺_𝐚Γ_𝛼=𝐼. (33)

Proposition 10 (see [7]). If Rang(𝐺_𝐚)=𝑍, then sup_𝛼>0𝛼𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹≤1. (34)

Remark 11. The proof of the following theorem follows from foregoing characterization of dense range linear operators and the classical unique continuation for elliptic equations (see [14]), and it is similar to the one given in Theorem 4.1 in [6].

Theorem 12. System (21) is approximately controllable on [0,𝜏]. Moreover, a sequence of controls steering the system (21) from initial state 𝑧₀ to an 𝜖 neighborhood of the final state 𝑧₁ at time 𝜏>0 is given by 𝑢_𝛼(𝑡)=𝐵^∗_𝜔𝑒^{𝑎(𝜏−𝑡)}𝑇^∗(𝜏−𝑡)𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹×𝑧₁−𝑒^𝑎𝜏𝑇(𝜏)𝑧₀, (35) and the error of this approximation 𝐸_𝛼 is given by 𝐸_𝛼=𝛼𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝑧₁−𝑒^𝑎𝜏𝑇(𝜏)𝑧₀. (36)

4. Controllability of the Semilinear System

In this section, we shall prove the main result of this paper, the interior approximate controllability of the semilinear 𝑛𝐷 heat equation given by (1), which is equivalent to prove the approximate controllability of the system (22). To this end, for all 𝑧₀∈𝑍 and 𝑢∈𝐿²(0,𝜏;𝑈), the initial value problem 𝑧^′=−𝐴𝑧+𝐵_𝜔𝑢+𝑎𝑧+𝑔^𝑒()𝑧(0)𝑡,𝑧,𝑢,𝑧∈𝑍,𝑡≥0=𝑧₀ (37) admits only one mild solution given by 𝑧_𝑢(𝑡)=𝑒^𝑎𝑡𝑇(𝑡)𝑧₀+^𝑡₀𝑒^{𝑎(𝑡−𝑠)}𝑇(𝑡−𝑠)𝐵_𝜔𝑢(𝑠)+𝑑𝑠_𝑡0𝑒^{𝑎(𝑡−𝑠)}𝑇(𝑡−𝑠)𝑔^𝑒𝑠,𝑧_𝑢(𝑠)(𝑠)[].,𝑢𝑑𝑠,𝑡∈0,𝜏 (38)

Definition 13. For the system (22), we define the following concept: the nonlinear controllability map (for 𝜏>0) 𝐺_𝑔∶𝐿²(0,𝜏;𝑈)→𝑍 is given by 𝐺_𝑔𝑢=^𝜏₀𝑒^{𝑎(𝜏−𝑠)}𝑇(𝜏−𝑠)𝐵_𝜔𝑢(𝑠)+𝑑𝑠_𝜏0𝑒^{𝑎(𝜏−𝑠)}𝑇(𝜏−𝑠)𝑔^𝑒𝑠,𝑧_𝑢(𝑠),𝑢(𝑠)𝑑𝑠=𝐺_𝐚(𝑢)+𝐻(𝑢), (39) where 𝐻∶𝐿²(0,𝜏;𝑈)→𝑍 is the nonlinear operator given by 𝐻(𝑢)=^𝜏₀𝑒^{𝑎(𝜏−𝑠)}𝑇(𝜏−𝑠)𝑔^𝑒𝑠,𝑧_𝑢(𝑠),𝑢(𝑠)𝑑𝑠,𝑢∈𝐿²(0,𝜏;𝑈). (40)

The following lemma is trivial.

Lemma 14. Equation (22) is approximately controllable on [0,𝜏] if and only if Rang(𝐺_𝑔)=𝑍.

Definition 15. The following equation shall be called the controllability equation associated with the nonlinear equation (22): 𝑢_𝛼=Γ_𝛼𝑧−𝐻𝑢_𝛼=𝐺^∗_𝐚𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝑧−𝐻𝑢_𝛼,(0<𝛼≤1). (41)

Now, we are ready to present and prove the main result of this paper, which is the interior approximate controllability of the semilinear 𝑛𝐷 heat equation (1)

Theorem 16. The system (22) is approximately controllable on [0,𝜏]. Moreover, a sequence of controls steering the system (22) from initial state 𝑧₀ to an 𝜖-neighborhood of the final state 𝑧₁ at time 𝜏>0 is given by 𝑢_𝛼(𝑡)=𝐵^∗_𝜔𝑒^{𝑎(𝜏−𝑡)}𝑇^∗(𝜏−𝑡)𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹×𝑧₁−𝑒^𝑎𝜏𝑇(𝜏)𝑧₀−𝐻𝑢_𝛼, (42) and the error of this approximation 𝐸_𝛼𝑧 is given by 𝐸_𝛼𝑧=𝛼𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝑧₁−𝑒^𝑎𝜏𝑇(𝜏)𝑧₀−𝐻𝑢_𝛼. (43)

Proof. For each 𝑧∈𝑍 fixed, we shall consider the following family of nonlinear operators 𝐾_𝛼∶𝐿²(0,𝜏;𝑈)→𝐿²(0,𝜏;𝑈) given by 𝐾_𝛼(𝑢)=Γ_𝛼(𝑧−𝐻(𝑢))=𝐺^∗_𝐚𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹((𝑢,().𝑧−𝐻))0<𝛼≤1 (44) First, we shall prove that for all 𝛼∈(0,1] the operator 𝐾_𝛼 has a fixed point 𝑢_𝛼. In fact, since 𝑔^𝑒 is smooth and satisfies (12) and the semigroup {𝑇(𝑡)}_𝑡≥0 given by (19) is compact (see [19, 20]), then using the result from [1], we obtain that the operator 𝐻 is compact, which implies that the operator 𝐾_𝛼 is compact.
Moreover, lim_{‖𝑢‖𝐿2→∞}𝐾_𝛼(𝑢)_𝐿2‖𝑢‖_𝐿2=0. (45) In fact, putting 𝐶=2𝑐𝜇(Ω)^{𝛽/(2𝛽−1/2)} and 𝐵=2𝑏𝜇(Ω), we get from Proposition 6 that 𝑔^𝑒(𝑡,𝑧,𝑢)_𝑍≤𝐶‖𝑢‖^𝛽_𝑍+𝐵,‖𝑧‖,‖𝑢‖≥𝑅𝜇(Ω), (46) and from the definition of the operator 𝐻(𝑢), Proposition 3, and (19) we have, for 𝑢∈𝐿²(0,𝜏;𝑈), the following estimate: ‖𝐻(𝑢)‖≤^𝜏₀𝑒^{(𝑎+𝜆1)(𝜏−𝑠)}𝑔^𝑒𝑠,𝑧_𝑢(𝑠)(𝑠)≤,𝑢𝑑𝑠_𝜏0𝑒^{2(𝑎+𝜆1)(𝜏−𝑠)}𝑑𝑠^1/2×^𝜏₀𝑔^𝑒𝑠,𝑧_𝑢(𝑠),𝑢(𝑠)²𝑑𝑠^1/2=𝑁^𝜏₀𝑔^𝑒𝑠,𝑧_𝑢(𝑠),𝑢(𝑠)²𝑑𝑠^1/2≤𝑁^𝜏₀𝐶‖𝑢(𝑠)‖^𝛽+𝐵²𝑑𝑠^1/2≤𝑁^𝜏₀4𝐶²‖𝑢(𝑠)‖^2𝛽+4𝐵²𝑑𝑠^1/2≤2𝑁𝐶^𝜏₀‖𝑢(𝑠)‖^2𝛽1/2+2𝑁𝐵√𝜏≤2𝑁𝐶^𝜏₀‖𝑢(𝑠)‖^{2𝛽1/2𝛽𝛽}+2𝑁𝐵√𝜏=2𝐶𝑁‖𝑢‖^𝛽_𝐿2𝛽+2𝑁𝐵√𝜏. (47) Now, since 1/2≤𝛽<1⇔1≤2𝛽<2, applying Proposition 3, we obtain that ‖𝐻(𝑢)‖≤2𝐶𝑁𝜏^{(1−𝛽)/𝛽}‖𝑢‖^𝛽_𝐿2+2𝑁𝐵√𝜏. (48) Therefore, lim_{‖𝑢‖𝐿2→∞}‖𝐻(𝑢)‖_𝐿2‖𝑢‖_𝐿2=0. (49) Consequently, lim_{‖𝑢‖𝐿2→∞}𝐾_𝛼(𝑢)‖𝑢‖_𝐿2=0. (50) Then, from condition (45), we obtain that, for a fixed 0<𝜌<1, there exists 𝑅_𝛼>0 big enough such that 𝐾_𝛼(𝑢)_𝐿2≤𝜌‖𝑢‖_𝐿2,‖𝑢‖_𝐿2=𝑅_𝛼. (51) Hence, if we denote by 𝐵(0,𝑅_𝛼) the ball of center zero and radio 𝑅_𝛼>0, we get that 𝐾_𝛼(𝜕𝐵(0,𝑅_𝛼))⊂𝐵(0,𝑅_𝛼). Since 𝐾_𝛼 is compact and maps the sphere 𝜕𝐵(0,𝑅_𝛼) into the interior of the ball 𝐵(0,𝑅_𝛼), we can apply Rothe's fixed point Theorem 4 to ensure the existence of a fixed point 𝑢_𝛼∈𝐿₂(0,𝜏;𝑈) such that 𝑢_𝛼=𝐾_𝛼𝑢_𝛼=Γ_𝛼𝑧−𝐻𝑢_𝛼=𝐺^∗𝛼𝐼+𝐺𝐺^∗−1𝑧−𝐻𝑢_𝛼,(0<𝛼≤1). (52)Claim. The family of fixed pint {𝑢_𝛼}_0<𝛼≤1 is bounded.
In fact, for the purpose of contradiction, let us assume the contrary. Then, there exists a subsequence {𝑢_𝛼𝑛}_𝑛≥1⊂{𝑢_𝛼}_0<𝛼≤1 such that lim_𝑛→∞𝑢_{𝛼𝑛𝐿2}=∞. (53) On the other hand, we have that 𝑢_𝛼𝑛=𝐾_𝛼𝑢_𝛼𝑛. So, 𝑢_{𝛼𝑛𝐿2}=𝐾_𝛼𝑢_{𝛼𝑛𝐿2}⟺𝐾_𝛼𝑢_{𝛼𝑛𝐿2}𝑢_{𝛼𝑛𝐿2}=1. (54) Hence, lim_𝑛→∞𝐾_𝛼𝑢_{𝛼𝑛𝐿2}𝑢_{𝛼𝑛𝐿2}=1, (55) which is evidently a contradiction. Then, the claim is true and there exists 𝛾>0 such that 𝑢_𝛼𝐿2≤𝛾,(0<𝛼≤1). (56) Therefore, without loss of generality, we can assume that the sequence 𝐻(𝑢_𝛼) converges to 𝑦∈𝑍. So, if 𝑢_𝛼=Γ_𝛼(𝑧−𝐻(𝑢))=𝐺^∗_𝐚𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝑧−𝐻𝑢_𝛼 (57) then 𝐺_𝐚𝑢_𝛼=𝐺_𝐚Γ_𝛼(𝑧−𝐻(𝑢))=𝐺_𝐚𝐺^∗_𝐚𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝑧−𝐻𝑢_𝛼=𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚−𝛼𝐼𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝑧−𝐻𝑢_𝛼=𝑧−𝐻𝑢_𝛼−𝛼𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝑧−𝐻𝑢_𝛼. (58) Hence, 𝐺_𝐚𝑢_𝛼+𝐻𝑢_𝛼=𝑧−𝛼𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝑧−𝐻𝑢_𝛼. (59) To conclude the proof of this theorem, it is enough to prove that lim_𝛼→0−𝛼𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝑧−𝐻𝑢_𝛼=0. (60) From Lemma 8 (d), we get that lim_𝛼→0−𝛼𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝑧−𝐻𝑢_𝛼=−lim_𝛼→0−𝛼𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝐻𝑢_𝛼=−lim_𝛼→0−𝛼𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝑦−lim_𝛼→0−𝛼𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝐻𝑢_𝛼−𝑦=lim_𝛼→0−𝛼𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝐻𝑢_𝛼−𝑦. (61) On the other hand, from Proposition 10, we get that 𝛼𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝐻𝑢_𝛼−𝑦≤𝐻𝑢_𝛼−𝑦. (62) Therefore, since 𝐻(𝑢_𝛼) converges to 𝑦, we get that lim_𝛼→0−𝛼𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝐻𝑢_𝛼−𝑦=0. (63) Consequently, lim_𝛼→0−𝛼𝛼𝐼+𝐺_𝐚𝐺^∗_𝐚⁻¹𝑧−𝐻𝑢_𝛼=0. (64) So, by putting 𝑧=𝑧₁−𝑒^𝑎𝜏𝑇(𝜏)𝑧₀ and using (38), we obtain the nice result: 𝑧₁=lim_𝛼→0+𝑒^𝑎𝜏𝑇(𝜏)𝑧₀+^𝜏₀𝑒^{𝑎(𝜏−𝑠)}𝑇(𝜏−𝑠)𝐵_𝜔𝑢_𝛼(𝑠)+𝑑𝑠_𝜏0𝑒^{𝑎(𝜏−𝑠)}𝑇(𝜏−𝑠)𝑔^𝑒𝑠,𝑧_𝑢𝛼(𝑠),𝑢_𝛼(𝑠)𝑑𝑠. (65)

5. Final Remark

Our technique is simple and can be applied to those systems involving compact semigroups like some control system governed by diffusion processes. For example, the Benjamin-Bona-Mahony equation, the strongly damped wave equations, beam equations, and so forth.

Example 17. The original Benjamin-Bona-Mahony equation is a nonlinear one in [21] the authors proved the approximate controllability of the linear part of this equation, which is the fundamental base for the study of the controllability of the nonlinear BBM equation. So, our next work is concerned on the controllability of nonlinear BBM equation: 𝑧_𝑡−𝑎Δ𝑧_𝑡−𝑏Δ𝑧=1_𝜔𝑢()((𝑡,()𝑧()𝑡,𝑥+𝑓𝑡,𝑧,𝑢))𝑡∈0,𝜏,𝑥∈Ω,𝑡,𝑥=0,𝑡≥0,𝑥∈𝜕Ω, (66) where 𝑎≥0 and 𝑏>0 are constants, Ω is a domain in ℝ^𝑁, 𝜔 is an open nonempty subset of Ω, 1_𝜔 denotes the characteristic function of the set 𝜔, the distributed control 𝑢∈𝐿²(0,𝜏;𝐿²(Ω)), and 𝑓(𝑡,𝑧,𝑢(𝑡)) is a nonlinear perturbation.

Example 18. We believe that this technique can be applied to prove the interior controllability of the strongly damped wave equation with Dirichlet boundary conditions: 𝑤_𝑡𝑡+𝜂(−Δ)^1/2𝑤_𝑡+𝛾(−Δ)𝑤=1_𝜔𝑢()((𝑡,()()𝑤()𝑡,𝑥+𝑓𝑡,𝑧,𝑢))in0,𝜏×Ω,𝑤=0,in0,𝜏×𝜕Ω,0,𝑥=𝑤₀(𝑥),𝑤_𝑡(0,𝑥)=𝑤₁(𝑥),inΩ, (67) in the space 𝑍_1/2=𝐷((−Δ)^1/2)×𝐿²(Ω), where Ω is a bounded domain in ℝ^𝑛, 𝜔 is an open nonempty subset of Ω, 1_𝜔 denotes the characteristic function of the set 𝜔, the distributed control 𝑢∈𝐿²(0,𝜏;𝐿²(Ω)), and 𝜂, 𝛾 are positive numbers.

Example 19. Another example where this technique may be applied is partial differential equations modeling the structural damped vibrations of a string or a beam: 𝑦_𝑡𝑡−2𝛽Δ𝑦_𝑡+Δ²𝑦=1_𝜔𝑢()((𝑡,()()𝑦()𝑡,𝑥+𝑓𝑡,𝑧,𝑢))on0,𝜏×Ω,𝑦=Δ𝑦=0,on0,𝜏×𝜕Ω,0,𝑥=𝑦₀(𝑥),𝑦_𝑡(0,𝑥)=𝑦₁(𝑥),inΩ, (68) where Ω is a bounded domain in ℝ^𝑛, 𝜔 is an open nonempty subset of Ω, 1_𝜔 denotes the characteristic function of the set 𝜔, the distributed control 𝑢∈𝐿²(0,𝜏;𝐿²(Ω)), and 𝑦₀∈𝐻²(Ω)∩𝐻¹₀,𝑦₁∈𝐿²(Ω).

Moreover, our result can be formulated in a more general setting. Indeed, we can consider the following semilinear evolution equation in a general Hilbert space 𝑍: 𝑧́=−𝐴𝑧+𝐵𝑢(𝑡)+𝑓^𝑒()(],𝑧(0)𝑡,𝑧,𝑢,𝑧∈𝑍,𝑡∈0,𝜏=𝑧₀, (69) where 𝐴∶𝐷(𝐴)⊂𝑍→𝑍 is an unbounded linear operator in 𝑍 with the following spectral decomposition: 𝐴𝑧=^∞_𝑗=1𝜆_𝑗^𝛾𝑗_𝑘=1𝑧,𝜙_𝑗,𝑘𝜙_𝑗,𝑘, (70) with the eigenvalues 0<𝜆₁<𝜆₂<⋯<⋯𝜆_𝑛→∞ of 𝐴 having finite multiplicity 𝛾_𝑗 equal to the dimension of the corresponding eigenspaces, and {𝜙_𝑗,𝑘} is a complete orthonormal set of eigenfunctions of 𝐴. The operator −𝐴 generates a strongly continuous compact semigroup {𝑇_𝐴(𝑡)}_𝑡≥0 given by 𝑇_𝐴(𝑡)𝑧=^∞_𝑗=1𝑒^{−𝜆𝑗𝑡𝛾𝑗}_𝑘=1𝑧,𝜙_𝑗,𝑘𝜙_𝑗,𝑘. (71) The control 𝑢∈𝐿²(0,𝜏;𝑈), with 𝑈=𝑍, 𝐵∶𝑍→𝑍 is a linear and bounded operator (linear and continuous), and the function 𝑓^𝑒∶[0,𝜏]×𝑍×𝑈→𝑍 is smooth enough, and 𝑓^𝑒(𝑡,𝑧,𝑢)−𝑎𝑧_𝑍≤𝑐‖𝑢‖^𝛽_𝑍‖𝑢‖,‖𝑧‖+𝑏,∀𝑢,𝑧∈𝑍,≥𝑅. (72) In this case, the characteristic function set is a particular operator 𝐵, and the following theorem is a generalization of Theorem 16.

Theorem 20. If vectors 𝐵^∗𝜙_𝑗,𝑘 are linearly independent in 𝑍, then the system (69) is approximately controllable on [0,𝜏].

Acknowledgments

This work has been supported by CDCHT-ULA-C-1796-12-05-AA and BCV.