A Characterization of Semilinear Dense Range Operators and Applications

Abstract

We characterize a broad class of semilinear dense range operators 𝐺_𝐻∶𝑊→𝑍 given by the following formula, 𝐺_𝐻𝑤=𝐺𝑤+𝐻(𝑤),𝑤∈𝑊, where 𝑍, 𝑊 are Hilbert spaces, 𝐺∈𝐿(𝑊,𝑍), and 𝐻∶𝑊→𝑍 is a suitable nonlinear operator. First, we give a necessary and sufficient condition for the linear operator 𝐺 to have dense range. Second, under some condition on the nonlinear term 𝐻, we prove the following statement: If Rang(𝐺)=𝑍, then Rang(𝐺_𝐻)=𝑍 and for all 𝑧∈𝑍 there exists a sequence {𝑤_𝛼∈𝑍∶0<𝛼≤1} given by 𝑤_𝛼=𝐺^∗(𝛼𝐼+𝐺𝐺^∗)⁻¹(𝑧−𝐻(𝑤_𝛼)), such that lim_𝛼→0⁺{𝐺𝑢_𝛼+𝐻(𝑢_𝛼)}=𝑧. Finally, we apply this result to prove the approximate controllability of the following semilinear evolution equation: 𝑧^′=𝐴𝑧+𝐵𝑢(𝑡)+𝐹(𝑡,𝑧,𝑢(𝑡)),𝑧∈𝑍,𝑢∈𝑈,𝑡>0, where 𝑍, 𝑈 are Hilbert spaces, 𝐴∶𝐷(𝐴)⊂𝑍→𝑍 is the infinitesimal generator of strongly continuous compact semigroup {𝑇(𝑡)}_𝑡≥0 in 𝑍,𝐵∈𝐿(𝑈,𝑍), the control function 𝑢 belongs to 𝐿²(0,𝜏;𝑈), and 𝐹∶[0,𝜏]×𝑍×𝑈→𝑍 is a suitable function. As a particular case we consider the controlled semilinear heat equation.

1. Introduction

It is well known from functional analysis that continuous linear surjective operators form an open set in the space of such operators; that is to say, if a surjective linear continuous operator is added to a linear continuous operator with a small enough norm, the resulting operator is still surjective; moreover, if a linear continuous surjective operator is perturbed by a nonlinear Lipschitz operator with a Lipschitz constant small enough, then the resulting operator is still surjective. This result is not true anymore for continuous linear operators that only have dense range; for instance, if a dense range continuous linear operator is perturbed by another linear operator with norm infinitely small, the resulting operator may not have dense range; in other words, the property of having dense range is not robust enough to be surjective. However, in this paper we proved the following statement: if a continuous linear operator with dense range is perturbed by a compact nonlinear operator with bounded range, then the resulting operator also has dense range. This result can have an unlimited number of applications, not only in the study of control theory for semilinear evolution equations, but it can also be used to find the approximate solution of functional equations in Hilbert spaces giving a formula for the error of this approximation, which is very important from the standpoint of numerical analysis. In addition, it is well known that approximate controllability is much more natural and common than exact controlabilidad, since most of the mechanical processes are diffusive, which implies that these systems can never be exactly controllable; and many years have passed to present a general result on semilinear operators with dense range to facilitate the study of the approximate controlabilidad for a large class of semilinear evolution equations whose dynamics are given by compact semigroups. However, in this work, by way of illustration, we only show how this result can be applied to study the approximate controllability of control systems governed by the semilinear heat equation.

Specifically, in this paper we characterize a broad class of semilinear dense range operators.

𝐺_𝐻∶𝑊→𝑍 given by the following formula: 𝐺_𝐻𝑤=𝐺𝑤+𝐻(𝑤),𝑤∈𝑊,(1) where 𝑍, 𝑊 are Hilbert spaces, 𝐺∶𝑊→𝑍 is a bounded linear operator (continuous and linear), and 𝐻∶𝑊→𝑍 is a suitable nonlinear operator. First, we give a necessary and sufficient condition for the linear operator 𝐺 to have dense range (Rang(𝐺)=𝑍). Second, we prove the following statement: If Rang(𝐺)=𝑍 and 𝐻 is smooth enough and Rang(𝐻) is compact, then Rang(𝐺_𝐻)=𝑍 and for all 𝑧∈𝑍 there exists a sequence {𝑤_𝛼∈𝑊∶0<𝛼≤1} given by 𝑤_𝛼=𝐺^∗𝛼𝐼+𝐺𝐺^∗−1𝑧−𝐻𝑤_𝛼,(2) such that lim_𝛼→0+𝐺𝑤_𝛼+𝐻𝑤_𝛼=𝑧,(3) and the error of this approximation 𝐸_𝛼𝑧 is given by 𝐸_𝛼𝑧=𝛼𝛼𝐼+𝐺𝐺^∗−1𝑧−𝐻𝑤_𝛼.(4) This result can be viewed as a generalization of the work done in [1–8].

We apply these results to prove the approximate controllability of the following semilinear evolution equation: 𝑧^′=𝐴𝑧+𝐵𝑢(𝑡)+𝐹(𝑡,𝑧,𝑢(𝑡)),𝑧∈𝑍,𝑢∈𝑈,𝑡>0,(5) where 𝑍, 𝑈 are Hilbert spaces, 𝐴∶𝐷(𝐴)⊂𝑍→𝑍 is the infinitesimal generator of strongly continuous compact semigroup {𝑇(𝑡)}_𝑡≥0 in 𝑍,𝐵∈𝐿(𝑈,𝑍), the control function 𝑢 belongs to 𝐿²(0,𝜏;𝑈), and 𝐹∶[0,𝜏]×𝑍×𝑈→𝑍 is a smooth enough function.

Remark 1 (see [2–4]). The function 𝐹 is smooth enough if(a)the mild solutions 𝑧(𝑢)=𝑧_𝑢 of (5) are unique, (b)the mild solutions 𝑧(𝑢)=𝑧_𝑢 depends continuously on 𝑢, (c)and if 𝐹 is a Lipschitz function, then 𝑧(𝑢)=𝑧_𝑢, as a function of 𝑢, is also a Lipchitz function.

As an application we consider the following example of controlled semilinear heat equation.

Example 2 (the interior controllability of the 𝑛𝐷 heat equation). The semilinear heat equation was studied in [8] where the authors prove the interior controllability of the following control system: 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,(6) 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]×𝔑×𝔑→𝔑 is smooth enough and there are constants 𝑎,𝑐∈𝔑, with 𝑐≠−1, such that sup_{(𝑡,𝑧,𝑢)∈𝑞𝜏}𝑓(𝑡,𝑧,𝑢)−𝑎𝑧−𝑐𝑢<∞,(7) where 𝑞_𝜏=[0,𝜏]×𝔑×𝔑.
We note that the interior approximate controllability of the linear heat equation, 𝑧_𝑡(𝑡,𝑥)=Δ𝑧(𝑡,𝑥)+1_𝜔𝑢()(]()𝑧()𝑡,𝑥in0,𝜏×Ω,𝑧=0,on0,𝜏×𝜕Ω,0,𝑥=𝑧₀(𝑥),𝑥∈Ω,(8) has been study by several authors, particularly by [9], and in a general fashion in [10].
The approximate controllability of the heat equation under nonlinear perturbation 𝑓(𝑧) independents of 𝑡 and 𝑢 variables, 𝑧_𝑡=Δ𝑧+1_𝜔𝑢()(𝑧)(]()𝑧()𝑡,𝑥+𝑓in0,𝜏×Ω,𝑧=0,in0,𝜏×𝜕Ω,0,𝑥=𝑧₀(𝑥),𝑥∈Ω,(9) has been studied by several authors, particularly in [11–13], depending on conditions imposed to the nonlinear term 𝑓(𝑧). For instance, in [12, 13] the approximate controllability of the system (9) is proved if 𝑓(𝑧) is sublinear at infinity; that is, 𝑓(𝑧)≤𝑑|𝑧|+𝑒.(10) Also, in the above reference, they mentioned that when 𝑓 is superlinear at the infinity, the approximate controllability of the system (9) fails.
Our result can be applied also to the semilinear Ornstein-Uhlenbeck equation, the Laguerre equation, and the Jacobi equation. Specifically, in [8], the following well-known example of reaction diffusion equations is studied.

Example 3 (see [14, 15]).  (1)The interior controllability of the semilinear Ornstein-Uhlenbeck equation 𝑧_𝑡=^𝑑_𝑖=112𝜕²𝑧𝜕𝑥²_𝑖−𝑥_𝑖𝜕𝑧𝜕𝑥_𝑖+1_𝜔𝑢()()𝑡,𝑥+𝑓𝑡,𝑧,𝑢𝑡>0,𝑥∈𝔑^𝑑,(11) where 𝑢∈𝐿²(0,𝜏;𝐿²(𝔑^𝑑,𝜇)), 𝜇(𝑥)=(1/𝜋^𝑑/2)∏^𝑑_𝑖=1𝑒^{−|𝑥𝑖|2}𝑑𝑥 is the Gaussian measure in 𝔑^𝑑, 𝜔 is an open nonempty subset of 𝔑^𝑑, and the nonlinear function 𝑓∶[0,𝜏]×𝔑×𝔑→𝔑 is smooth enough and there are constants 𝑎,𝑐∈𝔑, with 𝑐≠−1, such that sup_{(𝑡,𝑧,𝑢)∈𝑞𝜏}𝑓(𝑡,𝑧,𝑢)−𝑎𝑧−𝑐𝑢<∞,(12) where 𝑞_𝜏=[0,𝜏]×𝔑×𝔑.(2)The interior controllability of the semilinear Laguerre equation 𝑧_𝑡=^𝑑_𝑖=1𝑥_𝑖𝜕²𝑧𝜕𝑥²_𝑖+𝛼_𝑖+1−𝑥_𝑖𝜕𝑧𝜕𝑥_𝑖+1_𝜔𝑢()()𝑡,𝑥+𝑓𝑡,𝑧,𝑢,𝑡>0,𝑥∈𝔑^𝑑₊,(13) where 𝑢∈𝐿²(0,𝜏;𝐿²(𝔑^𝑑₊,𝜇_𝛼)), 𝜇_𝛼(𝑥)=∏^𝑑_𝑖=1(𝑥^𝛼𝑖_𝑖𝑒^−𝑥𝑖/Γ(𝛼_𝑖+1))𝑑𝑥 is the Gamma measure in 𝔑^𝑑₊, 𝜔 is an open nonempty subset of 𝔑^𝑑₊, and nonlinear function  𝑓∶[0,𝜏]×𝔑×𝔑→𝔑 is smooth enough and there are constant 𝑎,𝑐∈𝔑, with 𝑐≠−1, such that sup_{(𝑡,𝑧,𝑢)∈𝑞𝜏}𝑓(𝑡,𝑧,𝑢)−𝑎𝑧−𝑐𝑢<∞,(14) where 𝑞_𝜏=[0,𝜏]×𝔑×𝔑.(3)The interior controllability of the semilinear Jacobi equation 𝑧_𝑡=^𝑑_𝑖=11−𝑥²_𝑖𝜕²𝑧𝜕𝑥²_𝑖+𝛽_𝑖−𝛼_𝑖−𝛼_𝑖+𝛽_𝑖+2𝑥_𝑖𝜕𝑧𝜕𝑥_𝑖+1_𝜔𝑢(𝑡,𝑥)+𝑓(𝑡,𝑧,𝑢),(15) where 𝑡>0,𝑥∈[−1,1]^𝑑, 𝑢∈𝐿²(0,𝜏;𝐿²([−1,1]^𝑑,𝜇_𝛼,𝛽)), 𝜇_𝛼,𝛽(𝑥)=∏^𝑑_𝑖=1(1−𝑥_𝑖)^𝛼𝑖(1+𝑥_𝑖)^𝛽𝑖𝑑𝑥 is the Jacobi measure in [−1,1]^𝑑, 𝜔 is an open nonempty subset of [−1,1]^𝑑, and the nonlinear function  𝑓∶[0,𝜏]×𝔑×𝔑→𝔑 is smooth enough and there are constants 𝑎,𝑐∈𝔑, with 𝑐≠−1, such that sup_{(𝑡,𝑧,𝑢)∈𝑞𝜏}𝑓(𝑡,𝑧,𝑢)−𝑎𝑧−𝑐𝑢<∞,(16) where 𝑞_𝜏=[0,𝜏]×𝔑×𝔑.

2. Dense Range Linear Operators

In this section we shall present a characterization of dense range bounded linear operators. To this end, we denote by 𝐿(𝑊,𝑍) the space of linear and bounded operators mapping 𝑊 to 𝑍, endowed with the uniform convergence norm, and we will use the following lemma from [16] in Hilbert space.

Lemma 4. Let 𝐺^∗∈𝐿(𝑍,𝑊) be the adjoint operator of 𝐺∈𝐿(𝑊,𝑍). Then the following statements hold:(i)Rang(𝐺)=𝑍⇔∃𝛾>0 such that 𝐺^∗𝑧_𝑊≥𝛾‖𝑧‖_𝑍,𝑧∈𝑍,(17) (ii)Rang(𝐺)=𝑍⇔Ker(𝐺^∗)={0}.  The following lemma follows from Lemma 4 (ii).

Lemma 5 (see [1, 7, 8, 16–22]). The following statements are equivalent: (a)Rang(𝐺)=𝑍,(b)Ker(𝐺^∗)={0},(c)⟨𝐺𝐺^∗𝑧,𝑧⟩>0, 𝑧≠0 in 𝑍,(d)lim_𝛼→0⁺𝛼(𝛼𝐼+𝐺𝐺^∗)⁻¹𝑧=0,(e) for all 𝑧∈𝑍 we have 𝐺𝑤_𝛼=𝑧−𝛼(𝛼𝐼+𝐺𝐺^∗)⁻¹𝑧, where 𝑤_𝛼=𝐺^∗𝛼𝐼+𝐺𝐺^∗−1𝑧,𝛼∈(0,1].(18) So, lim_𝛼→0𝐺𝑤_𝛼=𝑧 and the error 𝐸_𝛼𝑧 of this approximation is given by the formula 𝐸_𝛼𝑧=𝛼𝛼𝐼+𝐺𝐺^∗−1𝑧,𝛼∈(0,1].(19)

Remark 6. Lemma 5 implies that the family of linear operators Γ_𝛼∶𝑍→𝑊, defined for 0<𝛼≤1 by Γ_𝛼𝑧=𝐺^∗𝛼𝐼+𝐺𝐺^∗−1𝑧,(20) is an approximate inverse for the right of the operator 𝐺, in the sense that lim_𝛼→0𝐺Γ_𝛼=𝐼(21) in the strong topology.

Proposition 7. If the Rang(𝐺)=𝑍, then sup_𝛼>0𝛼𝛼𝐼+𝐺𝐺^∗−1≤1.(22)

Proof. If Rang(𝐺)=𝑍, then from Lemma 4(ii) we have that 𝐺𝐺^∗𝑧,𝑧>0,𝑧≠0.(23) Therefore, 𝐺𝐺^∗+𝛼𝐼𝑧,𝑧≥𝛼‖𝑧‖²,𝑧≠0,𝛼∈(0,1].(24) Then, using the Cauchy Schwartz inequality, we obtain 𝐺𝐺^∗+𝛼𝐼𝑧≥𝛼‖𝑧‖,𝑧≠0,𝛼∈(0,1],(25) which is equivalents to 𝛼𝐺𝐺^∗+𝛼𝐼⁻¹𝑧≤‖𝑧‖,𝑧≠0,𝛼∈(0,1].(26) Consequently, sup_𝛼>0𝛼𝛼𝐼+𝐺𝐺^∗−1≤1.(27)

Proposition 8. If for some 𝛽∈(0,1] one has that 𝛽𝛽𝐼+𝐺𝐺^∗−1<1,(28) then Rang(𝐺)=𝑍.(29)

Proof. Suppose that ‖𝛽(𝛽𝐼+𝐺𝐺^∗)⁻¹‖<1. Then, from the following identity: 𝐺𝐺^∗=𝛽𝐼+𝐺𝐺^∗−𝛽𝐼,(30) we get that 𝐺𝐺^∗𝛽𝐼+𝐺𝐺^∗−1=𝐼−𝛽𝛽𝐼+𝐺𝐺^∗−1.(31) Since ‖𝛽(𝛽𝐼+𝐺𝐺^∗)⁻¹‖<1, we obtain that 𝐺𝐺^∗(𝛽𝐼+𝐺𝐺^∗)⁻¹ is a homeomorphism. Consequently, Rang(𝐺𝐺^∗(𝛽𝐼+𝐺𝐺^∗)⁻¹)=𝑍, which implies that Rang(𝐺)=𝑍.

Corollary 9. If  Rang(𝐺)=𝑍 and Rang(𝐺)≠𝑍, then 𝛼𝛼𝐼+𝐺𝐺^∗−1=1,∀𝛼∈(0,1].(32) Moreover, lim_𝛼→0+𝛼𝐼+𝐺𝐺^∗−1=∞.(33)

3. Dense Range Semilinear Operators

In this section we shall look for conditions under which the semilinear operator

𝐺_𝐻∶𝑊→𝑍, given by 𝐺_𝐻𝑤=𝐺𝑤+𝐻(𝑤),𝑤∈𝑊,(34) has dense range.

Theorem 10. If Rang(𝐺)=𝑍, 𝐻 is continuous, and Rang(𝐻) is compact, then Rang(𝐺_𝐻)=𝑍, and for all 𝑧∈𝑍 there exists a sequence {𝑤_𝛼∈𝑍∶0<𝛼≤1} given by 𝑤_𝛼=𝐺^∗𝛼𝐼+𝐺𝐺^∗−1𝑧−𝐻𝑤_𝛼,(35) such that lim_𝛼→0+𝐺𝑤_𝛼+𝐻𝑤_𝛼=𝑧,(36) and the error of this approximation 𝐸_𝛼𝑧 is given by 𝐸_𝛼𝑧=𝛼𝛼𝐼+𝐺𝐺^∗−1𝑧−𝐻𝑤_𝛼.(37)

Proof. For each 𝑧∈𝑍 fixed we shall consider the following family of nonlinear operators 𝐾_𝛼∶𝑊→𝑊 given by 𝐾_𝛼(𝑤)=Γ_𝛼(𝑧−𝐻(𝑤))=𝐺^∗𝛼𝐼+𝐺𝐺^∗−1(𝑧−𝐻(𝑤)),(0<𝛼≤1).(38) First, we shall prove that for all 𝛼∈(0,1] the operator 𝐾_𝛼 has a fix point 𝑤_𝛼. In fact, since 𝐻 is a continuous function, the set Rang(𝐻) is compact, and 𝐺 is a linear bounded operator, then there exists a constant 𝑀>0 such that 𝐾_𝛼(𝑤)≤Γ_𝛼(‖𝑧‖+𝑀),∀𝑤∈𝑊.(39) Therefore, the operator 𝐾_𝛼 maps the ball 𝐵_𝑟(0)⊂𝑊 of center zero and radio 𝑟≥‖Γ_𝛼‖(‖𝑧‖+𝑀) into itself. Hence, applying the Schauder fixed point theorem, we get that the operator 𝐾_𝛼 has a fixed point 𝑤_𝛼∈𝐵_𝑟(0)⊂𝑊.
Since Rang(𝐻) is compact, without loss of generality, we can assume that the sequence 𝐻(𝑤_𝛼) converges to 𝑦∈𝑍 as 𝛼→0. So, if we consider 𝑤_𝛼=Γ_𝛼𝑧−𝐻𝑤_𝛼=𝐺^∗𝛼𝐼+𝐺𝐺^∗−1𝑧−𝐻𝑤_𝛼,(40) then, 𝐺𝑤_𝛼=𝐺Γ_𝛼𝑧−𝐻𝑤_𝛼=𝐺𝐺^∗𝛼𝐼+𝐺𝐺^∗−1𝑧−𝐻𝑤_𝛼=𝛼𝐼+𝐺𝐺^∗−𝛼𝐼𝛼𝐼+𝐺𝐺^∗−1𝑧−𝐻𝑤_𝛼=𝑧−𝐻𝑤_𝛼−𝛼𝛼𝐼+𝐺𝐺^∗−1𝑧−𝐻𝑤_𝛼.(41) Hence, 𝐺𝑤_𝛼+𝐻𝑤_𝛼=𝑧−𝛼𝛼𝐼+𝐺𝐺^∗−1𝑧−𝐻𝑤_𝛼.(42) To conclude the proof of this theorem, it is enough to prove that lim_𝛼→0−𝛼𝛼𝐼+𝐺𝐺^∗−1𝑧−𝐻𝑤_𝛼=0.(43) From Lemma 5(d) we get that lim_𝛼→0−𝛼𝛼𝐼+𝐺𝐺^∗−1𝑧−𝐻𝑤_𝛼=−lim_𝛼→0−𝛼𝛼𝐼+𝐺𝐺^∗−1𝐻𝑤_𝛼=lim_𝛼→0𝛼𝛼𝐼+𝐺𝐺^∗−1𝐻𝑤_𝛼−𝑦+𝑦=lim_𝛼→0−𝛼𝛼𝐼+𝐺𝐺^∗−1𝐻𝑤_𝛼−𝑦.(44) On the other hand, from Proposition 7 we get that 𝛼𝛼𝐼+𝐺𝐺^∗−1𝐻𝑤_𝛼−𝑦≤𝐻𝑤_𝛼−𝑦.(45) Therefore, since 𝐻(𝑤_𝛼) converges to 𝑦 as 𝛼→0, we get that lim_𝛼→0−𝛼𝛼𝐼+𝐺𝐺^∗−1𝐻𝑤_𝛼−𝑦=0.(46) Consequently, lim_𝛼→0𝐺_𝐻𝑤_𝛼=𝑧.(47)

4. Controllability of Nonlinear Evolution Equations

In this section we shall apply the foregoing results to characterize the approximate controllability of the semilinear evolution equation 𝑧^′=𝐴𝑧+𝐵𝑢(𝑡)+𝐹(𝑡,𝑧,𝑢(𝑡)),𝑧∈𝑍,𝑢∈𝑈,𝑡>0,(48) where 𝑍, 𝑈 are Hilbert spaces, 𝐴∶𝐷(𝐴)⊂𝑍→𝑍 is the infinitesimal generator of strongly continuous compact semigroup {𝑇(𝑡)}_𝑡≥0 in 𝑍,𝐵∈𝐿(𝑈,𝑍), the control function 𝑢 belongs to 𝐿²(0,𝜏;𝑈), and 𝐹∶[0,𝜏]×𝑍×𝑈→𝑍 is smooth enough and there are constants 𝑎,𝑐∈𝔑 such that sup_{(𝑡,𝑧,𝑢)∈𝑍𝜏}𝐹(𝑡,𝑧,𝑢)−𝑎𝑧−𝑐𝐵₁𝑢_𝑍<∞,(49) where 𝑍_𝜏=[0,𝜏]×𝑍×𝑈 and 𝐵₁∶𝑈→𝑍 is a linear and bounded operator.

We observe that the controllability of semilinear systems has been studied by several authors, particularly interesting is the work done by [18–26].

Definition 11 (exact controllability). The system (48) is said to be exactly controllable on [0,𝜏] if for every 𝑧₀, 𝑧₁∈𝑍 there exists 𝑢∈𝐿²(0,𝜏;𝑈) such that the mild solution 𝑧(𝑡) of (48) corresponding to 𝑢 verifies 𝑧(0)=𝑧₀,𝑧(𝜏)=𝑧₁,(50) as shown in Figure 1.

Figure 1 

Definition 12 (approximate controllability). The system (48) is said to be approximately controllable on [0,𝜏] if for every 𝑧₀, 𝑧₁∈𝑍, 𝜀>0 there exists 𝑢∈𝐿²(0,𝜏;𝑈) such that the solution 𝑧(𝑡) of (48) corresponding to 𝑢 verifies 𝑧(0)=𝑧₀,𝑧(𝜏)−𝑧₁<𝜀,(51) as shown in Figure 2.

Figure 2 

Definition 13 (controllability to trajectories). The system (48) is said to be controllable to trajectories on [0,𝜏] if for every 𝑧₀,𝑧₀∈𝑍 and 𝑢∈𝐿²(0,𝜏;𝑈) there exists 𝑢∈𝐿²(0,𝜏;𝑈) such that the mild solution 𝑧(𝑡) of (48) corresponding to 𝑢 verifies: 𝑧𝜏,𝑧₀,𝑢=𝑧𝜏,𝑧₀,𝑢,(52) as shown in Figure 3.

Figure 3 

Definition 14 (null controllability). The system (48) is said to be null controllable on [0,𝜏] if for every 𝑧₀∈𝑍 there exists 𝑢∈𝐿²(0,𝜏;𝑈) such that the mild solution 𝑧(𝑡) of (48) corresponding to 𝑢 verifies: 𝑧(0)=𝑧₀,𝑧(𝜏)=0,(53) as shown in Figure 4.

Figure 4 

Remark 15. It is clear that exact controllability of the system (48) implies approximate controllability, null controllability, and controllability to trajectories of the system.  But, it is well known [27] 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 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 will concentrated only on the study of the approximate controllability of the system (48).
Now, we shall describe the strategy of this work:
First, we characterize the approximate controllability of the auxiliary linear system 𝑧^′=𝐴𝑧+𝐵𝑢(𝑡)+𝑎𝑧+𝑐𝐵₁𝑢(𝑡),𝑡∈[0,𝜏].(54) After that, we write the system (48) in the form 𝑧^′=𝐴𝑧+𝐵𝑢(𝑡)+𝑎𝑧+𝑐𝐵₁𝑢(𝑡)+𝐺(𝑡,𝑧,𝑢),𝑡∈[0,𝜏],(55) where 𝐺(𝑡,𝑧,𝑢)=𝐹(𝑡,𝑧,𝑢)−𝑎𝑧−𝑐𝐵₁𝑢 is a smooth enough and bounded function.
Finally, the approximate controllability of the system (55) follows from the controllability of (54), the compactness of the semigroup generated by the operator 𝐴, the uniform boundedness of the nonlinear term 𝐺, and applying Schauder fixed point theorem.

Remark 16. If 𝑐≠1 and 𝐵=𝐵₁, then the system 𝑧^′=𝐴𝑧+𝐵𝑢(𝑡) is approximately controllable if and only if the system (55) is approximately controllable.

4.1. The Linear System

First, we shall characterize the approximate controllability of the linear system (54), and to this end, for all 𝑧₀∈𝑍 and 𝑢∈𝐿²(0,𝜏;𝑈) the initial value problem 𝑧^′=𝐴𝑧+𝐵𝑢(𝑡)+𝑎𝑧+𝑐𝐵₁𝑢(𝑡)𝑧(0),𝑡>0=𝑧₀,(56) admits only one mild solution given by 𝑧(𝑡)=𝑒^𝑎𝑡𝑇(𝑡)𝑧₀+^𝑡₀𝑒^{𝑎(𝑡−𝑠)}𝑇(𝑡−𝑠)𝐵+𝑐𝐵₁𝑢(𝑠)𝑑𝑠,𝑡∈[0,𝜏].(57)

Definition 17. For the system (54) we define the following concept: the controllability map (for 𝜏>0) 𝐺_𝑎∶𝐿²(0,𝜏;𝑈)→𝑍 is given by 𝐺_𝑎𝑢=^𝜏₀𝑒^𝑎𝑠𝑇(𝑠)𝐵+𝑐𝐵₁𝑢(𝑠)𝑑𝑠,(58) whose adjoint operator 𝐺^∗_𝑎∶𝑍→𝐿²(0,𝜏;𝑍) is 𝐺^∗_𝑎𝑧(𝑠)=𝐵^∗+𝑐𝐵^∗₁𝑒^𝑎𝑠𝑇^∗(𝑠)𝑧,∀𝑠∈[0,𝜏],∀𝑧∈𝑍.(59)

The following lemma follows from Lemma 5.

Lemma 18. Equation (54) 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^∗+𝑐𝐵^∗₁)𝑒^𝑎𝑡𝑇^∗(𝑡)𝑧=0, ∀𝑡∈[0,𝜏],⇒𝑧=0, (f) for all 𝑧∈𝑍 one has 𝐺_𝑎𝑢_𝛼=𝑧−𝛼(𝛼𝐼+𝐺_𝑎𝐺^∗_𝑎)⁻¹𝑧, where 𝑢_𝛼=𝐺^∗_𝑎𝛼𝐼+𝐺_𝑎𝐺^∗_𝑎⁻¹𝑧,𝛼∈(0,1].(60) So, lim_𝛼→0𝐺_𝑎𝑢_𝛼=𝑧 and the error 𝐸_𝛼𝑧 of this approximation is given by the formula 𝐸_𝛼𝑧=𝛼𝛼𝐼+𝐺_𝑎𝐺^∗_𝑎⁻¹𝑧,𝛼∈(0,1].(61)

Remark 19. Lemma 5 implies that the family of linear operators Γ_𝛼∶𝑍→𝐿²(0,𝜏;𝑈), defined for 0<𝛼≤1 by Γ_𝛼𝑧=𝐵^∗+𝑐𝐵^∗₁𝑒^{𝑎(𝜏−⋅)}𝑇^∗(𝜏−⋅)𝛼𝐼+𝐺_𝑎𝐺^∗_𝑎⁻¹𝑧=𝐺^∗_𝑎𝛼𝐼+𝐺_𝑎𝐺^∗_𝑎⁻¹𝑧,(62) is an approximate inverse for the right of the operator 𝐺_𝑎, in the sense that lim_𝛼→0𝐺_𝑎Γ_𝛼=𝐼(63) in the strong topology.

4.2. The Semilinear System

Now, we are ready to characterize the approximate controllability of the semilinear system (48), which is equivalent to proof of the approximate controllability of the system (55). To this end, we notice that, for all 𝑧₀∈𝑍 and 𝑢∈𝐿²(0,𝜏;𝑈) the initial value problem 𝑧^′=𝐴𝑧+𝐵𝑢+𝑎𝑧+𝑐𝐵₁()𝑧(0)𝑢+𝐺𝑡,𝑧,𝑢,𝑧∈𝑍,𝑡≥0,=𝑧₀(64) admits only one mild solution given by 𝑧_𝑢(𝑡)=𝑒^𝑎𝑡𝑇(𝑡)𝑧₀+^𝑡₀𝑒^{𝑎(𝑡−𝑠)}𝑇(𝑡−𝑠)𝐵+𝑐𝐵₁𝑢(𝑠)+𝑑𝑠_𝑡0𝑒^{𝑎(𝑡−𝑠)}𝑇(𝑡−𝑠)𝐺𝑠,𝑧_𝑢(𝑠),𝑢(𝑠)𝑑𝑠,𝑡∈[0,𝜏].(65)

Definition 20. For the system (55) 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+𝑐𝐵₁𝑢(𝑠)+𝑑𝑠_𝜏0𝑒^{𝑎(𝜏−𝑠)}𝑇((𝜏−𝑠))𝐺𝑠,𝑧_𝑢(𝑠),𝑢(𝑠)𝑑𝑠=𝐺_𝑎(𝑢)+𝐻(𝑢),(66) where 𝐻∶𝐿²(0,𝜏;𝑈)→𝑍 is the nonlinear operator given by 𝐻(𝑢)=^𝜏₀𝑒^{𝑎(𝜏−𝑠)}𝑇((𝜏−𝑠))𝐺𝑠,𝑧_𝑢(𝑠),𝑢(𝑠)𝑑𝑠,𝑢∈𝐿²(0,𝜏;𝑈).(67) The following lemma is trivial.

Lemma 21. Equation (55) is approximately controllable on [0,𝜏] if and only if Rang(𝐺_𝑔)=𝑍.

Definition 22. The following equation will be called the controllability equations associated to the nonlinear equation (55) 𝑢_𝛼=Γ_𝛼𝑧−𝐻𝑢_𝛼=𝐺^∗_𝑎𝛼𝐼+𝐺_𝑎𝐺^∗_𝑎⁻¹𝑧−𝐻𝑢_𝛼,(0<𝛼≤1).(68)
Now, we are ready to present a result on the approximate controllability of the semilinear evolutions equation (48).

Theorem 23. If the linear system (54) is approximately controllable, then system (55) is approximately controllable on [0,𝜏]. Moreover, a sequence of controls steering the system (55) from initial state 𝑧₀ to an 𝜖 neighborhood of the final state 𝑧₁ at time 𝜏>0 is given by the formula 𝑢_𝛼(𝑡)=𝐵^∗+𝑐𝐵^∗₁𝑒^{𝑎(𝜏−𝑡)}𝑇^∗()×𝜏−𝑡𝛼𝐼+𝐺_𝑎𝐺^∗_𝑎⁻¹𝑧₁−𝑒^𝑎𝜏𝑇(𝜏)𝑧₀−𝐻𝑢_𝛼,(69) and the error of this approximation 𝐸_𝛼 is given by 𝐸_𝛼=𝛼𝛼𝐼+𝐺_𝑎𝐺^∗_𝑎⁻¹𝑧₁−𝑒^𝑎𝜏𝑇(𝜏)𝑧₀−𝐻𝑢_𝛼.(70)

Proof. From Theorem 10, it is enough to prove that the function 𝐻 given by (103) is continuous and Rang(𝐻) is a compact set, which follows from the compactness of the semigroup {𝑇(𝑡)}_𝑡≥0,  the smoothness and the boundedness of the nonlinear term 𝐺 (see [8, 27]).
So, putting 𝑧=𝑧₁−𝑒^𝑎𝜏𝑇(𝜏)𝑧₀ and using (65), we obtain the desired result 𝑧₁=lim_𝛼→0+𝑇(𝜏)𝑧₀+^𝜏₀𝑇(𝜏−𝑠)𝐵𝑢_𝛼(𝑠)+𝑑𝑠_𝜏0𝑇(𝜏−𝑠)𝐹𝑠,𝑧_𝑢𝛼(𝑠),𝑢_𝛼(𝑠)𝑑𝑠.(71)

5. Application to the Nonlinear Heat Equation

As an application of this result we shall prove the controllability of the semilinear 𝑛𝐷 heat equation (6). To this end, we shall use the following strategy:

first, we prove that the auxiliary linear system 𝑧_𝑡(𝑡,𝑥)=Δ𝑧(𝑡,𝑥)+1_𝜔𝑢()()(]()𝑧()𝑡,𝑥+𝑎𝑧+𝑐𝑢𝑡,𝑥in0,𝜏×Ω,𝑧=0,on0,𝜏×𝜕Ω,0,𝑥=𝑧₀(𝑥),𝑥∈Ω,(72) is approximately controllable.

After that, we write the system(6) as follows: 𝑧_𝑡(𝑡,𝑥)=Δ𝑧(𝑡,𝑥)+1_𝜔𝑢()()()(]()𝑧()𝑡,𝑥+𝑎𝑧+𝑐𝑢𝑡,𝑥+𝑔𝑡,𝑧,𝑢in0,𝜏×Ω,𝑧=0,on0,𝜏×𝜕Ω,0,𝑥=𝑧₀(𝑥),𝑥∈Ω,(73) where 𝑔(𝑡,𝑧,𝑢)=𝑓(𝑡,𝑧,𝑢)−𝑎𝑧−𝑐𝑢 is a smooth and bounded function.

Then to prove the controllability of the linear equation (72), we use the classical Unique Continuation Principle for Elliptic Equations (see [28]) and the following results.

Lemma 24 (see Lemma 3.14 from [16, 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,𝑡₁,𝑖=1,2,…,𝑚,(74) iff 𝛽_𝑖,𝑗=0,𝑖=1,2,…,𝑚;𝑗=1,2,…,∞.(75)
Finally, the approximate controllability of the system (73) follows from the controllability of (72), the compactness of the semigroup generated by the Laplacean operator Δ, and the uniform boundedness of the nonlinear term 𝑔 by applying Theorem 23.

5.1. Abstract Formulation of the Problem

In this part we choose a Hilbert space where system (6) can be written as an abstract differential equation; to this end, we consider the following notations.

Let us consider the Hilbert space 𝑍=𝐿²(Ω) and 0=𝜆₁<𝜆₂<⋯<𝜆_𝑗→∞ the eigenvalues of −Δ, 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,(76) where ⟨⋅,⋅⟩ is the inner product in 𝑍 and 𝐸_𝑛𝑧=^𝛾𝑗_𝑘=1𝑧,𝜙_𝑗,𝑘𝜙_𝑗,𝑘.(77) So, {𝐸_𝑗} is a family of complete orthogonal projections in 𝑍 and 𝑧=∑^∞_𝑗=1𝐸_𝑗𝑧,𝑧∈𝐻.(iii) −𝐴 generates a compact analytic semigroup {𝑇(𝑡)} given by 𝑇(𝑡)𝑧=^∞_𝑗=1𝑒^{−𝜆𝑗𝑡}𝐸_𝑗𝑧.(78)

Consequently, systems (6), (72), and (73) can be written, respectively, as an abstract differential equations in 𝑍: 𝑧^′=−𝐴𝑧+𝐵_𝜔𝑢+𝑓^𝑒()𝑧𝑡,𝑧,𝑢,𝑧∈𝑍,𝑡≥0,(79)_′=−𝐴𝑧+𝐵_𝜔𝑧𝑢+𝑎𝑧+𝑐𝑢,𝑧∈𝑍,𝑡≥0,(80)_′=−𝐴𝑧+𝐵_𝜔𝑢+𝑎𝑧+𝑐𝑢+𝑔^𝑒(𝑡,𝑧,𝑢),𝑧∈𝑍,𝑡≥0,(81) 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)),∀𝑥∈Ω, and 𝑔^𝑒(𝑡,𝑧,𝑢)=𝑓^𝑒(𝑡,𝑧,𝑢)−𝑎𝑧−𝑐𝑢.

On the other hand, the hypothesis (7) implies that sup_{(𝑡,𝑧,𝑢)∈𝑍𝜏}𝑓^𝑒(𝑡,𝑧,𝑢)−𝑎𝑧−𝑐𝑢_𝑍<∞,(82) where 𝑍_𝜏=[0,𝜏]×𝑍×𝑈. Therefore, 𝑔^𝑒∶[0,𝜏]×𝑍×𝑈→𝑍 is bounded and smooth enough.

5.2. The Linear Heat Equation

In this part we shall prove the interior controllability of the linear system (80). To this end, we notice that for all 𝑧₀∈𝑍 and 𝑢∈𝐿²(0,𝜏;𝑈) the initial value problem, 𝑧^′=−𝐴𝑧+𝐵_𝜔𝑢(𝑡)(𝑡)(𝑡)𝑧(0)+𝑎𝑧+𝑐𝑢,𝑧∈𝑍,=𝑧₀,(83) admits only one mild solution given by 𝑧(𝑡)=𝑒^𝑎𝑡𝑇(𝑡)𝑧₀+^𝑡₀𝑒^{𝑎(𝑡−𝑠)}𝑇(𝑡−𝑠)𝐵_𝜔+𝑐𝐼𝑢(𝑠)𝑑𝑠,𝑡∈[0,𝜏].(84)

Definition 25. For the system (80) we define the following concept: the controllability map (for 𝜏>0) 𝐺_𝑎∶𝐿²(0,𝜏;𝑈)→𝑍 is given by 𝐺_𝑎𝑢=^𝜏₀𝑒^𝑎𝑠𝑇(𝑠)𝐵_𝜔+𝑐𝐼𝑢(𝑠)𝑑𝑠,(85) whose adjoint operator 𝐺^∗_𝑎∶𝑍→𝐿²(0,𝜏;𝑍) is given by 𝐺^∗_𝑎𝑧(𝑠)=𝐵^∗_𝜔+𝑐𝐼𝑒^𝑎𝑠𝑇^∗(𝑠)𝑧,∀𝑠∈[0,𝜏],∀𝑧∈𝑍.(86) As a consequence of Lemma 18 and (101) one can prove the following result.

Lemma 26. Equation (80) 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^∗_𝜔+𝑎𝐼)𝑒^𝑎𝑡𝑇^∗(𝑡)𝑧=0, ∀𝑡∈[0,𝜏], ⇒𝑧=0, (f) for all 𝑧∈𝑍 one has 𝐺𝑢_𝛼=𝑧−𝛼(𝛼𝐼+𝐺_𝑎𝐺^∗_𝑎)⁻¹𝑧, where 𝑢_𝛼=𝐺^∗_𝑎𝛼𝐼+𝐺_𝑎𝐺^∗_𝑎⁻¹𝑧,𝛼∈(0,1].(87) 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].(88)

Theorem 27. The system (80) is approximately controllable on [0,𝜏]. Moreover, a sequence of controls steering the system (80) from initial state 𝑧₀ to an 𝜖 neighborhood of the final state 𝑧₁ at time 𝜏>0 is given by 𝑢_𝛼(𝑡)=𝐵^∗_𝜔+𝑐𝐼𝑒^𝑎𝑡𝑇^∗()×𝜏−𝑡𝛼𝐼+𝐺_𝑎𝐺^∗_𝑎⁻¹𝑧₁−𝑇(𝜏)𝑧₀,(89) and the error of this approximation 𝐸_𝛼 is given by 𝐸_𝛼=𝛼𝛼𝐼+𝐺_𝑎𝐺^∗_𝑎⁻¹𝑧₁−𝑇(𝜏)𝑧₀.(90)

Proof. It is enough to show that the restriction 𝐺_𝑎,𝜔=𝐺_𝑎|_{𝐿2(0,𝜏;𝐿2(𝜔))} of 𝐺_𝑎 to the space 𝐿²(0,𝜏;𝐿²(𝜔)) has range dense; that is, Rang(𝐺_𝑎,𝜔)=𝑍 or Ker(𝐺^∗_𝑎,𝜔)={0}. Consequently, 𝐺_𝑎,𝜔∶𝐿²(0,𝜏;𝐿²(𝜔))→𝑍 takes the following form: 𝐺_𝑎,𝜔𝑢=^𝜏₀𝑒^𝑎𝑠𝑇(𝑠)(1+𝑐𝐼)𝐵_𝜔𝑢(𝑠)𝑑𝑠,(91) whose adjoint operator 𝐺^∗_𝑎,𝜔∶𝑍→𝐿²(0,𝜏;𝐿²(𝜔)) is given by 𝐺_𝑎,𝜔𝑧(𝑠)=(1+𝑐)𝐵^∗_𝜔𝑒^𝑎𝑠𝑇^∗(𝑠)𝑧,∀𝑠∈[0,𝜏],∀𝑧∈𝑍.(92) To this end, we observe that 𝐵_𝜔=𝐵^∗_𝜔 and 𝑇^∗(𝑡)=𝑇(𝑡). Suppose that (1+𝑐)𝐵^∗_𝜔𝑒^𝑎𝑡𝑇^∗(𝑡)𝑧=0,∀𝑡∈[0,𝜏].(93) Then, since 1+𝑐≠0, this is equivalent to 𝐵^∗_𝜔𝑇^∗(𝑡)𝑧=0,∀𝑡∈[0,𝜏].(94) On the other hand, 𝐵^∗_𝜔𝑇^∗(𝑡)𝑧=^∞_𝑗=1𝑒^{−𝜆𝑗𝑡}𝐵^∗_𝜔𝐸_𝑗𝑧=^∞_𝑗=1𝑒^{−𝜆𝑗𝑡𝛾𝑗}_𝑘=1𝑧,𝜙_𝑗,𝑘1_𝜔𝜙_𝑗,𝑘⟺=0,_∞𝑗=1𝑒^{−𝜆𝑗𝑡𝛾𝑗}_𝑘=1𝑧,𝜙_𝑗,𝑘1_𝜔𝜙_𝑗,𝑘(𝑥)=0,∀𝑥∈𝜔.(95) Hence, from Lemma 24, we obtain that 𝐸_𝑗𝑧(𝑥)=^𝛾𝑗_𝑘=1𝑧,𝜙_𝑗,𝑘𝜙_𝑗,𝑘(𝑥)=0,∀𝑥∈𝜔,𝑗=1,2,3,….(96) Now, putting 𝑓(𝑥)=∑^𝛾𝑗_𝑘=1⟨𝑧,𝜙_𝑗,𝑘⟩𝜙_𝑗,𝑘(𝑥), ∀𝑥∈Ω, we obtain that Δ+𝜆_𝑗𝐼𝑓(𝑥)𝑓≡0inΩ,=0∀𝑥∈𝜔.(97) Then, from the classical Unique Continuation Principle for Elliptic Equations (see [28]), it follows that 𝑓(𝑥)=0, ∀𝑥∈Ω. So, ^𝛾𝑗_𝑘=1𝑧,𝜙_𝑗,𝑘𝜙_𝑗,𝑘(𝑥)=0,∀𝑥∈Ω.(98) On the other hand, {𝜙_𝑗,𝑘} is a complete orthonormal set in 𝑍=𝐿²(Ω), which implies that ⟨𝑧,𝜙_𝑗,𝑘⟩=0. Hence, 𝑧=0. So, Rang(𝐺_𝑎,𝜔)=𝑍, and consequently Rang(𝐺_𝑎)=𝑍. Hence, the system (80) is approximately controllable on [0,𝜏], and the remainder of the proof follows from Lemma 26.

5.3. The Semilinear Heat Equation

In this part we shall prove the interior controllability of the semilinear 𝑛𝐷 heat equation given by (6), which is equivalent to the proof of the approximate controllability of the system (81). To this end, for all 𝑧₀∈𝑍 and 𝑢∈𝐿²(0,𝜏;𝑈) the initial value problem, 𝑧^′=−𝐴𝑧+𝐵_𝜔𝑢+𝑎𝑧+𝑐𝑢+𝑔^𝑒()𝑧(0)𝑡,𝑧,𝑢,𝑧∈𝑍,𝑡≥0=𝑧₀,(99) admits only one mild solution given by 𝑧_𝑢(𝑡)=𝑒^𝑎𝑡𝑇(𝑡)𝑧₀+^𝑡₀𝑒^{𝑎(𝑡−𝑠)}𝑇(𝑡−𝑠)𝐵_𝜔𝑢(𝑠)++𝑐𝐼𝑑𝑠_𝑡0𝑒^{𝑎(𝑡−𝑠)}𝑇(𝑡−𝑠)𝑔^𝑒𝑠,𝑧_𝑢(𝑠),(𝑠)𝑑𝑠,𝑡∈[0,𝜏].(100)

Definition 28. For the system (81) 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)++𝑐𝐼𝑑𝑠(101)_𝜏0𝑒^{𝑎(𝜏−𝑠)}𝑇(𝜏−𝑠)𝑔^𝑒𝑠,𝑧_𝑢(𝑠),(𝑠)𝑑𝑠=𝐺_𝑎(𝑢)+𝐻(𝑢),(102) 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),(𝑠)𝑑𝑠,𝑢∈𝐿²(0,𝜏;𝑈).(103)

The following lemma is trivial.

Lemma 29. Equation (81) is approximately controllable on [0,𝜏] if and only if Rang(𝐺_𝑔)=𝑍.

Definition 30. The following equation will be called the controllability equations associated to the nonlinear equation (81): 𝑢_𝛼=Γ_𝛼𝑧−𝐻𝑢_𝛼=𝐺^∗_𝑎𝛼𝐼+𝐺_𝑎𝐺^∗_𝑎⁻¹𝑧−𝐻𝑢_𝛼,(0<𝛼≤1).(104)

Now, we are ready to present a result on the interior approximate controllability of the semilinear 𝑛𝐷 heat equation (6).

Theorem 31. The system (81) is approximately controllable on [0,𝜏]. Moreover, a sequence of controls steering the system (81) from initial state 𝑧₀ to an 𝜖 neighborhood of the final state 𝑧₁ at time 𝜏>0 is given by 𝑢_𝛼(𝑡)=𝐵^∗_𝜔+𝑐𝐼𝑒^{𝑎(𝜏−𝑡)}𝑇^∗(𝜏−𝑡)𝛼𝐼+𝐺_𝑎𝐺^∗_𝑎⁻¹×𝑧₁−𝑇(𝜏)𝑧₀−𝐻𝑢_𝛼,(105) and the error of this approximation 𝐸_𝛼 is given by 𝐸_𝛼=𝛼𝛼𝐼+𝐺_𝑎𝐺^∗_𝑎⁻¹𝑧₁−𝑇(𝜏)𝑧₀−𝐻𝑢_𝛼.(106)

6. Conclusion

We believe that these results can be applied to a broad class of reaction diffusion equation like the following well-known systems of partial differential equations.

Example 32. The thermoelastic plate equation 𝑤_𝑡𝑡+Δ²𝑤+𝛼Δ𝑤=1_𝜔𝑢₁(𝑡,𝑥)+𝑓₁𝑡,𝑤,𝑤_𝑡()𝜃,𝑢,in0,𝜏×Ω,_𝑡−𝛽Δ𝜃−𝛼Δ𝑤_𝑡=1_𝜔𝑢₂(𝑡,𝑥)+𝑓₂𝑡,𝑤,𝑤_𝑡()(),𝑢,in0,𝜏×Ω,𝜃=𝑤=Δ𝑤=0,on0,𝜏×𝜕Ω,(107) where 𝛼≠0, 𝛽>0, Ω is a sufficiently regular bounded domain in 𝔑³, 𝜔 is an open nonempty subset of Ω, 1_𝜔 denotes the characteristic function of the set 𝜔, the distributed control 𝑢_𝑖∈𝐿²([0,𝜏];𝐿²(Ω)), 𝑖=1, 2, 𝑤, 𝜃 denote the vertical deflection and the temperature of the plate, respectively, and the nonlinear terms 𝑓_𝑖(𝑡,𝑧,𝑢), 𝑖=1,2, are smooth enough and there are constants 𝑎_𝑖,𝑐_𝑖∈𝔑, with 𝑐_𝑖≠−1, 𝑖=1,2, such that sup_{(𝑡,𝑤,𝑣,𝑢)∈𝑞𝜏}𝑓_𝑖(𝑡,𝑤,𝑣,𝑢)−𝑎_𝑖𝑤−𝑐_𝑖𝑢<∞,𝑖=1,2,(108) where 𝑞_𝜏=[0,𝜏]×𝔑×𝔑×𝔑.

Example 33. The equation modelling the damped flexible beam: 𝜕²𝑧𝜕²𝑡𝜕=−⁴𝑧𝜕⁴𝑥𝜕+2𝛼³𝑧𝜕𝑡𝜕²𝑥+1_𝜔𝑢(𝑡,𝑥)+𝑓𝑡,𝑧,𝑧_𝑡𝑧()()=𝜕,𝑢𝑡≥0,0≤𝑥≤1,𝑡,1=𝑧𝑡,0²𝑧𝜕²𝑥()=𝜕0,𝑡₂𝑧𝜕²𝑥()𝑧()1,𝑡=0,0,𝑥=𝜙₀(𝑥),𝜕𝑧()𝜕𝑡0,𝑥=𝜓₀(𝑥),0≤𝑥≤1,(109) where 𝛼>0, 𝑢∈𝐿²([0,𝑟];𝐿²[0,1]), 𝜔 is an open nonempty subset of [0,1], 𝜙₀,𝜓₀∈𝐿²[0,1], and nonlinear function 𝑓∶[0,𝜏]×𝔑×𝔑→𝔑 is smooth enough and there are constant 𝑎,𝑐∈𝔑, with 𝑐≠−1, such that sup_{(𝑡,𝑧,𝑣,𝑢)∈𝑞𝜏}𝑓(𝑡,𝑧,𝑣,𝑢)−𝑎𝑧−𝑐𝑢<∞,(110) where 𝑞_𝜏=[0,𝜏]×𝔑×𝔑×𝔑.

Example 34. The strongly damped wave equation with Dirichlet boundary conditions: 𝜕²𝑤𝜕²𝑡+𝜂(−Δ)^1/2𝜕𝑤()𝑤𝜕𝑡+𝛾−Δ=1_𝜔𝑢(𝑡,𝑥)+𝑓𝑡,𝑤,𝑤_𝑡𝑤()𝑤(),𝑢,𝑡≥0,𝑥∈Ω,𝑡,𝑥=0,𝑡≥0,𝑥∈𝜕Ω,0,𝑥=𝜙₀(𝑥),𝜕𝑧()𝜕𝑡0,𝑥=𝜓₀(𝑥),𝑥∈Ω,(111) where Ω is a sufficiently smooth bounded domain in 𝔑^𝑁, 𝑢∈𝐿²([0,𝑟];𝐿²(Ω)), 𝜔 is an open nonempty subset of Ω, 𝜙₀,𝜓₀∈𝐿²(Ω), and nonlinear function 𝑓∶[0,𝜏]×𝔑×𝔑→𝔑 is smooth enough and there are constants 𝑎,𝑐∈𝔑, with 𝑐≠−1, such that sup_{(𝑡,𝑤,𝑣,𝑢)∈𝑞𝜏}𝑓(𝑡,𝑤,𝑣,𝑢)−𝑎𝑤−𝑐𝑢<∞,(112) where 𝑞_𝜏=[0,𝜏]×𝔑×𝔑×𝔑.

Acknowledgments

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