Approximate Boundary Controllability for Semilinear Delay Differential Equations

Abstract

This paper considers the approximate controllability for a class of semilinear delay control systems described by a semigroup formulation with boundary control. Sufficient conditions for approximate controllability are established provided the approximate controllability of corresponding linear systems.

1. Introduction

In this paper, we consider the boundary control system described by the following delay differential equation: 𝑦′(𝑡)=𝜎𝑦(𝑡)+𝑓𝑡,𝑦_𝑡,𝜏𝑦(𝑡)=𝐵₁𝑢(𝑡)for𝑡∈𝐼=[0,𝑇],𝑦₀=𝜉, (1.1) where system state 𝑦(𝑡) takes values in a Banach space 𝐸; control function 𝑢(𝑡) takes values in another Banach space 𝑈 and 𝑢(⋅)∈𝐿^𝑝(𝐼;𝑈) for 𝑝≥1; 𝜎∶𝐷(𝜎)→𝐸 is a closed, densely defined linear operator; 𝜏∶𝐸→𝑋 is a linear operator from 𝐸 to a Banach space 𝑋; 𝐵₁∶𝑈→𝑋 is a linear bounded operator; 𝑓∶𝐼×𝐶→𝐸 is a nonlinear perturbation function, where 𝐶∶=𝐶([−Δ,0];𝐸) is the Banach space of all continuous functions from [−Δ,0] to 𝐸 endowed with the supremum norm. For any 𝑦∈𝐶([−Δ,𝑏];𝐸) and 𝑡∈𝐼, 𝑦_𝑡∈𝐶 is defined by 𝑦_𝑡(𝜃)=𝑦(𝑡+𝜃) for 𝜃∈[−Δ,0].

In most applications, the state space 𝐸 is a space of functions on some domain Ω of the Euclidean space ℝ^𝑛, 𝜎 is a partial differential operator on Ω, and 𝜏 is a partial differential operator acting on the boundary Γ of Ω.

Several abstract settings have been developed to describe control systems with boundary control; see Barbu [1], Fattorini [2], Lasiecka [3], and Washburn [4]. In this paper, we use the setting developed in [2] to discuss the approximate controllability of system (1.1).

The norms in spaces 𝐸 and 𝐶 are denoted by ‖⋅‖ and |⋅|, respectively. In other spaces, we use the norm notation with a space name in the subindex such as ‖⋅‖_𝑈, ‖⋅‖_𝑋, and ‖⋅‖_𝐿^𝑝.

Let 𝐴∶𝐸→𝐸 be the linear operator defined by 𝐷(𝐴)=𝑦∈𝐷(𝜎)∶𝜏𝑦=0,𝐴𝑦=𝜎𝑦,∀𝑦∈𝐷(𝐴). (1.2) We impose the following assumptions throughout the paper. (H1)𝐷(𝜎)⊂𝐷(𝜏) and the restriction of 𝜏 to 𝐷(𝜎) is continuous relative to the graph norm of 𝐷(𝜎).(H2)The operator 𝐴 is the infinitesimal generator of an analytic semigroup 𝑆(𝑡) for 𝑡≥0 on 𝐸.(H3)There exists a linear continuous operator 𝐵∶𝑈→𝐸 and a positive constant 𝐾 such that 𝜎𝐵∈𝐿(𝑈,𝐸),𝜏(𝐵𝑢)=𝐵₁‖‖𝑢,∀𝑢∈𝑈,𝐵𝑢_𝐸≤𝐾𝐵₁𝑢_𝑋,∀𝑢∈𝑈. (1.3)(H4)For each 𝑡∈(0,𝑇] and 𝑢∈𝑈, one has 𝑆(𝑡)𝐵𝑢∈𝐷(𝐴). Also, there exists a positive function 𝛾(⋅)∈𝐿^𝑞(𝐼) with 1/𝑝+1/𝑞=1 such that ‖𝐴𝑆(𝑡)𝐵‖_{𝐿(𝑈,𝐸)}≤𝛾(𝑡)a.e.𝑡∈(0,𝑇]. (1.4)(H5)There exists a positive number 𝐿 such that 𝑓𝑡,𝜂₁−𝑓𝑡,𝜂₂≤𝐿𝜂₁−𝜂₂ (1.5) for all 𝜂₁,𝜂₂∈𝐶 and 𝑡∈𝐼.

Based on the discussions in [2], system (1.1) can be reformulated as 𝑦(𝑡)=𝑆(𝑡)𝜉(0)+^𝑡₀𝑆(𝑡−𝑠)𝜎𝐵𝑢(𝑠)+𝑓𝑠,𝑦_𝑠𝑑𝑠+^𝑡₀()(𝑠)𝑦𝐴𝑆𝑡−𝑠𝐵𝑢𝑑𝑠,𝑡∈𝐼,₀=𝜉. (1.6)

The following system is called the corresponding linear system of (1.6) 𝑦(𝑡)=𝑆(𝑡)𝜉(0)+^𝑡₀𝑆(𝑡−𝑠)𝜎𝐵𝑢(𝑠)𝑑𝑠+^𝑡₀𝐴𝑆(𝑡−𝑠)𝐵𝑢(𝑠)𝑑𝑠. (1.7)

Approximate controllability for semilinear control systems with distributed controls has been extensively studied in the literature under different conditions; see Fabre et al. [5], Fernandez and Zuazua [6], Li and Yong [7], Mahmudov [8], Naito [9], Seidman [10], Wang [11, 12], and many other papers. However, only a few papers dealt with approximate boundary controllability for semilinear control systems, in particular, semilinear delay control systems; the main difficulty is encountered in the construction of suitable integral equation to apply for different versions of fixed-point theorem. Balachandran and Anandhi [13] considered the controllability of boundary control integrodifferential system, Han and Park [14] studied the boundary controllability of nonlinear systems with nonlocal initial condition. MacCamy et al. [15] discussed the approximate controllability for the heat equations. The purpose of this paper is to study the approximate controllability for a class of semilinear delay systems with boundary control.

2. Mild Solutions

By solutions of system (1.6) we mean mild solutions, that is, solutions in the space 𝐶([−Δ,𝑏];𝐸). In the following, we provide an existence and uniqueness theorem for (1.6).

Theorem 2.1. If (H1)–(H5) are satisfied, then system (1.6) has a unique solution for each control 𝑢(⋅)∈𝐿^𝑝(𝐼;𝑈).

Proof. Define 𝜉̂(𝑡)=𝑆(𝑡)𝜉(0)𝜉(𝑡)[],,𝑡∈𝐼,,𝑡∈−Δ,0 (2.1) and define 𝑦(𝑡)=𝑥(𝑡)+𝜉̂(𝑡). It is easy to know that 𝑥 satisfies 𝑥(𝑡)=^𝑡₀𝑆(𝑡−𝑠)𝜎𝐵𝑢(𝑠)+𝑓𝑠,𝑥_𝑠+𝜉̂_𝑠𝑑𝑠+^𝑡₀()(𝑠)𝑥𝐴𝑆𝑡−𝑠𝐵𝑢𝑑𝑠,𝑡∈𝐼,₀=0. (2.2) Let 𝑌={𝑥∈𝐶([−Δ,𝑏];𝐸)∶𝑥(𝑡)=0,∀𝑡∈[−Δ,0]}. Then, 𝑌 is a Banach space with supremum norm. For any 𝑢(⋅)∈𝐿^𝑝(𝐼;𝑈), define an operator 𝐽∶𝑌→𝑌 as follows: (𝐽𝑥)(𝑡)=^𝑡₀𝑆(𝑡−𝑠)𝜎𝐵𝑢(𝑠)+𝑓𝑠,𝑥_𝑠+𝜉̂_𝑠𝑑𝑠+^𝑡₀()(𝑠)[].𝐴𝑆𝑡−𝑠𝐵𝑢𝑑𝑠,𝑡∈𝐼,0,𝑡∈−Δ,0 (2.3) We need to show that 𝐽 is well defined. First, we show that (𝐽𝑥)(𝑡)∈𝐸 for any 𝑥∈𝑌 and 𝑡∈𝐼. Indeed, we have from (H5) that ‖𝑓(𝑡,𝜂)‖≤𝐿|𝜂|+𝑀₁, where 𝑀₁=sup_𝑡∈𝐼‖𝑓(𝑡,0)‖. For any 𝑠∈𝐼 and 𝜃∈[−Δ,0], we have ‖𝑥(𝑠+𝜃)‖≤sup_{𝑡∈[−Δ,𝑇]}‖𝑥(𝑡)‖≤‖𝑥‖_𝑌, (2.4) and ‖𝜉̂(𝑠+𝜃)‖≤max(|𝜉|,𝑀‖𝜉(0)‖), where 𝑀=max_𝑡∈𝐼‖𝑆(𝑡)‖.
Note that 𝐴𝑆(^𝑡₀𝑡−𝑠)𝐵𝑢(𝑠)𝑑𝑠≤^𝑡₀‖𝐴𝑆(𝑡−𝑠)𝐵‖^{𝑞1/𝑞𝑡}₀‖𝑢(𝑠)‖^𝑝1/𝑝≤𝛾_𝐿𝑞‖𝑢‖_𝐿^𝑝, (2.5) and that ^𝑡₀𝑆(𝑡−𝑠)𝜎𝐵𝑢(𝑠)+𝑓𝑠,𝑥_𝑠+𝜉̂_𝑠𝑑𝑠≤𝑀^𝑡₀‖𝜎𝐵𝑢(𝑠)‖+𝐿𝑥_𝑠+𝜉̂_𝑠+𝑀₁‖‖𝑑𝑠≤𝑀𝜎𝐵√_{𝐿(𝑈,𝐸)}^𝑞𝑇‖𝑢‖_𝐿^𝑝+𝑀𝑀₁𝑇+𝑀𝐿𝑇max𝜉,𝑀𝜉(0)+‖𝑥‖_𝑌. (2.6) Combining (2.5) and (2.6), we prove that (𝐽𝑥)(𝑡)∈𝐸 for any 𝑥∈𝑌 and 𝑡∈𝐼.
Next, we show that 𝐽 maps 𝑌 into 𝑌, in other words, 𝐽𝑥∈𝑌 for any 𝑥∈𝑌. Taking 𝑡, 𝑡+𝛿∈𝐼 with 𝛿>0, then )−(𝑡=‖(𝐽𝑥)(𝑡+𝛿𝐽𝑥)()‖_0𝑡+𝛿𝑆(𝑡+𝛿−𝑠)𝜎𝐵𝑢(𝑠)+𝑓𝑠,𝑥_𝑠+𝜉̂_𝑠𝑑𝑠+_𝑡^𝑡+𝛿𝐴𝑆(𝑡+𝛿−𝑠)𝐵𝑢(𝑠)𝑑𝑠−^𝑡₀𝑆(𝑡−𝑠)𝜎𝐵𝑢(𝑠)+𝑓𝑠,𝑥_𝑠+𝜉̂_𝑠𝑑𝑠−^𝑡₀()(𝑠)≤𝐴𝑆𝑡−𝑠𝐵𝑢𝑑𝑠_𝑡0[𝑆(𝑡+𝛿−𝑠)−𝑆(𝑡−𝑠)]𝜎𝐵𝑢(𝑠)+𝑓𝑠,𝑥_𝑠+𝜉̂_𝑠+𝑑𝑠_𝑡0[()(𝑠)()(𝑠+𝐴𝑆𝑡+𝛿−𝑠𝐵𝑢−𝐴𝑆𝑡−𝑠𝐵𝑢)]𝑑𝑠_{𝑡𝑡+𝛿}𝑆(𝑡+𝛿−𝑠)𝜎𝐵𝑢(𝑠)+𝑓𝑠,𝑥_𝑠+𝜉̂_𝑠+𝑑𝑠_{𝑡𝑡+𝛿}𝐴𝑆(𝑡+𝛿−𝑠)𝐵𝑢(𝑠)𝑑𝑠∶=𝐼₁+𝐼₂+𝐼₃+𝐼₄. (2.7) Since 𝑆(𝑡) is an analytic semigroup, (2.6) implies that as 𝛿→0  𝐼₁=(𝑆(𝛿)−𝐼)^𝑡₀𝑆(𝑡−𝑠)𝜎𝐵𝑢(𝑠)+𝑓𝑠,𝑥_𝑠+𝜉̂_𝑠𝑑𝑠⟶0. (2.8) Also, from (2.5), we have 𝐼₂=(𝑆(𝛿)−𝐼)^𝑡₀𝐴𝑆(𝑡−𝑠)𝐵𝑢(𝑠)𝑑𝑠⟶0. (2.9) Notice that 𝐼₃→0 and 𝐼₄→0 as 𝛿→0 follow from estimates 𝐼₃≤𝑀_𝑡^𝑡+𝛿‖𝜎𝐵‖‖𝑢(𝑠)‖+𝐿𝑥_𝑠+𝜉̂_𝑠+𝑀₁‖‖𝑑𝑠≤𝑀𝜎𝐵√_{𝐿(𝑈,𝐸)}^𝑞𝛿‖𝑢‖_𝐿^𝑝+𝑀𝐿max𝜉,𝑀𝜉(0)+‖𝑥‖_𝑌𝛿+𝑀𝑀₁𝐼𝛿,₄≤_𝑡^𝑡+𝛿𝛾^𝑞(𝑠)𝑑𝑠^1/𝑞‖𝑢‖_𝐿^𝑝. (2.10) We have ‖(𝐽𝑥)(𝑡+𝛿)−(𝐽𝑥)(𝑡)‖→0 as 𝛿→0 and, hence, 𝐽𝑥∈𝑌.
Now, we prove that 𝐽^𝑛 is a contraction mapping for sufficiently large 𝑛. In fact, for any 𝑥₁,𝑥₂∈𝑌, 𝑥_1𝑠−𝑥_2𝑠=sup_{𝜃∈[−Δ,0]}𝑥₁(𝑠+𝜃)−𝑥₂(𝑠+𝜃)≤𝑥₁−𝑥_2𝑌. (2.11) Therefore, 𝐽𝑥₁(𝑡)−𝐽𝑥₂(𝑡)=^𝑡₀𝑆(𝑡−𝑠)𝑓𝑠,𝑥_1𝑠+𝜉̂_𝑠−𝑓𝑠,𝑥_1𝑠+𝜉̂_𝑠𝑑𝑠≤𝑀𝐿^𝑡₀𝑥_1𝑠−𝑥_2𝑠𝑑𝑠≤𝑀𝐿𝑡𝑥₁−𝑥_2𝑌,𝐽𝑥_1𝑠−𝐽𝑥_2𝑠=sup_{[]𝜃∈−Δ,0}𝐽𝑥₁(𝑠+𝜃)−𝐽𝑥₂(𝑠+𝜃)≤𝑀𝐿𝑠𝑥₁−𝑥_2𝑌. (2.12) Similarly, 𝐽²𝑥₁(𝑡)−𝐽²𝑥₂(𝑡)≤𝑀𝐿^𝑡₀𝐽𝑥_1𝑠−𝐽𝑥_2𝑠𝑀𝑑𝑠≤²𝐿²𝑡²2𝑥₁−𝑥_2𝑌. (2.13) By mathematical induction, we have 𝐽^𝑛𝑥₁(𝑡)−𝐽^𝑛𝑥₂(𝑡≤())𝑀𝐿𝑇^𝑛𝑥𝑛!¹−𝑥_2𝑌. (2.14) Hence, 𝐽^𝑛𝑥₁−𝐽^𝑛𝑥_2𝑌≤(𝑀𝐿𝑇)^𝑛𝑥𝑛!¹−𝑥_2𝑌, (2.15) and 𝐽^𝑛 is a contraction mapping for sufficiently large 𝑛. The contraction mapping principle implies that 𝐽 has a unique fixed-point in 𝑌, which is the unique solution of (1.6). The proof of the theorem is complete.

3. Approximate Controllability

The solution of (1.6) is denoted by 𝑦(𝑡;𝑡₀,𝜉,𝑢) to emphasize the initial time 𝑡₀, initial state 𝜉∈𝐶, and control function 𝑢(⋅). 𝑦(𝑡₁;𝑡₀,𝜉,𝑢) is called the system state at time 𝑡₁ corresponding to initial pair (𝑡₀,𝜉) and the control function 𝑢. The set 𝑅𝑡₁;𝑡₀,𝜉(𝑁)=𝑦𝑡₁;𝑡₀,𝜉,𝑢∶𝑢(⋅)∈𝐿^𝑝𝑡₀,𝑇;𝑈 (3.1) is called the reachable set of system (1.6) at time 𝑡₁ corresponding to initial pair (𝑡₀,𝜉). 𝑅(𝑡₁;𝑡₀,𝜉)(𝑁) is the closure of 𝑅(𝑡₁;𝑡₀,𝜉)(𝑁) in 𝐸.

Definition 3.1. System (1.6) is said to be approximately controllable on [𝑡₀,𝑡₁] if 𝑅(𝑡₁;𝑡₀,𝜉)(𝑁)=𝐸 for any 𝜉∈𝐶.

Definition 3.2. System (1.6) is said to be approximately null controllable on [𝑡₀,𝑡₁] if for any 𝜉∈𝐶 and 𝜖>0, there is a control function 𝑢(⋅)∈𝐿^𝑝(𝑡₀,𝑡₁;𝑈) such that ‖𝑦(𝑡₁;𝑡₀,𝜉,𝑢)‖<𝜖.

Similar to nonlinear system (1.6), we define the reachable set of system (1.7) at time 𝑡₁ corresponding to the initial pair (𝑡₀,𝑦₀) as 𝑅(𝑡₁;𝑡₀,𝑦₀)(𝐿). The approximate controllability and approximate null controllability for system (1.7) can also be defined similarly.

To consider the approximate controllability of system (1.6), we need two new operators. For any 𝑡₁,𝑡₂∈𝐼 with 𝑡₂>𝑡₁, 𝐸(𝑡₁,𝑡₂)∶𝐿^𝑝(𝑡₁,𝑡₂;𝑈)→𝐸, and 𝑁(𝑡₁,𝑡₂)∶𝐿^𝑝(𝑡₁,𝑡₂;𝑈)→𝐸 are defined as: 𝐸𝑡₁,𝑡₂𝑢=^𝑡2_𝑡1𝑆𝑡₂−𝑠𝜎𝐵𝑢(𝑠)𝑑𝑠+^𝑡2_𝑡1𝐴𝑆𝑡₂(𝑠)𝑁𝑡−𝑠𝐵𝑢𝑑𝑠,₁,𝑡₂𝑢=^𝑡2_𝑡1𝑆𝑡₂−𝑠𝑓𝑠,𝑦_𝑠𝑑𝑠, (3.2) where 𝑦(𝑡;𝑢) is the solution of (1.6) with the initial pair (𝑡₁,𝜉) and control function 𝑢(⋅)∈𝐿^𝑝(𝑡₁,𝑡₂;𝑈) in the definition of 𝑁(𝑡₁,𝑡₂).

The following result provides sufficient conditions for the approximate controllability of system (1.6).

Theorem 3.3. Assume that system (1.7) is approximately controllable on the interval [𝑏,𝑇] for any 𝑏≥0. If there exists a function 𝑄(⋅)∈𝐿¹(𝐼) such that 𝑓(𝑡,𝑧)≤𝑄(𝑡),∀(𝑡,𝑧)∈𝐼×𝐶, (3.3) then system (1.6) is approximately controllable on 𝐼.

Proof. We need to show that the reachable set of system (1.6) at time 𝑇 is dense in Banach space 𝐸, in other words, 𝑅(𝑇;0,𝜉)(𝑁)=𝐸 (3.4) for any 𝜉∈𝐶. To this end, given any 𝜖>0 and 𝑥∈𝐸. Since (1.7) is approximately controllable on [0,𝑇], there exists a control function 𝑣₀(⋅)∈𝐿^𝑝(0,𝑇;𝑈) such that 𝑆(𝑇)𝜉(0)+𝐸(0,𝑇)𝑣₀−𝑥<𝜖2. (3.5) Note that 𝑄(⋅)∈𝐿¹(𝐼), we can select a sequence 𝑡_𝑛∈𝐼 such that 𝑡_𝑛>𝑡_𝑛−1 and ^𝑇_𝑡𝑛𝑄(𝑡)𝑑𝑡⟶0,as𝑛⟶∞. (3.6) Let 𝑦₁∶=𝑦(𝑡₁;0,𝜉,𝑣₀). Again, the approximate controllability of (1.7) on [𝑡₁,𝑇] implies that a control 𝑣₁(⋅)∈𝐿^𝑝(𝑡₁,𝑇;𝑈) exists such that 𝑆𝑇−𝑡₁𝑦₁+𝐸𝑡₁,𝑇𝑣₁−𝑥<𝜖2. (3.7) Define 𝑢₁(𝑡)=𝑣₀(𝑡),0≤𝑡≤𝑡₁,𝑣₁(𝑡),𝑡₁<𝑡≤𝑇. (3.8) Then 𝑢₁(⋅)∈𝐿^𝑝(0,𝑇;𝑈). Repeating the procedure, we have three sequences 𝑦_𝑛, 𝑣_𝑛, and 𝑢_𝑛 such that 𝑣_𝑛(⋅)∈𝐿^𝑝(𝑡_𝑛,𝑇;𝑈), 𝑢_𝑛(⋅)∈𝐿^𝑝(0,𝑇;𝑈), 𝑢_𝑛(𝑡)=𝑢_𝑛−1(𝑡),0≤𝑡≤𝑡_𝑛,𝑣_𝑛(𝑡),𝑡_𝑛𝑦<𝑡≤𝑇,_𝑛=𝑦𝑡_𝑛;0,𝜉,𝑢_𝑛−1,𝑆𝑇−𝑡_𝑛𝑦_𝑛+𝐸𝑡_𝑛,𝑇𝑣_𝑛−𝑥<𝜖2. (3.9) The solution of (1.6) under the control function 𝑢_𝑛(⋅) is 𝑦𝑡;0,𝜉,𝑢_𝑛=𝑆𝑡−𝑡_𝑛𝑆𝑡_𝑛𝜉(0)+𝐸0,𝑡_𝑛𝑢_𝑛+𝑁0,𝑡_𝑛𝑢_𝑛+𝐸𝑡_𝑛,𝑡𝑢_𝑛+𝑁𝑡_𝑛,𝑡𝑢_𝑛=𝑆𝑡−𝑡_𝑛𝑆𝑡_𝑛𝜉(0)+𝐸0,𝑡_𝑛𝑢_𝑛−1+𝑁0,𝑡_𝑛𝑢_𝑛−1+𝐸𝑡_𝑛,𝑡𝑢_𝑛+𝑁𝑡_𝑛,𝑡𝑢_𝑛=𝑆𝑡−𝑡_𝑛𝑦_𝑛+𝐸𝑡_𝑛,𝑡𝑣_𝑛+𝑁𝑡_𝑛,𝑡𝑢_𝑛. (3.10) Therefore, 𝑦𝑇;0,𝜉,𝑢_𝑛−𝑥≤𝑆𝑇−𝑡_𝑛𝑦_𝑛+𝐸𝑡_𝑛,𝑇𝑣_𝑛−𝑥+𝑁𝑡_𝑛,𝑇𝑢_𝑛<𝜖2+^𝑇_𝑡𝑛𝑆(𝑇−𝑠)𝑓𝑠,𝑦_𝑠≤𝜖𝑑𝑠2+𝑀^𝑇_𝑡𝑛𝑄(𝑠)𝑑𝑠<𝜖 (3.11) for a sufficient large 𝑛 such that 𝑀∫^𝑇_𝑡𝑛𝑄(𝑠)𝑑𝑠<𝜖/2. Hence, (3.4) follows, and the proof is complete.

The next theorem is about the approximate null controllability of system (1.6).

Theorem 3.4. Assume that system (1.7) is approximately null controllable on the interval [𝑏,𝑇] for any 𝑏≥0, and (3.3) is satisfied. Then system (1.6) is approximately null controllable on 𝐼.

Proof. For any 𝜖>0 and 𝜉∈𝐶, we need to show that there exists a control function 𝑢(⋅)∈𝐿^𝑝(𝐼;𝑈) such that ‖𝑆(𝑇)𝜉(0)+𝐸(0,𝑇)𝑢+𝑁(0,𝑇)𝑢‖<𝜖. Since system (1.7) is approximately null controllable on [0,𝑇], there is a control function 𝑣₀(⋅)∈𝐿^𝑝(0,𝑇;𝑈) such that ‖𝑆(𝑇)𝜉(0)+𝐸(0,𝑇)𝑣₀‖<𝜖/2. Select a sequence 𝑡_𝑛 as in the proof of Theorem 3.3. Let 𝑦₁∶=𝑦(𝑡₁;0,𝜉,𝑣₀). There exists a control function 𝑣₁(⋅)∈𝐿^𝑝(𝑡₁,𝑇;𝑈) such that 𝑆𝑇−𝑡₁𝑦₁+𝐸𝑡₁,𝑇𝑣₁<𝜖2 (3.12) due to the assumption that (1.7) is approximately null controllable on [𝑡₁,𝑇].
Similar to the proof of Theorem 3.3, we obtain three sequences 𝑦_𝑛, 𝑣_𝑛, and 𝑢_𝑛 such that 𝑣_𝑛(⋅)∈𝐿^𝑝(𝑡_𝑛,𝑇;𝑈), 𝑢_𝑛(⋅)∈𝐿^𝑝(𝐼;𝑈), 𝑢_𝑛(𝑡)=𝑢_𝑛−1(𝑡),0≤𝑡≤𝑡_𝑛,𝑣_𝑛(𝑡),𝑡_𝑛𝑦<𝑡≤𝑇,_𝑛=𝑦𝑡_𝑛;0,𝜉,𝑢_𝑛−1,𝑆𝑇−𝑡_𝑛𝑦_𝑛+𝐸𝑡_𝑛,𝑇𝑣_𝑛<𝜖2. (3.13) Note that 𝑦𝑡;0,𝜉,𝑢_𝑛=𝑆(𝑡)𝜉(0)+𝐸(0,𝑡)𝑢_𝑛+𝑁(0,𝑡)𝑢_𝑛=𝑆𝑡−𝑡_𝑛𝑦_𝑛+𝐸𝑡_𝑛,𝑡𝑣_𝑛+𝑁𝑡_𝑛,𝑡𝑢_𝑛, (3.14) we have 𝑦𝑇;0,𝜉,𝑢_𝑛≤𝑆𝑇−𝑡_𝑛𝑦_𝑛+𝐸𝑡_𝑛,𝑇𝑣_𝑛+𝑁𝑡_𝑛,𝑇𝑢_𝑛<𝜖2+𝑀^𝑇_𝑡𝑛𝑄(𝑠)𝑑𝑠<𝜖. (3.15) The proof of the theorem is complete.

4. Example

In this section, we provide an example to illustrate the application of the results established in Section 3.

Example 4.1. Consider the following heat control system: 𝑦_𝑡()()()()𝑦()(𝑡)𝑦()()[]𝑡,𝑥=Δ𝑦𝑡,𝑥+𝑓𝑡,𝑦𝑡,𝑥,𝑦𝑡−Δ,𝑥,𝑡∈𝐼,𝑥∈Ω,𝑡,𝑥=𝑢,𝑡∈𝐼,𝑥∈Γ,𝑡,𝑥=𝜉𝑡,𝑥,𝑡∈−Δ,0,𝑥∈Ω, (4.1) where Ω is a bounded and open subset of the Euclidean space ℝ^𝑛 with a sufficiently smooth boundary Γ.
To formulated this system as a boundary control system (1.1), we let 𝐸=𝐿²(Ω), 𝑋=𝐻^−1/2(Γ), 𝑈=𝐿²(Γ), 𝐵₁=𝐼, 𝐷(𝜎)={𝑦∈𝐿²(Ω)∶Δ𝑦∈𝐿²(Ω)}, and 𝜎=Δ. The operator 𝐴 is given by 𝐷(𝐴)=𝐻¹₀(Ω)∩𝐻²(Ω), 𝐴=Δ. Then 𝐴 generates an analytic semigroup 𝑆(𝑡) in 𝐸. The operator 𝜏 is the trace operator 𝛾₀𝑦 which is well defined and belongs to 𝐻^−1/2(Γ) for each 𝑦∈𝐷(𝜎). Clearly, assumptions (H1) and (H2) are satisfied. Define the linear operator 𝐵∶𝐿²(Γ)→𝐿²(Ω) by 𝐵𝑢=𝑤_𝑢, where 𝑤_𝑢∈𝐿²(Ω) is the unique solution to the Dirichlet boundary-value problem Δ𝑤_𝑢𝑤=0inΩ,_𝑢=𝑢onΓ. (4.2) It is proved in [1] that for every 𝑢∈𝐻^−1/2(Γ), (4.2) has a unique solution 𝑤_𝑢∈𝐿²(Ω) satisfying ‖𝐵𝑢‖_𝐿²_(Ω)=‖𝑤_𝑢‖_𝐿²_(Ω)≤𝐶₁‖𝑢‖_𝐻^−1/2_(Γ). This shows that (H3) is satisfied. It is proved in [4] that there exists a positive constant 𝐾₁ independent of 𝑢 and 𝑡 such that ‖𝐴𝑆(𝑡)𝐵𝑢‖_𝐿²_(Ω)≤𝐾₁𝑡^−3/4‖𝑢‖_𝐿²_(Γ) (4.3) for all 𝑢∈𝐿²(Γ) and 𝑡>0. In other words, (H4) holds with 𝛾(𝑡)=𝐾₁𝑡^−3/4. Therefore, system (4.1) can be formulated to the form (1.6). Since the corresponding linear system of (4.1) 𝑦_𝑡()()𝑦()(𝑡)𝑦()()𝑡,𝑥=Δ𝑦𝑡,𝑥,𝑡∈𝐼,𝑥∈Ω,𝑡,𝑥=𝑢,𝑡∈𝐼,𝑥∈Γ,0,𝑥=𝜉0,𝑥,𝑥∈Ω (4.4) is approximately controllable on any interval [𝑏,𝑇] with 𝑏≥0; see [15]. It follows from Theorem 3.3 that system (4.1) is approximately controllable on 𝐼 if the nonlinear perturbation function 𝑓 satisfies (H5).