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:

()

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 ℝⁿ, 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 .

Let A : E → E be the linear operator defined by

()

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
()
(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
()
(H5)
There exists a positive number L such that
()
for all η₁, η₂ ∈ C and t ∈ I.

Based on the discussions in [2], system (1.1) can be reformulated as

()

The following system is called the corresponding linear system of (1.6)

()

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

()

and define

. It is easy to know that x satisfies

()

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:

()

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

()

and

, where M = max _t∈I∥S(t)∥.

Note that

()

and that

()

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

()

Since S(t) is an analytic semigroup, (2.6) implies that as δ → 0

()

Also, from (2.5), we have

()

Notice that I₃ → 0 and I₄ → 0 as δ → 0 follow from estimates

()

We have ∥(Jx)(t + δ)−(Jx)(t)∥→0 as δ → 0 and, hence, Jx ∈ Y.

Now, we prove that Jⁿ is a contraction mapping for sufficiently large n. In fact, for any x₁, x₂ ∈ Y,

()

Therefore,

()

Similarly,

()

By mathematical induction, we have

()

Hence,

()

and Jⁿ 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

()

is called the reachable set of system (1.6) at time t₁ corresponding to initial pair (t₀, ξ).

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 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:

()

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

()

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,

()

for any ξ ∈ C. To this end, given any ϵ > 0 and

. Since (1.7) is approximately controllable on [0, T], there exists a control function v₀(·) ∈ L^p(0, T; U) such that

()

Note that Q(·) ∈ L¹(I), we can select a sequence t_n ∈ I such that t_n > t_n−1 and

()

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

()

Define

()

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),

()

The solution of (1.6) under the control function u_n(·) is

()

Therefore,

()

for a sufficient large n such that

. 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

()

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),

()

Note that

()

we have

()

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:

()

where Ω is a bounded and open subset of the Euclidean space ℝⁿ 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 , 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

()

It is proved in [1] that for every u ∈ H^−1/2(Γ), (4.2) has a unique solution w_u ∈ L²(Ω) satisfying

. 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

()

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)

()

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).

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Approximate Boundary Controllability for Semilinear Delay Differential Equations

Abstract

1. Introduction

2. Mild Solutions

3. Approximate Controllability

4. Example

References

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Approximate Boundary Controllability for Semilinear Delay Differential Equations

Abstract

1. Introduction

2. Mild Solutions

3. Approximate Controllability

4. Example

References

Citing Literature

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