# Approximate Boundary Controllability for Semilinear Delay Differential Equations

Authors: Lianwen Wang
Journal: Journal of Applied Mathematics
 (2011)

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## 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.


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## Body

## 1. Introduction

In this paper, we consider the boundary control system described by the following delay differential equation:(1.1)y′(t)=σy(t)+f(t,yt),τy(t)=B1u(t)fort∈I=[0,T],y0=ξ,
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(·)∈Lp(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, yt∈C is defined by yt(θ)=y(t+θ) for θ∈[-Δ,0].In most applications, the state spaceE 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 spacesE 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 ∥·∥Lp.LetA:E→E be the linear operator defined by
(1.2)D(A)={y∈D(σ):τy=0},Ay=σy,∀y∈D(A).
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 operatorA is the infinitesimal generator of an analytic semigroup S(t) for t≥0 on E.(H3)
There exists a linear continuous operatorB:U→E and a positive constant K such that
(1.3)σB∈L(U,E),τ(Bu)=B1u,∀u∈U,‖Bu‖E≤K‖B1u‖X,∀u∈U.(H4)
For eacht∈(0,T] and u∈U, one has S(t)Bu∈D(A). Also, there exists a positive function γ(·)∈Lq(I) with 1/p+1/q=1 such that
(1.4)‖AS(t)B‖L(U,E)≤γ(t)a.e.t∈(0,T].(H5)
There exists a positive numberL such that
(1.5)‖f(t,η1)-f(t,η2)‖≤L|η1-η2|
for all η1,η2∈C and t∈I.Based on the discussions in [2], system (1.1) can be reformulated as
(1.6)y(t)=S(t)ξ(0)+∫0tS(t-s)[σBu(s)+f(s,ys)]ds+∫0tAS(t-s)Bu(s)ds,t∈I,y0=ξ.The following system is called the corresponding linear system of (1.6)
(1.7)y(t)=S(t)ξ(0)+∫0tS(t-s)σBu(s)ds+∫0tAS(t-s)Bu(s)ds.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(·)∈Lp(I;U).Proof.
Define(2.1)ξ̂(t)={S(t)ξ(0),t∈I,ξ(t),t∈[-Δ,0],
and define y(t)=x(t)+ξ̂(t). It is easy to know that x satisfies
(2.2)x(t)=∫0tS(t-s)[σBu(s)+f(s,xs+ξ̂s)]ds+∫0tAS(t-s)Bu(s)ds,t∈I,x0=0.
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(·)∈Lp(I;U), define an operator J:Y→Y as follows:
(2.3)(Jx)(t)={∫0tS(t-s)[σBu(s)+f(s,xs+ξ̂s)]ds+∫0tAS(t-s)Bu(s)ds,t∈I,0,t∈[-Δ,0].
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=supt∈I∥f(t,0)∥. For any s∈I and θ∈[-Δ,0], we have
(2.4)‖x(s+θ)‖≤sup⁡t∈[-Δ,T]‖x(t)‖≤‖x‖Y,
and ∥ξ̂(s+θ)∥≤max(|ξ|,M∥ξ(0)∥), where M=maxt∈I∥S(t)∥.
Note that(2.5)‖∫0tAS(t-s)Bu(s)ds‖≤(∫0t‖AS(t-s)B‖q)1/q(∫0t‖u(s)‖p)1/p≤‖γ‖Lq‖u‖Lp,
and that
(2.6)‖∫0tS(t-s)[σBu(s)+f(s,xs+ξ̂s)]ds‖≤M∫0t[‖σBu(s)‖+L|xs+ξ̂s|+M1]ds≤M‖σB‖L(U,E)Tq‖u‖Lp+MM1T+MLT(max⁡(|ξ|,M‖ξ(0)‖)+‖x‖Y).
Combining (2.5) and (2.6), we prove that (Jx)(t)∈E for any x∈Y and t∈I.
Next, we show thatJ maps Y into Y, in other words, Jx∈Y for any x∈Y. Taking t, t+δ∈I with δ>0, then
(2.7)‖(Jx)(t+δ)-(Jx)(t)‖=‖∫0t+δS(t+δ-s)[σBu(s)+f(s,xs+ξ̂s)]ds+∫tt+δAS(t+δ-s)Bu(s)ds-∫0tS(t-s)[σBu(s)+f(s,xs+ξ̂s)]ds-∫0tAS(t-s)Bu(s)ds‖≤‖∫0t[S(t+δ-s)-S(t-s)][σBu(s)+f(s,xs+ξ̂s)]ds‖+‖∫0t[AS(t+δ-s)Bu(s)-AS(t-s)Bu(s)]ds‖+‖∫tt+δS(t+δ-s)[σBu(s)+f(s,xs+ξ̂s)]ds‖+‖∫tt+δAS(t+δ-s)Bu(s)ds‖:=I1+I2+I3+I4.
Since S(t) is an analytic semigroup, (2.6) implies that as δ→0
(2.8)I1=(S(δ)-I)∫0tS(t-s)[σBu(s)+f(s,xs+ξ̂s)]ds⟶0.
Also, from (2.5), we have
(2.9)I2=(S(δ)-I)∫0tAS(t-s)Bu(s)ds⟶0.
Notice that I3→0 and I4→0 as δ→0 follow from estimates
(2.10)I3≤M∫tt+δ(‖σB‖‖u(s)‖+L|xs+ξ̂s|+M1)ds≤M‖σB‖L(U,E)δq‖u‖Lp+ML(max⁡(|ξ|,M‖ξ(0)‖)+‖x‖Y)δ+MM1δ,I4≤(∫tt+δγq(s)ds)1/q‖u‖Lp.
We have ∥(Jx)(t+δ)-(Jx)(t)∥→0 as δ→0 and, hence, Jx∈Y.
Now, we prove thatJn is a contraction mapping for sufficiently large n. In fact, for any x1,x2∈Y,
(2.11)|x1s-x2s|=sup⁡θ∈[-Δ,0]‖x1(s+θ)-x2(s+θ)‖≤‖x1-x2‖Y.
Therefore,
(2.12)‖(Jx1)(t)-(Jx2)(t)‖=‖∫0tS(t-s)[f(s,x1s+ξ̂s)-f(s,x1s+ξ̂s)]ds‖≤ML∫0t|x1s-x2s|ds≤MLt‖x1-x2‖Y,|(Jx1)s-(Jx2)s|=sup⁡θ∈[-Δ,0]‖(Jx1)(s+θ)-(Jx2)(s+θ)‖≤MLs‖x1-x2‖Y.
Similarly,
(2.13)‖(J2x1)(t)-(J2x2)(t)‖≤ML∫0t|(Jx1)s-(Jx2)s|ds≤M2L2t22‖x1-x2‖Y.
By mathematical induction, we have
(2.14)‖(Jnx1)(t)-(Jnx2)(t)‖≤(MLT)nn!‖x1-x2‖Y.
Hence,
(2.15)‖Jnx1-Jnx2‖Y≤(MLT)nn!‖x1-x2‖Y,
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
(3.1)R(t1;t0,ξ)(N)={y(t1;t0,ξ,u):u(⋅)∈Lp(t0,T;U)}
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]ifR(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(·)∈Lp(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):Lp(t1,t2;U)→E, and N(t1,t2):Lp(t1,t2;U)→E are defined as:
(3.2)E(t1,t2)u=∫t1t2S(t2-s)σBu(s)ds+∫t1t2AS(t2-s)Bu(s)ds,N(t1,t2)u=∫t1t2S(t2-s)f(s,ys)ds,
where y(t;u) is the solution of (1.6) with the initial pair (t1,ξ) and control function u(·)∈Lp(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
(3.3)‖f(t,z)‖≤Q(t),∀(t,z)∈I×C,
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,
(3.4)R(T;0,ξ)(N)¯=E
for any ξ∈C. To this end, given any ϵ>0 and x̅∈E. Since (1.7) is approximately controllable on [0,T], there exists a control function v0(·)∈Lp(0,T;U) such that
(3.5)‖S(T)ξ(0)+E(0,T)v0-x̅‖<ϵ2.
Note that Q(·)∈L1(I), we can select a sequence tn∈I such that tn>tn-1 and
(3.6)∫tnTQ(t)dt⟶0,asn⟶∞.
Let y1:=y(t1;0,ξ,v0). Again, the approximate controllability of (1.7) on [t1,T] implies that a control v1(·)∈Lp(t1,T;U) exists such that
(3.7)‖S(T-t1)y1+E(t1,T)v1-x̅‖<ϵ2.
Define
(3.8)u1(t)={v0(t),0≤t≤t1,v1(t),t1<t≤T.
Then u1(·)∈Lp(0,T;U). Repeating the procedure, we have three sequences yn, vn, and un such that vn(·)∈Lp(tn,T;U), un(·)∈Lp(0,T;U),
(3.9)un(t)={un-1(t),0≤t≤tn,vn(t),tn<t≤T,yn=y(tn;0,ξ,un-1),‖S(T-tn)yn+E(tn,T)vn-x̅‖<ϵ2.
The solution of (1.6) under the control function un(·) is
(3.10)y(t;0,ξ,un)=S(t-tn)[S(tn)ξ(0)+E(0,tn)un+N(0,tn)un]+E(tn,t)un+N(tn,t)un=S(t-tn)[S(tn)ξ(0)+E(0,tn)un-1+N(0,tn)un-1]+E(tn,t)un+N(tn,t)un=S(t-tn)yn+E(tn,t)vn+N(tn,t)un.
Therefore,
(3.11)‖y(T;0,ξ,un)-x̅‖≤‖S(T-tn)yn+E(tn,T)vn-x̅‖+‖N(tn,T)un‖<ϵ2+∫tnT‖S(T-s)f(s,ys)‖ds≤ϵ2+M∫tnTQ(s)ds<ϵ
for a sufficient large n such that M∫tnTQ(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 anyϵ>0 and ξ∈C, we need to show that there exists a control function u(·)∈Lp(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(·)∈Lp(0,T;U) such that ∥S(T)ξ(0)+E(0,T)v0∥<ϵ/2. Select a sequence tn as in the proof of Theorem 3.3. Let y1:=y(t1;0,ξ,v0). There exists a control function v1(·)∈Lp(t1,T;U) such that
(3.12)‖S(T-t1)y1+E(t1,T)v1‖<ϵ2
due to the assumption that (1.7) is approximately null controllable on [t1,T].
Similar to the proof of Theorem3.3, we obtain three sequences yn, vn, and un such that vn(·)∈Lp(tn,T;U), un(·)∈Lp(I;U),
(3.13)un(t)={un-1(t),0≤t≤tn,vn(t),tn<t≤T,yn=y(tn;0,ξ,un-1),‖S(T-tn)yn+E(tn,T)vn‖<ϵ2.
Note that
(3.14)y(t;0,ξ,un)=S(t)ξ(0)+E(0,t)un+N(0,t)un=S(t-tn)yn+E(tn,t)vn+N(tn,t)un,
we have
(3.15)‖y(T;0,ξ,un)‖≤‖S(T-tn)yn+E(tn,T)vn‖+‖N(tn,T)un‖<ϵ2+M∫tnTQ(s)ds<ϵ.
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 Section3.Example 4.1.
Consider the following heat control system:(4.1)yt(t,x)=Δy(t,x)+f(t,y(t,x),y(t-Δ,x)),t∈I,x∈Ω,y(t,x)=u(t),t∈I,x∈Γ,y(t,x)=ξ(t,x),t∈[-Δ,0],x∈Ω,
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 D(A)=H01(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=wu, where wu∈L2(Ω) is the unique solution to the Dirichlet boundary-value problem
(4.2)Δwu=0inΩ,wu=uon⁡Γ.
It is proved in [1] that for every u∈H-1/2(Γ), (4.2) has a unique solution wu∈L2(Ω) satisfying ∥Bu∥L2(Ω)=∥wu∥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
(4.3)‖AS(t)Bu‖L2(Ω)≤K1t-3/4‖u‖L2(Γ)
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)
(4.4)yt(t,x)=Δy(t,x),t∈I,x∈Ω,y(t,x)=u(t),t∈I,x∈Γ,y(0,x)=ξ(0,x),x∈Ω
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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