In this paper, we are concerned with the global existence of solution and the approximate controllability for the semilinear neutral system in a Hilbert space H:

(1.1)ddt[x(t)+g(t,∫0t a(t,s,x(s))ds)]+Ax(t)=f(t,x(t))+k(t), t∈0,T,x(0)=x0.

Here, -A generates an analytic semigroup in H(see [21, Theorem 3.6.1]). The nonlinear operator f is assumed to be locally Lipschitz continuous with respect to the second variable, and g is Lipschitz continuous. This kind of equation arises in heat conduction in material with memory, in population dynamics, and in control systems with hereditary feed back control governed by an integro-differential law.

In the first part of this paper, we establish the well-posedness and regularity property for (1.1). The solvability for a class of semilinear functional differential equations has been studied by many authors as seen in Section 4.3.1 of Barbu [1] and [11, 13, 17]. Our approach is to obtain the L²-regularity under the above formulation of the semilinear neutral problem (1.1) using the contraction mapping principle (see the linear cases of [4]). Recently, the existence of solutions for mild solutions for neutral differential equations with state-dependence delay has been studied in the literature [8, 9, 10].

Next, based on the regularity for (1.1), we intend to establish the approximate controllability for the following semilinaer neutral control system with control part in place of a forcing term:

(1.2)ddt[x(t)+g∫0t a(t,s,x(s))ds)]+Ax(t)=f(t,x(t))+Bu(t), t∈(0,T],x(0)=x0,

namely that the reachable set of trajectories of (1.2) is a dense subset of H. Here, the controller operator B is a bounded linear operator from a Banach space of control variables into H and u is a control. This kind of equations arise naturally in physics, in biology, control engineering problem, etc.

As for the approximate controllability for semilinear control systems, we refer to [2, 3, 5, 7, 20, 23]. The controllability for neutral equations has been studied by many authors, for example, the controllability of neutral functional differential systems with unbounded delay in [5, 6, 15], neutral evolution integrodifferential systems with state dependent delay in [14, 18], impulsive neutral functional evolution integrodifferential systems with infinite delay in [19]. However, there are few literature works treating the systems with local Lipschitz continuity. As a sufficient condition for the approximate controllabilityl, Wang [24] assumed that the semigroup S(t) generated by A is compact in order to guarantee the compactness of the solution mapping(see also [16]).

In this paper, we no longer require the compact property of the semigroup and the uniform boundedness of the nonlinear term, but instead we need properties fractional power of operators and conditions for the range of the controller without the inequality condition as in previous results.

The paper is organized as follows. In Section 2, the results of general linear evolution equations besides notations and assumptions are stated. In Section 3, we will obtain that the regularity for parabolic linear equations can also be applicable to (1.1) with nonlinear terms satisfying local Lipschitz continuity. The approach used here is similar to that developed in [11, 12, 16] on the general semilinear evolution equations, which is an important role to extend the theory of practical nonlinear partial differential equations. Thereafter, we investigate the approximate controllability for the problem (1.2) in Section 4. In the proofs of the main theorems, we need conditions on the range of the controller without the inequality condition as in previous results(see [16, 25]) without conditions of the compact property of a semigroup and the uniform boundedness. Finally we give a simple example to which our main result can be applied.

If H is identified with its dual space we may write V⊂H⊂V* densely and the corresponding injections are continuous. The norm on V, H and V^* will be denoted by ||⋅||, |⋅| and ||⋅||*, respectively. The duality pairing between the element v₁ of V^* and the element v₂ of V is denoted by (v1,v2), which is the ordinary inner product in H if v1,v2∈H.

For l∈V* we denote (l,v) by the value l(v) of l at v ∈ V. The norm of l as element of V^* is given by

||l||*=supv∈V|(l,v)|||v||.

Therefore, we assume that V has a stronger topology than H and, for brevity, we may regard that

(2.1)||u||*≤|u|≤||u||, ∀u∈V.

Let b(⋅,⋅) be a bounded sesquilinear form defined in V×V and satisfying Gårding's inequality

(2.2)Re b(u,u)≥ω1||u||2−ω2|u|2,

where ω1>0 and ω2 is a real number. Let A be the operator associated with this sesquilinear form:

(Au,v)=b(u,v), u, v∈V.

Then -A is a bounded linear operator from V to V^* by the Lax-Milgram Theorem. The realization of A in H which is the restriction of A to

D(A)={u∈V:Au∈H}

is also denoted by A. From the following inequalities

ω1||u||2≤Rea(u,u)+ω2|u|2≤|Au||u|+ω2|u|2≤max{1,ω2}||u||D(A)|u|,

where

||u||D(A)=(|Au|2+|u|2)1/2

is the graph norm of D(A), it follows that there exists a constant C₀>0 such that

(2.3)||u||≤C0||u||D(A)1/2|u|1/2.

Thus we have the following sequence

(2.4)D(A)⊂V⊂H⊂V*⊂D(A)*,

where each space is dense in the next one which continuous injection.

Lemma 2.1. With the notations (2.3), (2.4), we have

(V,V*)1/2,2=H,(D(A),H)1/2,2=V,

where (V,V*)1/2,2 denotes the real interpolation space between V and V^*(Section 1.3.3 of [22]).

It is also well known that -A generates an analytic semigroup S(t) in both H and V^*. For the sake of simplicity we assume that ω₂=0 and hence the closed half plane {λ:Reλ≥0} is contained in the resolvent set of A.

If X is a Banach space, L2(0,T;X) is the collection of all strongly measurable square integrable functions from (0,T) into X and W1,2(0,T;X) is the set of all absolutely continuous functions on [0,T] such that their derivative belongs to L²(0,T;X). C([0,T];X) will denote the set of all continuously functions from [0,T] into X with the supremum norm. If X and Y are two Banach spaces, L(X,Y) is the collection of all bounded linear operators from X into Y, and L(X,X) is simply written as L(X). Let the solution spaces W(T) and W1(T) of strong solutions be defined by

W(T):=L2(0,T;D(A))∩W1,2(0,T;H),W1(T):=L2(0,T;V)∩W1,2(0,T;V*).

Here, we note that by using interpolation theory, we have

W(T)⊂C([0,T];V), W1(T)⊂C([0,T];H).

The semigroup generatedby -A is denoted by S(t) and there exists a constant M such that

|S(t)|≤M, ||S(t)||*≤M.

The following Lemma is from Lemma 3.6.2 of [21].

Lemma 2.2. There exists a constant M>0 such that the following inequalities hold for all t>0 and every x ∈ H or V^*:

(2.5)|S(t)x|≤Mt−1/2||x||*, ||S(t)x||≤Mt−1/2|x|.

First of all, consider the following linear system

(2.6)x'(t)+Ax(t)=k(t),x(0)=x0.

By virtue of Theorem 3.3 of [4](or Theorem 3.1 of [11], [21]), we have the following result on the corresponding linear equation of (2.6).

Proposition 2.3. Suppose that the assumptions for the principal operator A stated above are satisfied. Then the following properties hold:

(1) For x0∈V=(D(A),H)1/2,2(see Lemma 2.1) and k∈L2(0,T;H), T>0, there exists a unique solution x of (2.6) belonging to W(T)⊂C([0,T];V) and satisfying

(2.7)||x||W(T)≤C1(||x0||+||k||L2(0,T;H)),

where C₁ is a constant depending on T.

(2) Let x0∈H and k∈L2(0,T;V*) , T>0. Then there exists a unique solution x of (2.6) belonging to W1(T)⊂C([0,T];H) and satisfying

(2.8)||x||W1(T)≤C1(|x0|+||k||L2(0,T;V*)),

where C₁ is a constant depending on T.

Corollary 2.4. Suppose that k∈L2(0,T;H) and x(t)=∫0tS(t−s)k(s)ds for 0≤t≤T. Then there exists a constant C₂ such that

(2.9)||x||L2(0,T;D(A))≤C1||k||L2(0,T;H), 
(2.10)||x||L2(0,T;H)≤C2T||k||L2(0,T;H),

and

(2.11)||x||L2(0,T;V)≤C2T||k||L2(0,T;H).

Proof. The assertion (2.9) is immediately obtained by (2.7). Since

||x||L2(0,T;H)2=∫0T |∫0tS(t−s)k(s)ds|2dt≤M∫0T ( ∫ 0 t| k(s)|ds) 2 dt≤M∫0Tt∫0t |k(s)|2dsdt    ≤MT2 2∫0T |k(s)|2ds

it follows that

||x||L2(0,T;H)≤TM/2||k||L2(0,T;H).

From (2.3), (2.9), and (2.10) it holds that

||x||L2(0,T;V)≤C0C1T(M/2)1/4||k||L2(0,T;H).

So, if we take a constant C2>0 such that

C2=max{M/2,C0C1(M/2)1/4},

the proof is complete.

From now on, we establish the following results on the local solvability of the following equation;

(3.1)ddt[x(t)+g(t,∫0tta(t,s,x(s))ds)]+Ax(t)=f(t,x(t))+k(t), t∈(0,T]x(0)=x0,

where A is the operator mentioned in Section 2 and f is a nonlinear mapping from [0,T]×V into H which will be assumed later. It is also well known that Aα is a closed operator with its domain dense and D(Aα)⊃D(Aβ) for 0<α<β. Due to the well known fact that A−α is a bounded operator, we can assume that there is a constant Cα>0 such that

(3.2)||A−α||L(H)≤C−α, ||A−α||L(V*,V)≤C−α. 

Lemma 3.1. For any T>0, there exists a positive constant C_α such that the following inequalities hold for all t>0:

(3.3)||AαS(t)||L(H)≤Cαtα, ||AαS(t)||L(V,H)≤Cαt3α/2. 

Proof. The inequality (2.5) implies (3.3) by properties of fractional power of A and the definition of W(t). For more details about the above lemma, we refer to [21, 17].

We give the following assumptions.

Assumption(A). Let a:ℝ+×ℝ+×V→H

be a continuous function. Then there exists a constant L_a such that

|a(t,s,0)|≤La, |a(t,s,x(s))−a(t,s,y(s))|≤La||x(s)−y(s)||

Assumption(F). Let f:[0,T]×V→H be a nonlinear mapping such that There exists a function L:ℝ+→ℝ such that

|f(t,x)|≤L(r), |f(t,x)−f(t,y)|≤L(r)||x−y||

hold for any t∈[0,T], ||x||≤r and ||y||≤r.

Assumption(G). Let g:[0,T]×H→H be a nonlinear mapping such that there exist constants β>1/3 and L_g satisfying the following conditions hold:

(i) For any x∈H, the mapping g(⋅,x) is strongly measurable;

(ii) There exist positive constants L_g and β>1/3 such that

g(0,0)=0, |Aβg(t,0)|≤Lg, |Aβg(t,x)−Aβg(t,x^)|≤Lg|x−x^|,

for all t∈[0,T], and x,x^∈H.

(iii) ∂ig is measurable in t∈[0,T] for each x∈H and continuous in x∈H for a.e. t∈[0,T], where ∂ig is the partial derivative with respect to i-th coordinate and the value ∂ig(t,x) is the Gateau derivative of g(t, x) for each i=1,2 , and

|∂ig(t,0)|≤Lg, |∂ig(t,x1)−∂ig(t,x2)|≤Lg|x1−x2|

for t≤T and x1,x2∈H.

Let us rewrite (Fx)(t)=f(t,x(t)) for each x∈L2(0,T;V). Then by Assumption (F), there is a constant, denoted again by L(r), such that

||Fx||L2(0,T;H)≤L(r)T, ||Fx1−Fx2||L2(0,T;H)≤L(r)||x1−x2||L2(0,T;V)

hold for x1,x2∈Br(T)={x∈L2(0,T;V):||x||L2(0,T;V)≤r}.

Lemma 3.2. Let us assume Assumptions (F),(G) and (A) for 0<s≤t, we have

(3.4)|Aβg(s,∫0sa(s,τ,x(τ))dτ)|≤Lg(Lat||x||L2(0,t;V)+Lat+1),

and

(3.5)|Aβg(s,∫0sa(s,τ,x(τ))dτ)−Aβg(s,∫0sa(s,τ,y(τ))dτ)|≤LgLat||x−y||L2(0,t;V).

Proof. From Assumptions (G), (A) and using Hölder inequality we have

|Aβg(s,∫0sa(s,τ,x(τ))dτ)|=|Aβg(s,∫0sa(s,τ,x(τ))dτ)−Aβg(s,0)|+|Aβg(s,0)|≤Lg(∫0s|a(s,τ,x(τ))|dτ+1)≤Lg(∫0s|a(s,τ,x(τ))−a(s,τ,0)|dτ+∫0s|a(s,τ,0)|dτ+1)≤Lg(La∫0s||x(τ)||dτ+Lat+1)≤Lg(Lat||x||L2(0,T;V)+Lat+1).

Moreover, we have

|Aβg(s,∫0sa(s,τ,x(τ))dτ)−Aβg(s,∫0sa(s,τ,y(τ))dτ)|≤Lg(∫0s|a(s,τ,x(τ))−a(s,τ,y(τ))|dτ≤LgLa∫0s||x(τ)−y(τ)||dτ≤LgLat||x−y||L2(0,t;V).

Theorem 3.3. Let Assumptions (F), (G) and (A) be satisfied. Assume that x0∈H, k∈L2(0,T;V*). Then, there exists a time T0∈(0,T) such that the equation (3.1) admits a solution

(3.6)x∈L2(0,T0;V)∩W1,2(0,T0;V*)⊂C([0,T0];H).

Proof. For a solution of (3.1) in the wider sense, we are going to find a solution of the following integral equation

(3.7)x(t)=−g(t,∫0ta(t,s,x(s))ds)+S(t)x0+∫0tS(t−s){(Fx)(s)+k(s)}ds+∫0tAS(t−s)g(s,∫0sa(s,τ,x(τ))dτ)ds.

To provea local solution, we will use the successive iteration method. First, put

x0(t)=S(t)x0+∫0tS(t−s)k(s)ds

and define xj+1(t) as

(3.8)xj+1(t)=x0(t)−g(t,∫0ta(t,s,xj(s))ds)+∫0tS(t−s)(Fxj)(s)ds+∫0tAS(t−s)g(s,∫0sa(s,τ,xj(τ))dτ)ds.

By virtue of Proposition 2.3, we have x0(⋅)∈W1(t), so

that

(3.9)||x0||W1(t)≤C1(|x0|+||k||L2(0,t;V*)),

where C₁ isa constant in Proposition 2.3. Choose

r>C1(|x0|+||k||L2(0,t;V*)).

Putting

p1(t)=∫0tS(t−s)(Fx0)(s)ds,

by (2.11) of Corollary 2.4, we have

(3.10)||p1||L2(0,t;V)≤C2t||Fx0||L2(0,t;H)≤C2L(r)t.

Let

p2(s)=∫0sAS(s−τ)g(τ,∫0τa(τ,σ,x0(σ))dσ))dτ.

Then From Assumption (G), (A), (3.3) and (3.4), we have

(3.11)||p2||L2(0,t;V)=[∫0t‖∫0sAS(s−τ)g(τ,∫0τa(τ,σ,x0(σ))dσ))dτ‖2ds]12≤[∫0t{∫0sC1−β(s−τ)3(1−β)/2Lg(Lat||x0||L2(0,t;V)+Lat+1)dτ}2ds]12≤C1−βLg(Lat||x0||L2(0,t;V)+Lat+1)(∫0t(∫0s1(s−τ)3(1−β)/2dτ)2ds)12=23β−113βC1−βLg(Lat||x0||L2(0,t;V)+Lat+1)t3β2

Set

p3(s)=g(s,∫0sa(s,τ,x0(τ))dτ).

Then by Assumption (G), (3.2) and (3.4),

(3.12)|p3||L2(0,t;V)=(∫0t||g(s,∫0sa(s,τ,x0(τ))dτ||2ds)12=(∫0t||A−βAβg(s,∫0sa(s,τ,x0(τ))dτ||2ds)12≤C−βLg(∫0t(Lat||x0||L2(0,t;V)+Lat+1)2ds)12≤C−βLg(Lat||x0||L2(0,t;V)+Lat+1)t

Put

(3.13)M1:=max{C2L(r)t,    23β−113βC1−βLg(Lat||x0||L2(0,t;V)+Lat+1)t3β2,    C−βLg(Lat||x0||L2(0,t;V)+Lat+1)t}.

Then for any t satisfying M1<r, from(3.6) and (3.7),

||x1||L2(0,t;V)≤r+C2L(r)t+23β−113βC1−βLg(Lat||x0||L2(0,t;V)+Lat+1)t3β2+C−βLg(Lat||x0||L2(0,t;V)+Lat+1)t≤4r.

By induction, it can be shown that for all j=1,2,...,

||xj||L2(0,t;V)≤4r.

Hence, from the equation

xj+1(t)−xj(t)=−g(t,∫0ta(t,s,xj(s))ds)+g(t,∫0ta(t,s,xj−1(s))ds)+∫0tS(t−s){f(t,xj(s))−f(t,xj−1(s))}ds+∫0tAS(t−s){g(s,∫0sa(s,τ,xj(τ))dτ)ds−g(s,∫0sa(s,τ,xj−1(τ)))}dτ)ds.

Put

(3.14)M2:=C−βLgLat+C2L(4r)t+C1−βLgLa(23(3β−2)(3β+3))12t3β+34.

In a similar way to (3.11) and (3.12) and Assumption (F), we can observe that the inequality

|xj+1−xj||L2(0,t;V)≤M2||xj−xj−1||L2(0,t;V)      ≤(M2)j||x1−x0||L2(0,t;V).

Choose T0>0 satisfying max{M1,M2}<1, Then {xj} is strongly convergent to a function x in L2(0,T0;V) uniformly on 0≤t≤T0 and so is in W1,2(0,T;V*) by (iii) of Assumption (G). By letting j→∞ in (3.8) has a unique solution x in W1(T0).

From now on, we give a norm estimation of the solution of (3.1) and establish the global existence of solutions with the aid of norm estimations by similar argument using (3.1) and (iii) of Assumption (G).

Theorem 3.4. Under the Assumptions (A), (F) and (G), there exists a unique solution x of (3.1) such that

x∈W1(T)≡L2(0,T;V)∩W1,2(0,T;V*)⊂C([0,T];H), T>0.

for any

x0∈H, k∈L2(0,T;V*). Moreover, there exists a constant C₃ such that

(3.15)||x||W1≤C3(1+|x0|+||k||L2(0,T;V*)),

where C₃ is a constant depending on T.

Proof. Let x be a solution of (3.1) on [0,T₀], T₀>0 satisfies max{M1,M2}<1. Here, M₁ and M₂ be constants in (3.13) and (3.14), respectively. Then by virtue of Theorem 3.1, the solution x is represented as

x(t)=x0(t)−g(t,∫0ta(t,s,x(s))ds)+∫0tS(t−s)(Fx)(s)ds+∫0tAS(t−s)g(s,∫0sa(s,τ,x(τ))dτ)ds.

where

x0(t)=S(t)x0+∫0tS(t−s)k(s)ds

By (3.9), we have x0(⋅)∈W1(T0), so

that

|x0||W1(T0)≤C1(|x0|+||k||L2(0,T0;V*)),

where C₁ is a constant in Proposition 2.3. Moreover, from (3.9)-(3.12), it follows that

(3.16)||x||W1(T0)≤C1(|x0|+||k||L2(0,T0;V*))+max{M1,M2}||x||W1(T0).

Thus, Moreover, there exists a constant C₃ such that

|x||W1(T0)≤C3(1+|x0|+||k||L2(0,T0;V*)).

Now from

|S(T0)x0+∫0T0S(T0−s){(Fx)(s)+k(s)}ds|≤M|x0|+MT0L(r)+MT0 ||k||L2(0,T0;H),|−g(t,∫0ta(t,s,x(s))ds)|≤Lg(Lat||x||L2(0,t;V)+Lat+1),

and

|∫0tAS(t−s)g(s,∫0sa(s,τ,x(τ))dτ)ds|≤∫0t|C1−β(t−s)(1−β)/2Lg(Lat||x||L2(0,t;V)+Lat+1)|ds=2(β+1)−1t(β+1)/2C1−βLg(Lat||x||L2(0,t;V)+Lat+1).

it follows that

|x(T0)|≤M|x0|+MT0L(r)+MT0||k||L2(0,T0;H)+Lg(Lat||x||L2(0,t;V)+Lat+1)+2((β+1)−1t(β+1)/2C1−βLg(Lat||x||L2(0,t;V)+Lat+1)<∞.

Hence, we cansolve the equation in [T₀,2T₀] with the initial value x(T₀) and obtain an analogous estimate to (3.16). Since the condition (3.13), (3.14) is independent of initial values, the solution can be extended to the interval [0,nT₀] for any natural number n, i.e., for the initial u(nT₁) in the interval [nT1,(n+1)T1], as analogous estimate (3.16) holds for the solution in [0,(n+1)T1].

From the following result, we obtain that the solution mapping is continuous, which is useful for physical applications of the given equation.

Corollary 3.5. Let the Assumptions (A), (F) and (G) be satisfied and (x0,k)∈H×L2(0,T;V*) for each T>0.

Then the solution x of the equation (3.1) belongs to x∈W1(T)≡L2(0,T;V)∩W1,2(0,T;V*)⊂C([0,T];H) and the mapping

H×L2(0,T;V*)∋(x0,k)↦x∈W1(T)

is continuous.

Proof. From Theorem 3.4, it follows that if (x0,k)∈H×L2(0,T;V*) then x belongs to W1(T). Let (x0i,ki)∈H×L2(0,T;V*) and xi∈W1(T) be the solution of (3.1) with (x0i,ki) in place of (x0,k) for i=1,2. Hence, we assume that x_i belongs to a ball Br(T)={y∈W(T):||y||W1(T)≤r}.

Let

(pxj)(t)=−g(t,∫0ta(t,s,xj(s))ds)+∫0tS(t−s)(Fxj)(s)ds+∫0tAS(t−s){g(s,∫0sa(s,τ,xj(τ))dτ)ds.

Then, by virtue of 2) of Proposition 2.1, we get

(3.17)||x1−x2||W1(T)≤C1{|x01−x02|+||k1−k2||L2(0,T;V*+||px1−px2||L2(0,T;V*)}.

Set ||⋅||L2(0,T0;V)=||⋅||L2 forbrevity, where T0>0 satisfies max{M1,M2}<1.

Then, we have

(3.18)||px1−px2||L2(0,T;V*)≤||px1−px2||L2=||−g(t,∫0ta (t,s,xj(s))ds)+g(t,∫0ta (t,s,xj−1(s))ds)||L2+‖∫0tS (t−s){(Fx1)(s))−(Fx2)(s))}ds‖L2+‖∫0tA S(t−s){g(s,∫0sa (s,τ,x1(τ))dτ)ds−g(s,∫0sa (s,τ,x2(τ)))}dτ)ds‖L2≤M2||x1−x2||L2.

Hence, by (3.17), (3.18) and (iii) of Assumption (G), we see that

xn↦x∈W1(T0)≡L2(0,T0;V))∩W1,2(0,T0;V*).

This implies that (xn(T0),(xn)T0) ↦(x(T0),xT0) in H×L2(0,T;V*). Hence the same argument shows that xn↦x in

L2(T0,min{2T0,T};V))∩W1,2(T2,min{2T0,T};V*).

Repeating this processwe conclude that xn↦x in W1(T).

Let U be a Banach space of control variables, and let B be an operator from U to H, called controller. In this paper, we are concerned with the approximate controllability for the following the semilinear neutral control system with a control part Bu in place of k of (3.1):

(4.1)ddt[x(t)+g(t,∫0ta(t,s,x(s))ds)]+Ax(t)=f(t,x(t))+Bu(t), t∈(0,T],x(0)=x0.

Let x(T;f, u) be a state value of the system (4.1) at time T corresponding to the nonlinear term f and the control u.

Definition 4.1 The system (4.1) is said to be approximately controllable in the time interval [0,T] if for every desired final state x1∈H and ϵ>0 there exists a control function u∈L2(0,T;U) such that the solutionx(T;f,u) of (4.1) satisfies |x(T;f,u)−x1|<ϵ.

In order to obtain results of controllability, we need the stronger hypotheses than those of Section 3:

Assumption (A1). Let a:ℝ+×ℝ+×H→H

be a continuous function. Then there exists a constant L_a such that

|a(t,s,0)|≤La, |a(t,s,x(s))−a(t,s,y(s))|≤La|x(s)−y(s)|.

Assumption (F1). Let f:[0,T]×H be a nonlinear mapping such that

(i) t→f(t,x) is measurable;

(ii) f is locally Lipschitz continuous respect to x, that is, for each r>0, there exists a constant Lf=L(r)>0 such that

|f(t,x)−f(t,y)|≤Lf|x−y|

hold for any t∈[0,T], |x|≤r and |y|≤r.

Assumption (G1). Let g:[0,T]×H→D(A) be a nonlinear mapping such that there exists L_g satisfying the following conditions hold:

(i) (i) and (iii) of Assumption (G) in Section 3 are satisfied.

(ii) There exists positive constants L_g such that

g(0,0)=0, |Ag(t,0)|≤Lg, |Ag(t,x)−Ag(t,x^)|≤Lg|x−x^|,

for all t∈[0,T], and x,x^∈H.

We define the linear operator S^ from L2(0,T;H) to H by

S^p=∫0T S(T−s)p(s)ds for p∈L2(0,T;H).

Assumption (S). For any ε>0 and p∈L2(0,T;H) there exists a u∈L2(0,T;U) such that

     |S^p−S^Φu|<ε,     ||Bu||L2(0,t;H)≤q||p||L2(0,t;H), 0≤t≤T.

where q is a constant independent of p.

Here, we remark that Assumptions (A1), (F1) and (G1) are actually sufficient conditions for Assumptions (A), (F) and (G), respectively. So, if (x0,k)∈H×L2(0,T;V*) then from Theorem 3.2 and Corollary 3.1, it follows that the solution x of the equation (4.1) belongs to x∈L2(0,T;V)∩W1,2(0,T;V*)⊂C([0,T];H) and the mapping

H×L2(0,T;U)∋(x0,u)↦x∈W1(T)

is continuous.

Lemma 4.2. Let u₁ and u₂ be in L2(0,T;U). Then under the Assumptions (A1), (F1), (G1), and ||x(⋅;f,u1)||C([0,T;H])<r, we have

(4.2)||x(⋅;f,u1)−x(⋅;f,u2)||C([0,T];H)≤MeM3t||Bu1−Bu2||L2(0,T;H)

for 0≤t≤T, where

(4.3)M3:=(||A−1||LgLa+MLf+MLaLgT)T.

Proof. For brevity, we set xi(t)=x(t;f,ui)(i=1,2). Let

(pxi)(t)=−g(t,∫0ta(t,s,xi(s))ds)+∫0tS(t−s)f(t,xj(s))ds+∫0tAS(t−s){g(s,∫0sa(s,τ,xj(τ))dτ)ds.

Then, we see

(4.4)|x1(t)−x2(t)|≤|px1−px2|+|∫0tS(t−s)B(u1(s)−u2(s))ds|.

Here, by Assumptions (F1) and(G1), he following inequalities hold:

(4.5)|px1−px2|≤(||A−1||LgLa+MLf+MLaLgt)∫0t|x1(s)−x2(s)|ds,

and

(4.6)|∫0tS(t−s)B(u1(s)−u2(s))ds|≤Mt||Bu1−Bu2||L2(0,T;H).

Thus from (4.4)-(4.6) and using Gronwall's inequality, it follows that

|x1(t)−x2(t)|≤Mte(||A−1||LgLa+MLf+MLaLgT)T||Bu1−Bu2||L2(0,t;H).

Therefore, (4.2) holds.

Theorem 4.3. Under the Assumptions (A1), (F1), (G1) and (S), the system (4.1) is approximately controllable on [0,T].

Proof. Let us define a reachable set for the system (4.1):

RT={x(T;f,u):u∈L2(0,T;U)}.

Then we will show that D(A)⊂RT¯, i.e., for given ε>0 and ξT∈D(A) there exists u∈L2(0,T;U) such that

|ξT−x(T;f,u)|<ε.

Noting that

g(t,∫0ta(t,s,x(s))ds)=∫0tS(t−s){g(t,∫0ta(t,s,x(s))ds)        +sAg(t,∫0ta(t,s,x(s))ds)/t}ds,

the solution of (4.1) is represented as

x(t;f,u)=S(t)x0+∫0tS(t−s)G(s,x(s;f,u))ds+∫0tS(t−s)Bu(s)}ds, t≤T,

where

(4.7)G(s,x(s))=g(t,∫0ta(t,s,x(s))ds)+sAg(t,∫0ta(t,s,x(s))ds)/t+f(s,x(s))    +Ag(s,∫0sa(s,τ,x(τ))dτ).

As ξT∈D(A) there exists a h∈L2(0,T;H) such that

S^h=ξT−S(T)x0,

for instance, take h(s)=(ξT+sAξT)−S(s)x0/T.

Let u1∈L2(0,T;U) be arbitrary fixed. Since by Assumption (S) there exists u2∈L2(0,T;U) such that

|S^(h−G(⋅,x(⋅,f,u1)))−S^Bu2|<ε4,

it follows

|ξT−S(T)x0−S^G(⋅,x(⋅f,u1))−S^Bu2|<ε4.

We can also choose w2∈L2(0,T;U) by Assumption (S) such that

(4.8)|S^(G(⋅x(⋅;g,f,u2))−G(⋅x(⋅;g,f,u1))−S^Bw2|<ε8 

and by Assumption (S)

‖Bw2||L2(0,t;H≤q||G(⋅,x(⋅;f,u1))−G(⋅,x(⋅;f,u2))||L2(0,t;H)

for 0≤t≤T.

Choose a constant r₁ satisfying

||x(⋅;f,u1)||C([0,t];H)≤r1, ||x(⋅;f,u2)||C([0,t];H)≤r1.

According to a simple calculation of (4.7), from Lemma 4.1 we have

(4.9)|G(s,x(s;f,u1))−G(s,x(s;f,u2))|≤{(||A−1||+2)LaLgT+Lf}||x(⋅;f,u1)−x(⋅;f,u2)||C([0,t];H), s∈(0,t],≤{(||A−1||+2)LaLgT+Lf}MeM3(r1)t||Bu1−Bu2||L2(0,T;H)

where M₃ is the constant of (4.2). For the sake of simplicity, set

L^:={(||A−1||+2)LaLgT+Lf}MeM3, r>0.

Thus, in view of (4.9) and Assumption (S), we see

||Bw2||L2(0,t;H)≤q{∫0t|G(τ,x(τ;f,u2))−G(τ,x(τ;f,u1))|2dτ}12      ≤qL^{∫0tτ||Bu2−Bu1||L2(0,τ;H)2dτ}12      ≤qL^(∫0tτdτ)12||Bu2−Bu1||L2(0,t;H)      =qL^(t22)12||Bu2−Bu1||L2(0,t;H).

Put u3=u2−w2. We determine w₃ such that

|S^(G(⋅,x(⋅;f,u3))−G(⋅,x(⋅;f,u2)))−S^Bw3|<ε8,||Bw3||L2(0,t;H)≤q||G(⋅,x(⋅;f,u3))−G(⋅,x(⋅;f,u2))||L2(0,t;H)

for 0≤t≤T. Let r₂ be a constant satisfying r2≥r1 and

||x(⋅;f,u3)||C([0,t];H)≤r2.

Then, we have

||Bw3||L2(0,t;H)≤q{∫0t||G(τ,x(τ;f,u3))−G(τ,x(τ;f,u2))||2dτ}12≤qL^{∫0tτ||Bu3−Bu2||L2(0,τ;H)2dτ}12≤qL^{∫0tτ||Bw2||L2(0,τ;H)2dτ}12≤qL^{∫0tτ(qL^)2τ22||Bu2−Bu1||L2(0,τ;H)2dτ}12≤q2L^2(∫0tτ32dτ)12||Bu2−Bu1||L2(0,t;H)=q2L^2(t42⋅4)12||Bu2−Bu1||L2(0,t;H).

By proceeding this process, the following holds

||B(un−un+1)||L2(0,t;H)=||Bwn||L2(0,t;H)≤qn−1L^n−1(t2n−22⋅4⋅⋅⋅(2n−2))12||Bu2−Bu1||L2(0,t;H)=(tq2)n−1L^n−11(n−1)!||Bu2−Bu1||L2(0,t;H),

it follows that

∑n=1∞ B u n+1−b un L 2(0,T;h)≤∑n=0∞ ( tq 2)nL^n)1 n! B u2−B u1 L 2(0,T;h)<∞

Therefore, by virtue of Assumption (F1), there exists u*∈L2(0,T;U) such that

(4.10)limn→∞Bun=u*  in L2(0,T;H).

From (4.8), (4.9) it follows that

|ξT−S(T)g−S^G(⋅,x(⋅;f,u2))−S^Bu3|=|ξT−S(T)g−S^G(⋅,x(⋅;f,u1))−S^Bu2+S^Bw2−S^[G(⋅,x(⋅;f,u2))−G(⋅,x(⋅;f,u1))]|<(122+123)ε.

By choosing choose wn∈L2(0,T;U) by the Assumption (S) such that

|S^(G(⋅x(⋅;f,un))−G(⋅x(⋅;f,un−1))−S^Bwn|<ε2n+1,

putting un+1=un−wn, we have

|ξT−S(T)g−S^G(⋅,x(⋅;f,un))−S^Bun+1|<(122+⋅⋅⋅+12n+1)ε, n=12,⋅⋅⋅.

Therefore, for ε>0 there exists integer N such that

||S^BuN+1−S^BuN||<ε2

and

|ξT−S(T)g−S^G(⋅,x(⋅;f,uN))−S^BuN|≤|ξT−S(T)g−S^G(⋅,x(⋅;f,uN))−S^BuN+1|+|S^BuN+1−S^BuN|<(122+⋅⋅⋅+12N+1)ε+ε2≤ε.

Thus the system (4.1) is approximately controllable on [0,T] as N tends to infinity.