KYUNGPOOK Math. J. 2019; 59(2): 241-258
Null Controllability of Semilinear Integrodiﬀerential Control Systems in Hilbert Spaces
Ah-ran Park and Jin-Mun Jeong∗
Department of Applied Mathematics, Pukyong National University, Busan 48513, Korea
e-mail : alanida@naver.com and jmjeong@pknu.ac.kr
Received: March 8, 2018; Revised: June 18, 2018; Accepted: June 26, 2018; Published online: June 23, 2019.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

In this paper, we deal with the null controllability of semilinear functional integrodifferential control systems under the Lipschitz continuity of nonlinear terms. Moreover, we establish the regularity and a variation of constant formula for solutions of the given control systems in Hilbert spaces.

Keywords: controllability, semilinear control system, regularity for solution, analytic semigroup, integrodiﬀerential control system.
1. Introduction

Let H and V be two complex Hilbert spaces. Assume that V is a dense subspace in H and the injection of V into H is continuous. The norm on V (resp. H) will be denoted by ||·|| (resp. |·|) respectively. Let A be a continuous linear operator from V into V* which is assumed to satisfy Gårding’s inequality, and generate an analytic semigroup (S(t))t≥0. We study the following the semilinear functional integrodifferential control systems:

${x′(t)=Ax(t)+∫0tk(t-s)g(s,x(s))ds+Bu(t),x(0)=x0.$

Here, a forcing term kL2(0, T; V*), x0H, and g: R+ × VH is a nonlinear mapping as detailed in Section 2.. The controller B is a bounded linear operator from L2(0, T;U) to L2(0, T;H), where U is some Banach space of control variables.

The existence of solutions for a class of semilinear functional integrodifferential control systems has been studied by many authors. For example, one finds parabolic type problems in [3, 17, 18], hyperbolic type problems in [11, 18], and linear cases in [4, 5, 7, 13, 14]. The background of these problems is to serve as an initial value problem for many partial integrodifferential equations which arise in problems connected with heat flow in materials, random dynamical systems, and other physical phenomena. For more details on applications of the theory we refer to the survey of Balachandran and Dauer [2] and the book by Curtain and Zwart [7].

In recent years, as for the controllability of semilinear differential equations, Carrasco and Leiva [6] discussed sufficient conditions for approximate controllability of parabolic equations with delay, Mahmudov [15] in the case that the semilinear equations with nonlocal conditions with condition on the uniform boundedness of the Frechet derivative of nonlinear term, and Sakthivel et al. [19] on impulsive and neutral functional differential equations. As for some considerations on the trajectory set of (1.1) and that of its corresponding linear system (in case g ≡ 0) as matters related to (1.1), we refer the reader to Naito [16], Sukavanam and Tomar [22] and references therein.

In [2, 10] the authors dealt with the approximate controllability of a semilinear control system as a particular case of sufficient conditions for the approximate solvability of semilinear equations by assuming that

• S(t) is compact operator, and

• the linear operator $STu:=∫0TS(t-s)u(s)ds$ has a bounded inverse operator.

The paper [15] replaces the above condition (2) with

• (2-1) The Frechet derivative of nonlinear term is uniformly bounded, and

• (2-2) the corresponding linear system (1.1) in case g ≡ 0 and x0 ≡ 0 is approximately controllable.

In [22] and [23] they studied the control problems of the semilinear equations by assuming conditions (1), (2-2), a Lipschitz continuity of the nonlinear term, and a range condition of the controller B with an inequality constraint.

In this paper we replace the condition (1) by the compactness of the embedding D(A) ⊂ V, and instead of (2.2) and the uniform boundedness f the nonlinear term, we require the following inequality constraint on the range condition of the controller B: for any pL2(0, T;H) there exists a uL2(0, T;U) such that

$STp=STBu.$

In Section 2, we will obtain that most parts of the regularity for parabolic linear equations can also be applicable to (1.1) with nonlinear perturbations. The approach used here is similar to that developed in [8, 9, 13, 15] on the general semilnear evolution equations. Moreover, in Section 3, we establish the null controllability of semilinear functional integrodifferential control systems (1.1) under the Lipschitz continuity instead of the uniform boundedness of the Frechet derivative of nonlinear term. It is useful for physical applications of the given equations.

2. Regularity for Solutions

If H is identified with its dual space we may write VHV* 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, v2H. For the sake of simplicity, we may consider

$‖u‖*≤ ∣u∣ ≤ ‖u‖, u∈V.$

For lV* we denote (l, v) by the value l(v) of l at vV. 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 the brevity, we may regard that

$‖u‖*≤ ∣u∣ ≤ ‖u‖, ∀u∈V.$

Let a(·, ·) be a bounded sesquilinear form defined in V × V and satisfying Gårding’s inequality

$Re a(u,u)≥c0‖u‖2-c1∣u∣2,$

where c0 > 0 and c1 is a real number. Let A be the operator associated with this sesquilinear form:

$(Au,v)=-a(u,v), u, v∈V.$

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. Moreover, for each T > 0, by using interpolation theory we have

$L2(0,T;V)∩W1,2(0,T;V*)⊂C([0,T];H).$

From the following inequalities

$c0‖u‖2≤Re a(u,u)+c1∣u∣2≤ ∣Au∣∣u∣+c1∣u∣2≤(∣Au∣+c1∣u∣)∣u∣ ≤max{1,c1}‖u‖D(A)∣u∣,$

where

$‖u‖D(A)=(∣Au∣2+∣u2∣)1/2$

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

$‖u‖ ≤C0‖u‖D(A)1/2∣u∣1/2.$

Thus we have the following sequence

$D(A)⊂V⊂H⊂V*⊂D(A)*$

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

### Lemma 2.1

With the notations (2.1)–(2.3), we have

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

where (V, V*)1/2,2denotes the real interpolation space between V and V*(Section 1.3.3 of [21]).

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 c1 = 0 and hence the closed half plane {λ: Re λ ≥ 0} is contained in the resolvent set of A.

### Lemma 2.2

Let T > 0. Then

$H={x∈V*:∫0T‖AetAx‖*2dt<∞}.$
Proof

Put u(t) = etAx for xH. Then,

$u′(t)=Au(t), u(0)=x.$

As in Theorem 4.1 of Chapter 4 of [14], the solution u belongs to L2(0, T; V ) ∩ W1,2(0, T; V*), hence we obtain that

$∫0T‖AetAx‖*2dt=∫0T‖u′(s)‖*2ds<∞.$

Conversely, suppose that xV* and $∫0T‖AetAx‖*2dt<∞$. Put u(t) = etAx. Then since A is an isomorphism operator from V to V* there exists a constant c > 0 such that

$∫0T‖u(t)‖2dt≤c∫0T‖Au(t)‖*2dt=c∫0T‖AetAx‖*2dt.$

From the assumptions and u′ (t) = AetAx it follows

$u∈L2(0,T;V)∩W1,2(0,T;V*)⊂C([0,T];H).$

Therefore, x = u(0) ∈ H.

By Lemma 2.1, from Theorem 3.5.3 of Butzer and Berens [5], we can see that

$(V,V*)1/2,2=H.$

Consider the following linear system

${x′(t)=Ax(t)+h(t),x(0)=x0.$

By virtue of Theorem 3.3 of [4](or Theorem 3.1 of [9]), we have the following result on the corresponding linear equation of (2.4).

### Proposition 2.1

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

• Let V = (D(A), H)1/2,2where (D(A), H)1/2,2is the real interpolation space between D(A) and H(see [[21]; section 1.3.3], or Lemma 2.1). For x0V and hL2(0, T;H), T > 0, there exists a unique solution x of (2.4) belonging to$L2(0,T;D(A))∩W1,2(0,T;H)⊂C([0,T];V)$

and satisfying$‖x‖L2(0,T;D(A))∩W1,2(0,T;H)≤C1(‖x0‖+‖h‖L2(0,T;H)),$

where C1is a constant depending on T.

• Let x0H and hL2(0, T; V*), T > 0. Then there exists a unique solution x of (2.4) belonging to$L2(0,T;V)∩W1,2(0,T;V*)⊂C([0,T];H)$

and satisfying$‖x‖L2(0,T;V)∩W1,2(0,T;V*)≤C1(∣x0∣+‖h‖L2(0,T;V*)),$

where C1is a constant depending on T.

For the sake of simplicity, we assume that solution semigroup S(t) generated by A is uniformly bounded:

$‖S(t)‖ ≤M t≥0.$

First, we consider the following inequalities.

### Lemma 2.3

Suppose that hL2(0, T;H) and$x(t)=∫0tS(t-s)h(s)ds$for 0 ≤ tT. Then there exists a constant C2such that

$‖x‖L2(0,T;D(A))≤C1‖h‖L2(0,T;H),$$‖x‖L2(0,T;H)≤C2T‖h‖L2(0,T;H),$

and

$‖x‖L2(0,T;V)≤C2T‖h‖L2(0,T;H).$
Proof

The assertion (2.7) is immediately obtained by (2.5). Since

$‖x‖L2(0,T;H)2=∫0T∣∫0tS(t-s)h(s)ds∣2dt≤M∫0T(∫0t∣h(s)∣ds)2dt≤M∫0Tt∫0t∣h(s)∣2dsdt≤MT22∫0T∣h(s)∣2ds$

it follows that

$‖x‖L2(0,T;H)≤TM/2‖h‖L2(0,T;H).$

From (2.3), (2.7), and (2.8) it holds that

$‖x‖L2(0,T;V)≤C0C1T(M/2)1/4‖h‖L2(0,T;H).$

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

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

the proof is complete.

Consider the following initial value problem for the abstract semilinear parabolic equation (1.1). Let U be a Banach space and the controller operator B be a bounded linear operator from U to H.

Let g: R+ × VH be a nonlinear mapping satisfying the following:

• (F1) For any xV, the mapping g(·, x) is strongly measurable;

• (F2) There exist positive constants L0, L1 such that $∣g(t,x)-g(t,x^)∣ ≤L1‖x-x^‖,∣g(t,0)∣ ≤L0$

for all tR+, and x, x̂V.

For xL2(0, T; V ), we set

$f(t,x)=∫0tk(t-s)g(s,x(s))ds,$

where k belongs to L2(0, T).

### Lemma 2.4

Let xL2(0, T; V ) for any T > 0. Then f(·, x) ∈ L2(0, T;H) and

$‖f(·,x)‖L2(0,T;H)≤L0‖k‖L2(0,T)T/2+‖k‖L2(0,T)L1T‖x‖L2(0,T;V).$

Moreover if x, x̂L2(0, T; V ), then

$‖f(·,x)-f(·,x^)‖L2(0,T;H)≤ ‖k‖L2(0,T)L1T‖x-x^‖L2(0,T;V).$
Proof

From (F1), (F2), and using the Hölder inequality, it is easily seen that

$‖f(·,x)‖L2(0,T;H)≤ ‖f(·,0)‖+‖f(·,x)-f(·,0)‖≤(∫0T∣∫0tk(t-s)g(s,0)ds∣2dt)1/2+(∫0T∣∫0tk(t-s){g(s,x(s))-g(s,0)}ds∣2dt)1/2≤L0‖k‖L2(0,T)T/2+‖k‖L2(0,T)T‖g(·,x)-g(·,0)‖L2(0,T;H)≤L0‖k‖L2(0,T)T/2+‖k‖L2(0,T)L1T‖x‖L2(0,T;V).$

The proof of (2.11) is similar.

### Theorem 2.1

Under the assumptions (F1), and (F2) for the nonlinear mapping f, as given by

$f(t,x)=∫0tk(t-s)g(s,x(s))ds,$

there exists a unique solution x of (1.1) such that

$x∈L2(0,T;V)∩W1,2(0,T;V*)⊂C([0,T];H)$

for any x0H. Moreover, there exists a constant C3such that

$‖x‖L2(0,T;V)∩W1,2(0,T;V*)≤C3(∣x0∣+‖u‖L2(0,T;U)).$
Proof

Let us fix T0 > 0 satisfying

$C2L1T0‖k‖L2(0,T)<1$

with the constant C2 in Lemma 2.3. Let y be the solution of

$y(t)=S(t)x0+∫0tS(t-s){f(s,x(s))+Bu(s)}ds.$

We are going to show that xy is strictly contractive from L2(0, T0; V ) to itself. Let y, ŷ belong to V with the same initial condition in [0, T0]. Then from assumption (2.9), (2.11) and

$y(t)-y^(t)=∫0tS(t-s){f(s,x(s))-f(s,x^(s))}ds$

we have

$‖y-y^‖L2(0,T0;V)≤C2T0‖f(·,x)-f(·,x^)‖L2(0,T0;H)≤C2L1T0‖k‖L2(0,T0)‖x(·)-x^(·)‖L2(0,T0;V).$

So by virtue of the condition (2.13) the contraction mapping principle gives that the solution of (1.1) exists uniquely in [0, T0]. Let x be a solution of (1.1) and x0H. Then there exists a constant C1 such that

$‖S(t)x0‖L2(0,T0;V)≤C1∣x0∣$

in view of Proposition 2.1. Let

$x1(t)=∫0tS(t-s){f(s,x(s))+Bu(s)}ds.$

Then from (2.11), it follows

$‖x1‖L2(0,T0;V)≤C2T0‖f(·,x)+Bu‖L2(0,T0;H)≤C2T0(L1T0‖k‖L2(0,T0)‖x‖L2(0,T0;V)+‖f(·,0)+Bu‖L2(0,T0;H)).$

Thus, combining (2.14) with (2.15) we have

$‖x‖L2(0,T0;V)≤(1-C2L1T0‖k‖L2(0,T0))-1(C1∣x0∣+C2T0‖f(·,0)+Bu‖L2(0,T0;H)).$

Hence, (2.12) holds. Now from

$∣x(T0)∣=∣S(T0)x0+∫0T0S(T0-s){f(s,x(s))+Bu(s)}ds∣≤M∣x0∣+ML1T0‖k‖L2(0,T0)‖x‖L2(0,T0;V)+MT0‖f(·,0)+Bu‖L2(0,T0;H),$

since the condition (2.13) is independent of initial values, the solution of (1.1) can be extended to the interval [0, nT0] for every natural number n. An analogous estimate to (2.12) holds for the solution in [0, nT0], and hence for the initial value xnT0 in the interval [nT0, (n + 1)T0].

3. Null Controllability of Semilinear Systems

Let S(t) be the analytic semigroup generated by the principal operator A. We define the linear operator Ŝ from L2(0, T;H) to H by

$S^Tp=∫0TS(T-s)p(s)ds$

for pL2(0, T;H). Let x(T; f, u) be a state value of the system (1.1) at time T corresponding to the nonlinear term f and the control u. Then the solution x(T; f, u) of (1.1) is represented by

$x(T;f,u)=S(T)x0+S^Tf(·,x)+S^TBu.$

### Definition 3.1

Equation (1.1) is said to be null controllable at time T > 0 if for a given x0H there exists a control uL2(0, T;U) such that x(T; f, u) = 0.

Let GT = ŜTB. Here, we remark that G is a bounded linear operator(see Proposition 2.1 or Theorem 2.1) but necessary one-to-one. Denote the orthogonal complement in L2(0, T;U) by [kerGT]. Let G: [kerGT] → ImGT be the restriction of GT to [kerGT]. Then we know that G is necessary a one-to-one operator.

For any (x, h) ∈ H × L2(0, T; V*), define

$N(x,h)=S(T)x0+∫0TS(T-s)h(s)ds:H×L2(0,T;V*)→H,W(x,h)≡(G)-1N=(G)-1(S(T)x0+∫0TS(T-s)h(s)ds).$

First, we consider the following linear control equation with a general forcing term h:

${x′(t)=Ax(t)+Bu(t)+h(t),x(0)=x0.$

The following is immediately seen from Definition 3.1.

### Lemma 3.1

The linear system (3.1) is null controllable at time T > 0 if

$Im G(=S^TB)⊃Im N.$

We need the following hypothesis:

• Let us assume the natural assumption that the embedding $D(A)⊂V is compact.$

• For any pL2(0, T;H) there exists a uL2(0, T;U) such that $S^Tp=S^TBu.$

• ### Remak 3.1

Denote the kernel of the operator ŜT by N, which is a closed subspace in L2(0, T;H), and its orthogonal space in L2(0, T;H) by N. Let ℬ be defined by (ℬu)(·) = Bu(·). Denote the range of the operator ℬ by R(ℬ) and its closure by $R(ℬ)¯$ in L2(0, T;H). As seen in [9, 16], it is easily known that the hypothesis (B) is equivalent to the following condition: $L2(0,T;H)=R(ℬ)¯+N$.

### Lemma 3.2

Let us assume the hypothesis (B). Then we have

$D(A)⊂Im G.$
Proof

Let x0D(A) and put p(s) = (x0sAx0)/T. Then pL2(0, T;H) and

$x0=∫0TS(T-s)p(s)ds(=S^Tp).$

Hence, from (B) we can choose a control u0L2(0, T;U) satisfying

$S^Tp=S^TBu0,$

which implies D(A) ⊂ ImG.

### Theorem 3.1

For uL2(0, T;U), let xu = GTu with xu(0) = 0. Under Assumption(A), we have the mapping GT: uxu is compact from L2(0, T;U) to L2(0, T; V ) ⊂ L2(0, T;H).

Proof

If uL2(0, T;U), with the aid of Proposition 2.1 (or Lemma 2.3), we have xuL2(0, T;D(A)) ∩ W1,2(0, T; V*) and satisfy the following inequality:

$‖xu‖L2(0,T;D(A))∩W1,2(0,T;H)≤C1‖u‖L2(0,T;U),$

where C1 is the constant in Proposition 2.1. Hence if u is bounded in L2(0, T;U), then so is xu in L2(0, T;D(A)) ∩ W1,2(0, T;H) by the above inequality. Since D(A) is compactly embedded in V by assumption, the embedding

$L2(0,T;D(A))∩W1,2(0,T;H)⊂L2(0,T;V)$

is compact in view of Theorem 2 of J. P. Aubin [1].

Therefore, if we define the operator xu = GTu, then GT is a compact mapping from L2(0, T;U) to L2(0, T; V).

The following lemma is obtained from the proof of Lemma 3 of [8].

### Lemma 3.3

Under Hypothesis (B), we have

$W(x,h):H×L2(0,T;V*)→L2(0,T;U).$

is bounded and the control

$u(t)=-W(x0,h)$

transfers the linear system (3.1) from x0D(A) to 0.

Proof

By the definition of G, let

$G:[kerGT]⊥→ImGT$

be the restriction of GT to [kerGT]. G is necessarily a one-to-one operator. Here, we remark that by Lemma 3.2, S(T)x0 ∈ ImG since S(T)x0D(A) for x0D(A). Define

$W(x,h):H×L2(0,T;V*)→L2(0,T;U)$

by W(x, h) ≡ (G)−1N(x, h). From Theorem 3.1, it follows that ImGT is closed and [kerGT] is obviously closed. Hence, the inverse mapping theorem says that G−1 is a bounded linear operator, and so is W.

Since the operator BW is bounded, for the sake of simplicity, we assume that

$∣BW(x0,f)∣ ≤ ‖BW‖(∣x0∣+∣f∣).$

### Lemma 3.4

For xL2(0, T; V ), we set

$K(s,x)=-BW(x0,f(s,x(s))+f(s,x(s)).$

Then we obtain the following:

$‖K(·,x)‖L2(0,T;H)≤(1+‖BW‖)‖k‖L2(0,T){L0T2+L1T‖x‖L2(0,T;V)}+2‖BW‖ ∣x0∣$

and

$‖K(·,x1)-K(·,x2)‖L2(0,T;H)≤(1+‖BW‖)‖k‖L2(0,T)L1T‖x1-x2‖L2(0,T;V).$
Proof

From (2.10), (2.11) it is easily seen that

$‖K(·,x)‖L2(0,T;H)≤ ‖K(·,0)‖+‖K(·,x)-K(·,0)‖≤ ‖-BW(x0,f(·,0))+f(·,0)‖+‖-BW(x0,f(·,x))+BW(x0,f(·,0))+f(·,x)-f(·,0)‖≤ ‖BW‖(∣x0∣+L0‖k‖L2(0,T)T2)+L0‖k‖L2(0,T)T2+‖BW‖(∣x0∣+L1‖k‖L2(0,T)T‖x‖L2(0,T;V))+L1‖k‖L2(0,T)T‖x‖L2(0,T;V)≤(1+‖BW‖)‖k‖L2(0,T){L0T2+L1T‖x‖L2(0,T;V)}+2‖BW‖ ∣x0∣.$

Moreover, we obtain

$‖K(·,x1)-K(·,x2)‖L2(0,T;H)=‖-BW(x0,f(s,x1(s)))+f(s,x1(s))+BW(x0,f(s,x2(s))-f(s,x2(s))‖L2(0,T;H)≤ ‖BW‖‖f(s,x1(s))-f(s,x2(s))‖L2(0,T;H)+‖f(s,x1(s))-f(s,x2(s))‖L2(0,T;H)≤ (1+‖BW‖)‖k‖L2(0,T)L1T‖x1-x2‖L2(0,T;V).$

### Theorem 3.2

Assume the assumptions (F1-2), (A), and (B) be satisfied. Then for the initial data x0D(A) the system (1.1) is null controllable at time T > 0.

Proof

Define the operator F on L2(0, T; V ) by

$Fx(t)={S(t)x0+∫0tS(t-s){-BW(x0,f)+f(s,x)}ds, 0

Let us fix T0 > 0 so that

$C0(T032)1/2L1M(1+‖BW‖‖k‖L2(0,T))<1,$

where C0 is constant in (2.3). We are going to show that xFx is strictly contractive from L2(0, T0; V ) to itself if the condition (3.3) is satisfied. Let Fx1, Fx2 be the solutions of the above equation with x replaced by x1, x2L2(0, T0; V) respectively. From (3.3) it follows that

$‖Fx1(t)-Fx2(t)‖L2(0,T0;D(A))∩W1,2(0,T0;H)≤ ‖∫0T0S(T0-s){K(s,x1)-K(s,x2)}ds‖≤M​T0‖K(·,x1)-K(·,x2)‖L2(0,T0;H)≤(1+‖BW‖)MT0L1‖k‖L2(0,T0)‖x1-x2‖L2(0,T0;V)$

and hence in view of (2.3) we have

$‖Fx1(t)-Fx2(t)‖L2(0,T0;V)≤C0‖Fx1(t)-Fx2(t)‖L2(0,T0;D(A))1/2‖Fx1(t)-Fx2(t)‖L2(0,T0;H)1/2≤C0‖Fx1(t)-Fx2(t)‖L2(0,T0;D(A))1/2(T02)1/2‖Fx1(t)-Fx2(t)‖W1,2(0,T0;H)1/2≤C0(T02)1/2‖Fx1(t)-Fx2(t)‖L2(0,T0;D(A))∩W1,2(0,T0;H)≤C0(T02)1/2MT0L1(1+‖BW‖)‖k‖L2(0,T0)‖x1-x2‖L2(0,T0;V).$

Here we used the following inequality

$‖Fx1(t)-Fx2(t)‖L2(0,T0;H)={∫0T0∣Fx1(t)-Fx2(t)∣2dt}1/2={∫0T0∣∫0t(F˙x1(τ)-F˙x2(τ))dτ∣2dt}1/2≤{∫0T0t∫0t∣F˙x1(τ)-F˙x2(τ)∣2dτdt}1/2≤T02‖Fx1(t)-Fx2(t)‖W1,2(0,T0;H).$

Hence, by virtue of (3.4) the contraction mapping principle gives that the operator F has unique solution in [0, T0], that is, x is the solution of the following equation:

${x(t)=S(t)x0+∫0tS(t-s){-BW(x0,f)+f(s,x)}ds, t≤T0,x(0)=x0, if t=0.$

Next we establish the estimates of solution. Let x(·) be the solution of (3.5) in the (0, T0) and y(·) be the solution of (3.1) with the control u(t) = −W(x0, k) as in (3.2), i.e., the solution y of (3.1) is represented by

$y(t)=S(t)x0-∫0tS(t-s)BW(x0,f)ds, t≥0.$

Thus, the arguing as in the proof of Lemmas 2.3, 2,4, we have

$‖x-y‖L2(0,T0;V)=‖∫0T0S(t-s){-BW(x0,f)+f(s,x)}-Bu(t)}ds‖L2(0,T0;V)≤‖∫0T0S(t-s){f(·,x)-f(·,0)+f(·,0)}ds‖L2(0,T0;V)≤C2T0‖f(·,x)-f(·,0)‖L2(0,T0;H)+C2T0‖f(·,0)‖L2(0,T0;H)≤C2T0‖k‖L2(0,T0)L1‖x‖L2(0,T0;V)+C2T03/22L0‖k‖L2(0,T0)≤C2T0L1‖k‖L2(0,T0)‖x-y‖L2(0,T0;V)+C2T0L1‖k‖L2(0,T0)‖y‖L2(0,T0;V)+C2T03/22L0‖k‖L2(0,T0).$

Therefore, we have

$‖x-y‖L2(0,T0;V)≤C2T0L1‖k‖L2(0,T0)1-C2T0L1‖k‖L2(0,T0){‖y‖L2(0,T0;V)+2T0L02L1}$

and hence with the aid of Lemma 2.3, or Theorem 2.1

$‖x‖L2(0,T0;V)≤11-C2T0L1‖k‖L2(0,T0){‖y‖L2(0,T0;V)+2T0L02L1}≤11-C2T0L1‖k‖L2(0,T0){C3(∣x0∣+‖u‖L2(0,T0;U))+2T0L02L1}.$

Thus, there exists a constant C4 such that

$‖x‖L2(0,T;V)∩W1,2(0,T;V*)≤C4(1+∣x0∣+‖u‖L2(0,T;U)).$

Now from

$∣x(T0)∣=∣S(T0)x0∣+∣∫0T0S(T0-s){-BW(x0,f)+f(s,x)}ds∣≤(M+2‖BW‖)∣x0∣+M(1+‖BW‖){L0‖k‖L2(0,T)T/2+‖k‖L2(0,T)L1T‖x‖L2(0,T;V)},$

since the condition (3.3) is independent of initial values, the solution of (1.1) can be extended to the interval [0, nT0] for every natural number n. That is, an analogous estimate to (3.6) holds for the solution in [0, nT0], and hence for the initial value xnT0 in the interval [nT0, (n + 1)T0], which means that the system (1.1) is null controllable at time T > 0 with the control u = −W(x0, f).

### Theorem 3.3

Let the assumption (F1), (F2) be satisfied and (x0, u) ∈ V × L2(0, T;U), Then the solution x of the equation (1.1) belongs to x ∈ L2(0, T;D(A)) ∩ W1,2(0, T;H) and the mapping

$V×L2(0,T;U)∋(x0,u)↦x∈L2(0,T;D(A))∩W1,2(0,T;H)⊂C([0,T];V)$

is Lipschitz continuous.

Proof

It is easy to show that if x0V and f(·, x) ∈ L2(0, T;H), then x belongs to L2(0, T;D(A)) ∩ W1,2(0, T;H). Let $(x0i,ui)∈H×L2(0,T;U)$ and xi be the solution of (1.1) with (x0, u) in place of ($x0i$, ui) for i = 1, 2. Then

${(x1-x2)′(t)=A(x1-x2)(t)+f(t,x1(t))-f(t,x2(t))+B(u1-u2)(t), t>0,(x1-x2)(0)=x01-x02.$

Hence in view of proposition 2.1 and lemma 2.4, we have

$‖x1-x2‖L2(0,T;D(A))∩W1,2(0,T;H)≤C1{‖x01-x02‖+‖u1-u2‖L2(0,T;U)+‖f(·,x1)-f(·,x2)‖L2(0,T;H)}≤C1{‖x01-x02‖+‖u1-u2‖L2(0,T;U)+‖k‖L2(0,T)L1T‖x1-x2‖L2(0,T;V)}$

Since

$x1(t)-x2(t)=x01-x02+∫0t(x˙1(s)-x˙2(s))ds$

We get

$‖x1-x2‖L2(0,T;H)≤T∣x01-x02∣+T2‖x1-x2‖W1,2(0,T;H)$

Hence arguing as in (3.4) we get

$‖x1-x2‖L2(0,T;V)≤C0‖x1-x2‖L2(0,T;D(A))1/2‖x1-x2‖L2(0,T;H)1/2≤C0‖x1-x2‖L2(0,T;D(A))1/2{T1/4∣x01-x02∣1/2+(T2)1/2‖x1-x2‖W1,2(0,T;H)1/2}≤C0T1/4‖x1-x2‖L2(0,T;D(A))1/2∣x01-x02∣1/2+C0(T2)1/2‖x1-x2‖L2(0,T;D(A))∩W1,2(0,T;H)≤2-7/4C0∣x01-x02∣+2C0(T2)1/2‖x1-x2‖L2(0,T;D(A))∩W1,2(0,T;H).$

Combining (3.6) and (3.7) we obtain

$‖x1-x2‖L2(0,T;D(A))∩W1,2(0,T;H)≤C1{‖x01-x02‖+‖k‖L2(0,T)L1T(2-7/4C0‖x01-x02‖+2C0(T2)1/2‖x1-x2‖L2(0,T;D(A))∩W1,2(0,T;H))+‖u1-u2‖L2(0,T;U)}=(C1+2-7/4C0‖k‖L2(0,T)L1T)‖x01-x02‖+C1‖u1-u2‖L2(0,T;U)+23/4C0TL1‖k‖L2(0,T)‖x1-x2‖L2(0,T;D(A))∩W1,2(0,T;H).$

Suppose that $x0n→x0$ in V and let xn and x be the solution (1.1) with $x0n$ and x0 respectively. Let 0 < T1T be such that

$23/4C0TL1‖k‖L2(0,T)<1.$

Then by virtue of (3.8) with T replaced by T1 we see that

$xn→x in L2(0,T;D(A))∩W1,2(0,T;H).$

This implies that (xn(T1), (xn)T1) → (x(T1), xT1) in V × L2(0, T;D(A)). Hence the same argument shows that

$xn→x in L2(T1,min{2T1,T1};D(A))∩W1,2(T1,min{2T1,T1};H).$

Repeating this process we conclude that xnx in L2(0, T;D(A)) ∩ W1,2(0, T;H) for any T > 0.

### Remark 3.2

Let us we assume the following hypothesis:

For any ɛ > 0 and pL2(0, T;H) there exists a uL2(0, T;U) such that

${∣S^p-S^Bu∣ <ɛ, ‖Bu‖L2(0,t;H)≤q1‖p‖L2(0,t;H),0≤t≤T,$

where q1 is a constant independent of p.

Then, as seen in [12], we note that for every desired final state x1H and ε > 0 there exists a control function uL2(0, T;U) such that the solution x(T; u) of (1.1) satisfies |x(T; u) − x1| < ε, i.e., the system (1.1) is said to be approximately controllable in the time interval [0, T].

### Example 3.1

We consider an application of the results obtained in the preceding sections to a class of partial functional integrodifferential systems with delay terms dealt with by Naito [16] and Zhou [23]:

${∂∂tu(x,t)=A(x,Dx)u(x,t)+∫0tk(t-s)g(s,u(x,s))ds+Bαw(t), (x,t)∈Ω×(0,T), 0<α

The boundary condition attached to (3.9) is given by Dirichlet boundary condition

$u∣∂Ω=0, 0

and k belongs to L2(0, T). Here, Ω ⊂ ℛn is a bounded domain with smooth boundary ∂Ω. We set H = L2(Ω) and $V=H01(Ω)$. Let b(u, v) be the sesquilinear form in $H01(Ω)×H01(Ω)$ defined by

$a(u,v)=∫Ω{∑i,j=1naij∂u∂xi∂v¯∂xj+∑i=1nβi∂u∂xiv¯+cuv¯}dx.$

Here, we assume that aij is a real-valued and smooth function for each i, j = 1, ···, n, and aij(x) = aji(x) for each x ∈ Ω̄ and {aij(x)} is positive definite uniformly in Ω, i.e., there exists a positive number c0 such that

$∑i,j=1naij(x)ξiξj≥c0∣ξ∣2$

for all x ∈ Ω̄ and all real vectors ξ. Let biL(Ω) and cL(Ω). As is well known this sesquilinear form a(·, ·) is bounded and satisfying the Gårding’s inequality (2.2)(see e.g. Tanabe [20]. Let

$A(x,Dx)=-∑i,j=1n∂∂xi(aij(x)∂∂xj)+∑i=1nβi(x)∂∂xi+c(x), x∈Ω$

be the associated uniformly elliptic differential operator of second order. Then the realization of In L2(Ω) under the Dirichlet boundary condition is exactly A, i.e.,

$D(A)=W2,2(Ω)∩H01(Ω),Au=-A(x,Dx)u, ∀u∈D(A).$

It is not difficult to verify that for $u∈H01(Ω)$ in the sense of distribution and u|∂Ω = 0 for $u∈H01(Ω)$ also in the sense of distribution(see Lions and Magenes [14]), and

$(Au,v)=-a(u,v), u,v∈H01(Ω).$

We consider the nonlinear term g given by

$g(t,u)=γ(t){‖Dxu(x,t)‖+φ(u(x,t))}, γ∈C([0,T]), φ∈C(H).$

Then g is not uniformly bounded and satisfies hypotheses (F1)and (F2). Let U = H be the space of control variables and let us define the intercept controller operator Bα(0 < α < T) on L2(0, T;H) by

$Bαw(t)={0,0≤t<α,w(t),α≤t≤T$

for wL2(0, T;H). For a given pL2(0, T;H) let us choose a control function w satisfying

$w(t)={0, 0≤t<α,p(t)+αT-αS(t-αT-α(t-α))p(αT-α(t-α)), α≤t≤T.$

Then wL2(0, T;H) and Ŝp = ŜBαw, which is that the controller Bα satisfies Assumption (B). Hence, for the initial data $u0∈W2,2(Ω)∩H01(Ω)$ the system (3.9) is null controllable.

Acknowledgements

Authors would like to thank the referees for their useful suggestions which have significantly improved the paper.

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