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.
© Kyungpook Mathematical Journal. All rights reserved.

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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*uu,         uV.

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*=supvV(l,v)v.

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

u*uu,         uV.

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

Rea(u,u)c0u2-c1u2,

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,vV.

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)={uV:AuH}

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

c0u2Re a(u,u)+c1u2Auu+c1u2(Au+c1u)umax{1,c1}uD(A)u,

where

uD(A)=(Au2+u2)1/2

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

uC0uD(A)1/2u1/2.

Thus we have the following sequence

D(A)VHV*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={xV*:0TAetAx*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

0TAetAx*2dt=0Tu(s)*2ds<.

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

0Tu(t)2dtc0TAu(t)*2dt=c0TAetAx*2dt.

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

uL2(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 toL2(0,T;D(A))W1,2(0,T;H)C([0,T];V)

    and satisfyingxL2(0,T;D(A))W1,2(0,T;H)C1(x0+hL2(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 toL2(0,T;V)W1,2(0,T;V*)C([0,T];H)

    and satisfyingxL2(0,T;V)W1,2(0,T;V*)C1(x0+hL2(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         t0.

First, we consider the following inequalities.

Lemma 2.3

Suppose that hL2(0, T;H) andx(t)=0tS(t-s)h(s)dsfor 0 ≤ tT. Then there exists a constant C2such that

xL2(0,T;D(A))C1hL2(0,T;H),xL2(0,T;H)C2ThL2(0,T;H),

and

xL2(0,T;V)C2ThL2(0,T;H).
Proof

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

xL2(0,T;H)2=0T0tS(t-s)h(s)ds2dtM0T(0th(s)ds)2dtM0Tt0th(s)2dsdtMT220Th(s)2ds

it follows that

xL2(0,T;H)TM/2hL2(0,T;H).

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

xL2(0,T;V)C0C1T(M/2)1/4hL2(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^)L1x-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)L0kL2(0,T)T/2+kL2(0,T)L1TxL2(0,T;V).

Moreover if x, x̂L2(0, T; V ), then

f(·,x)-f(·,x^)L2(0,T;H)kL2(0,T)L1Tx-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)(0T0tk(t-s)g(s,0)ds2dt)1/2+(0T0tk(t-s){g(s,x(s))-g(s,0)}ds2dt)1/2L0kL2(0,T)T/2+kL2(0,T)Tg(·,x)-g(·,0)L2(0,T;H)L0kL2(0,T)T/2+kL2(0,T)L1TxL2(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

xL2(0,T;V)W1,2(0,T;V*)C([0,T];H)

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

xL2(0,T;V)W1,2(0,T;V*)C3(x0+uL2(0,T;U)).
Proof

Let us fix T0 > 0 satisfying

C2L1T0kL2(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)C2T0f(·,x)-f(·,x^)L2(0,T0;H)C2L1T0kL2(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)x0L2(0,T0;V)C1x0

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

x1L2(0,T0;V)C2T0f(·,x)+BuL2(0,T0;H)C2T0(L1T0kL2(0,T0)xL2(0,T0;V)+f(·,0)+BuL2(0,T0;H)).

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

xL2(0,T0;V)(1-C2L1T0kL2(0,T0))-1(C1x0+C2T0f(·,0)+BuL2(0,T0;H)).

Hence, (2.12) holds. Now from

x(T0)=S(T0)x0+0T0S(T0-s){f(s,x(s))+Bu(s)}dsMx0+ML1T0kL2(0,T0)xL2(0,T0;V)+MT0f(·,0)+BuL2(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)Vis 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()¯ 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()¯+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:

    xuL2(0,T;D(A))W1,2(0,T;H)C1uL2(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)kL2(0,T){L0T2+L1TxL2(0,T;V)}+2BWx0

    and

    K(·,x1)-K(·,x2)L2(0,T;H)(1+BW)kL2(0,T)L1Tx1-x2L2(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+L0kL2(0,T)T2)+L0kL2(0,T)T2+BW(x0+L1kL2(0,T)TxL2(0,T;V))+L1kL2(0,T)TxL2(0,T;V)(1+BW)kL2(0,T){L0T2+L1TxL2(0,T;V)}+2BWx0.

    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)BWf(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)kL2(0,T)L1Tx1-x2L2(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<tT,x0,         if         t=0.

    Let us fix T0 > 0 so that

    C0(T032)1/2L1M(1+BWkL2(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)}dsMT0K(·,x1)-K(·,x2)L2(0,T0;H)(1+BW)MT0L1kL2(0,T0)x1-x2L2(0,T0;V)

    and hence in view of (2.3) we have

    Fx1(t)-Fx2(t)L2(0,T0;V)C0Fx1(t)-Fx2(t)L2(0,T0;D(A))1/2Fx1(t)-Fx2(t)L2(0,T0;H)1/2C0Fx1(t)-Fx2(t)L2(0,T0;D(A))1/2(T02)1/2Fx1(t)-Fx2(t)W1,2(0,T0;H)1/2C0(T02)1/2Fx1(t)-Fx2(t)L2(0,T0;D(A))W1,2(0,T0;H)C0(T02)1/2MT0L1(1+BW)kL2(0,T0)x1-x2L2(0,T0;V).

    Here we used the following inequality

    Fx1(t)-Fx2(t)L2(0,T0;H)={0T0Fx1(t)-Fx2(t)2dt}1/2={0T00t(F˙x1(τ)-F˙x2(τ))dτ2dt}1/2{0T0t0tF˙x1(τ)-F˙x2(τ)2dτdt}1/2T02Fx1(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,tT0,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,         t0.

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

    x-yL2(0,T0;V)=0T0S(t-s){-BW(x0,f)+f(s,x)}-Bu(t)}dsL2(0,T0;V)0T0S(t-s){f(·,x)-f(·,0)+f(·,0)}dsL2(0,T0;V)C2T0f(·,x)-f(·,0)L2(0,T0;H)+C2T0f(·,0)L2(0,T0;H)C2T0kL2(0,T0)L1xL2(0,T0;V)+C2T03/22L0kL2(0,T0)C2T0L1kL2(0,T0)x-yL2(0,T0;V)+C2T0L1kL2(0,T0)yL2(0,T0;V)+C2T03/22L0kL2(0,T0).

    Therefore, we have

    x-yL2(0,T0;V)C2T0L1kL2(0,T0)1-C2T0L1kL2(0,T0){yL2(0,T0;V)+2T0L02L1}

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

    xL2(0,T0;V)11-C2T0L1kL2(0,T0){yL2(0,T0;V)+2T0L02L1}11-C2T0L1kL2(0,T0){C3(x0+uL2(0,T0;U))+2T0L02L1}.

    Thus, there exists a constant C4 such that

    xL2(0,T;V)W1,2(0,T;V*)C4(1+x0+uL2(0,T;U)).

    Now from

    x(T0)=S(T0)x0+0T0S(T0-s){-BW(x0,f)+f(s,x)}ds(M+2BW)x0+M(1+BW){L0kL2(0,T)T/2+kL2(0,T)L1TxL2(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)xL2(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-x2L2(0,T;D(A))W1,2(0,T;H)C1{x01-x02+u1-u2L2(0,T;U)+f(·,x1)-f(·,x2)L2(0,T;H)}C1{x01-x02+u1-u2L2(0,T;U)+kL2(0,T)L1Tx1-x2L2(0,T;V)}

    Since

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

    We get

    x1-x2L2(0,T;H)Tx01-x02+T2x1-x2W1,2(0,T;H)

    Hence arguing as in (3.4) we get

    x1-x2L2(0,T;V)C0x1-x2L2(0,T;D(A))1/2x1-x2L2(0,T;H)1/2C0x1-x2L2(0,T;D(A))1/2{T1/4x01-x021/2+(T2)1/2x1-x2W1,2(0,T;H)1/2}C0T1/4x1-x2L2(0,T;D(A))1/2x01-x021/2+C0(T2)1/2x1-x2L2(0,T;D(A))W1,2(0,T;H)2-7/4C0x01-x02+2C0(T2)1/2x1-x2L2(0,T;D(A))W1,2(0,T;H).

    Combining (3.6) and (3.7) we obtain

    x1-x2L2(0,T;D(A))W1,2(0,T;H)C1{x01-x02+kL2(0,T)L1T(2-7/4C0x01-x02+2C0(T2)1/2x1-x2L2(0,T;D(A))W1,2(0,T;H))+u1-u2L2(0,T;U)}=(C1+2-7/4C0kL2(0,T)L1T)x01-x02+C1u1-u2L2(0,T;U)+23/4C0TL1kL2(0,T)x1-x2L2(0,T;D(A))W1,2(0,T;H).

    Suppose that x0nx0 in V and let xn and x be the solution (1.1) with x0n and x0 respectively. Let 0 < T1T be such that

    23/4C0TL1kL2(0,T)<1.

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

    xnx         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

    xnx         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,BuL2(0,t;H)q1pL2(0,t;H),0tT,

    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<T,u(x,0)=u0(x),

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

    uΩ=0,         0<tT,

    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=1naijuxiv¯xj+i=1nβiuxiv¯+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)ξiξjc0ξ2

    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=1nxi(aij(x)xj)+i=1nβi(x)xi+c(x),xΩ

    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,         uD(A).

    It is not difficult to verify that for uH01(Ω) in the sense of distribution and u|∂Ω = 0 for uH01(Ω) also in the sense of distribution(see Lions and Magenes [14]), and

    (Au,v)=-a(u,v),         u,vH01(Ω).

    We consider the nonlinear term g given by

    g(t,u)=γ(t){Dxu(x,t)+φ(u(x,t))},         γC([0,T]),φC(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)={0,0t<α,w(t),αtT

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

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

    Then wL2(0, T;H) and Ŝp = ŜBαw, which is that the controller Bα satisfies Assumption (B). Hence, for the initial data u0W2,2(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.

    References
    1. JP. Aubin. Un thèoréme de compacité. C R Acad Sci Paris., 256(1963), 5042-5044.
    2. K. Balachandran, and JP. Dauer. Controllability of nonlinear systems in Banach spaces: a survey. J Optim Theory Appl., 115(2002), 7-28.
      CrossRef
    3. V. Barbu. Nonlinear semigroups and differential equations in Banach spaces, , Nordhoff Leiden, Netherlands, 1976.
      CrossRef
    4. G. Di Blasio, K. Kunisch, and E. Sinestrari. L2–regularity for parabolic partial integrodifferential equations with delay in the highest-order derivatives. J Math Anal Appl., 102(1984), 38-57.
      CrossRef
    5. PL. Butzer, and H. Berens. Semi-groups of operators and approximation, , Springer-verlag, Belin-Heidelberg-Newyork, 1967.
      CrossRef
    6. A. Carrasco, and H. Leiva. Approximate controllability of a system of parabolic equations with delay. J Math Anal Appl., 345(2008), 845-853.
      CrossRef
    7. RF. Curtain, and H. Zwart. An introduction to infinite dimensional linear systems theory, , Springer-Verlag, New York, 1995.
      CrossRef
    8. JP. Dauer, and NI. Mahmudov. Exact null controllability of semilinear integrodifferential systems in Hilbert spaces. J Math Anal Appl., 299(2004), 322-332.
      CrossRef
    9. JM. Jeong, YC. Kwun, and JY. Park. Approximate controllability for semilinear retarded functional differential equations. J Dynam Control Systems., 5(3)(1999), 329-346.
      CrossRef
    10. JM. Jeong, and HH. Roh. Approximate controllability for semilinear retarded systems. J Math Anal Appl., 321(2006), 961-975.
      CrossRef
    11. JM. Jeong, JE. Ju, and KY. Lee. Controllability for variational inequalities of parabolic type with nonlinear perturbation. J Inequal Appl., (2010) Art. ID 768469, 16 pp.
      CrossRef
    12. JM. Jeong, JR. Kim, and HG. Kim. Regularity for solutions of nonlinear second order evolution equations. J Math Anal Appl., 338(2008), 209-222.
      CrossRef
    13. JL. Lions. Quelques méthodes de résolution des problems aux limites non-linéaires, , Dunnod, Gauthier-Villars, Paris, 1969.
    14. JL. Lions, and E. Magenes. Non-homogeneous boundary value problèmes and applications, , Springer-Verlag, Berlin-heidelberg-New York, 1972.
    15. NI. Mahmudov. Approximate controllability of evolution systems with nonlocal conditions. Nonlinear Anal., 68(2008), 536-546.
      CrossRef
    16. K. Naito. Controllability of semilinear control systems dominated by the linear part. SIAM J Control Optim., 25(1987), 715-722.
      CrossRef
    17. DG. Park, JM. Jeong, and SH. Park. Regularity of parabolic hemivariational inequalities with boundary conditions. J Inequal Appl., (2009) Art. ID 207873, 22 pp.
      CrossRef
    18. JY. Park, and SH. Park. On solutions for a hyperbolic system with differential inclusion and memory source term on the boundary. Nonlinear Anal., 57(2004), 459-472.
      CrossRef
    19. R. Sakthivel, NI. Mahmudov, and JH. Kim. Approximate controllability of nonlinear impulsive differential systems. Rep Math Phys., 60(2007), 85-96.
      CrossRef
    20. N. Sukavanam, and NK. Tomar. Approximate controllability of semilinear delay control systems. Nonlinear Funct Anal Appl., 12(2007), 53-59.
    21. H. Tanabe. Equations of evolution, , Pitman, Boston, Mass.-London, 1979.
    22. H. Triebel. Interpolation theory, function spaces, differential operators, , North-Holland, 1978.
    23. HX. Zhou. Approximate controllability for a class of semilinear abstract equations. SIAM J Control Optim., 21(1983), 551-565.
      CrossRef


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