Approximate controllability of neutral functional differential system with unbounded delay

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PII. S016117120100597X http://ijmms.hindawi.com © Hindawi Publishing Corp.

APPROXIMATE CONTROLLABILITY OF NEUTRAL FUNCTIONAL

DIFFERENTIAL SYSTEM WITH UNBOUNDED DELAY

JONG YEOUL PARK and SANG NAM KANG

(Received 9 October 2000)

Abstract.We consider a class of control systems governed by the neutral functional dif-ferential equation with unbounded delay and study the approximate controllability of the system. An example is given to illustrate the result.

2000 Mathematics Subject Classiﬁcation. 93B05, 93C20.

1. Introduction. LetᏮbe an abstract phase space. Consider the following nonlinear control equation:

d dt

x(t)+Ft, xt=Ax(t)+Gt, xt+Bv(t), 0< t < T , x0=ϕ∈Ω, (1.1)

whereF , G:[0, T ]×Ꮾ→Xare continuous functions,Ais the inﬁnitesimal generator of an analytic semigroupS(·)of bounded linear operators on a Banach spaceX, the state functionx(t), 0≤t≤T, takes values inX, and the control functionv(·)is given inL2₍₀_{, T} _:_{V )}_{, which is a Banach space of admissible control functions, with}_V _{as a}

Banach space. Also,Bis a bounded linear operator fromL2₍₀_{, T}_:_{V )}_into_L2₍₀_{, T}_:_X)_.

The theory of functional differential equations with unbounded delay has been stud-ied by many authors. Hale and Kato [1] have established the local existence and contin-uation of solutions for retarded eqcontin-uations with infinite delay with initial values in an abstract phase space. Henríquez [2] proved the existence of solutions and the periodic solutions of a class of partial functional differential equations. Recently, Hernández and Henríquez [3] have studied the existence problem for partial neutral functional differential equations with initial values in phase space.

In this paper, we study the approximate controllability of system (1.1) by using the results of Hernández and Henríquez [3]. Similar results on controllability and approximate controllability of linear and nonlinear control systems have been studied in [5,6,8].

To study the nonlinear system (1.1), we assume that the historiesxt:(−∞,0]→X, xt(θ):=x(t+θ), belong to some abstract phase spaceᏮ, that is, a phase space deﬁned

axiomatically. Here,Ꮾis a linear space of functions mapping(−∞,0]intoXendowed with a seminorm·ᏮandᏮsatisﬁes the following axioms (see [1]):

(A1) Ifx:(−∞, σ+a)→X,a >0, is continuous on[σ , σ+a),σis ﬁxed, andxσ∈Ꮾ,

then for every t∈[σ , σ+a)the following conditions hold: (i) xtis inᏮ,

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(iii) xtᏮ≤K(t−σ )sup{x(s):σ≤s≤t}+M(t−σ )xσᏮ, whereH≥0 is a constant;K, M:[0,∞)→[0,∞),Kis continuous andM is locally bounded, andH,K, andMare independent ofx(·).

(A2) For the functionx(·)in (A1),xtis aᏮ-valued continuous function on[σ , σ+a).

(A3) The spaceᏮis complete.

Denote by ˆᏮthe quotient Banach spaceᏮ/ · Ꮾand ifϕ∈Ꮾ, we write ˆϕfor the coset determined byϕ. Examples of phase space satisfying the above axioms can be found in [3,4].

2. Preliminaries. Let the norm of the spaceXbe denoted by·and for the other spaces we use·L2(0,T:X),·L2(0,T:V ),·∞, and so on.

We assume the following hypotheses:

(H1) −Ais the inﬁnitesimal generator of an analytic semigroupS(·)of bounded

linear operator onX, where the semigroupS(t)is uniformly bounded,S(t) ≤Mfor some constantM≥1 and for everyt≥0, and 0∈ρ(A).

(H2) There exist constantsβ∈(0,1)andL1≥0, such that the functionF:[0, T ]×Ꮾ→ XisXβ-valued and satisﬁes the Lipschitz condition

₍₋_A)β_F_{t, ψ} 1

−(−A)βFs, ψ2≤L1

|t−s|+ψ1−ψ2_Ꮾ

, (2.1)

for every 0≤s,t≤T, andψ1, ψ2∈Ꮾ, and

µ=1−L1(−A)−β·K∞ (2.2)

is positive.

(H3) The nonlinear operatorG:[0, T ]×Ꮾ→Xsatisﬁes the Lipschitz condition _G

s, ψ1

−Gs, ψ2≤L2ψ1−ψ2_Ꮾ

, (2.3)

for every 0≤s≤T, andψ1, ψ2∈Ꮾ,

(H4) Letϕ∈Ꮾbe a function such thatϕ(0)∈D(A)andF ([0, T )×Ꮾ)⊆D(A), a.e. t∈[0, T )and

Λ(t)= t

0(−A)S(t−s)F

s, xsds (2.4)

is diﬀerentiable a.e. on[0, T ), that is,Λ(t)∈D(A).

(H5) The operatorBis a bounded linear operator fromL2(0, T:V )toL2(0, T:X).

Under the above hypotheses it is well known [3] that for eachu∈L2₍₀_{, T}_:_X)_there

exists a unique mild solution

xt(u)=S(t)φ(0)+F (0, φ)−Ft, xt(u)− t

0AS(t−s)F

s, xs(u)ds

+

t

0S(t−s)G

s, xs(u)ds+ t

0S(t−s)u(s)ds.

(2.5)

The solution mappingWfromL2₍₀_{, T}_:_X)_to_C(₀_{, T}_:_X)_{can be deﬁned by}

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We also deﬁne the continuous linear operatorΦfromL2₍₀_{, T}_:_X)_to_X_by

Φp=

T

0

S(T−s)p(s)ds, forp∈L2₍₀_{, T}_:_X). _(2.7)

Deﬁnition_2.1. _{Let the reachable set of the system (}_1.1_{) at time}_T _be

KT(G)=xT(Bv); v∈L2(0, T:V ), (2.8)

wherext(Bv)is a mild solution which satisﬁes (2.5) withu=Bv.

Deﬁnition2.2. The system (1.1) is said to be approximate controllable on the interval[0, T ]ifKT(G)=X,that is, for every >0 andξ∈D(A)there exists a control v∈L2₍₀_{, T}_:_{V )}_{such that}

ξ−S(t)φ(0)+F (0, φ)+FT , xT(Bv)

+

T

0

AS(t−s)Fs, xs(Bv)

ds−ΦGs, xs(Bv)

−Bv(s)< ,

(2.9)

wherext(Bv)is a solution of (1.1) associated with the nonlinear termGand control Bvat the timet.

To simplify our task we consider the linear case ofF. We introduce the following assumptions.

For any given >0 and p(·)∈L2₍₀_{, T} _:_X)_{, there exists some} _v(_·₎_∈_L2₍₀_{, T} _:_{V )}

such that

(P1) Φp−ΦBvX< ,

(P2) Bv(·)L2_(0,T_:X)≤q1p(·)L2_(0,T_:X),whereq1is a positive constant independent

ofp(·),

(P3) the constantq1satisﬁes

µ−1q1L2K∞MTexp Cα_βL1Tβ+ML2T

K∞µ−1

<1. (2.10)

3. Approximate controllability. First, we show the approximate controllability of the corresponding system withG≡0.

Lemma_3.1. _{Under hypotheses (H}₁_{), (H}₂_{), and (P}₁_),_K_T₍₀₎=_X_.

Proof. Since the domainD(A)of the operatorAis dense inX(see [7]), it is

suf-ﬁcient to prove thatD(A)⊂KT(0), that is, for any given >0 andξ∈D(A)there

exists av(·)∈L2₍₀_{, T}_:_{V )}_{such that}

_ξ₋_{h(T , ϕ)}₋_Φ_Bv_{< ,}

(3.1)

h(T , ϕ)=S(T )ϕ(0)+F (0, ϕ)−FT , xT(Bv)

−

T

0 AS(T−s)F

s, xs(Bv)

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Letξ∈D(A), thenξ−h(T , ϕ)∈D(A). So there exists somep∈C1₍₀_{, T}_:_X)_{such that}

η= T

0S(T−s)p(s)ds, (3.2)

whereη=ξ−h(T , ϕ). For instance, if we takep(s)= {1−sA}{ξ−h(T , ϕ)}/T, then the ﬁrst equality of (3.3) holds, and by hypothesis (P1) there exists a functionv(·)∈ L2₍₀_{, T}_:_{V )}_{such that}

η= T

0S(T−s)p(s)ds= T

0 S(T−s)Bv(s)ds. (3.3)

Sinceη=ξ−h(T , ϕ),thenξ=h(T , ϕ)+T

0S(T−s)Bv(s)ds.

The denseness of the domainD(A)inXimplies the approximate controllability of the corresponding system withG≡0.

To prove the approximate controllability of system (1.1), we need the following lemma.

Lemma3.2. Letv₁andv₂be inL2(0, T:V ). Then under hypotheses (H₁), (H₂), (H₃), and (H5), the solution mappingW (Bv)(t)=xt(Bv)of (1.1) satisﬁes

_x_t

Bv1−xtBv2_∞

≤µ−1M√Texp CαL1 β T

β₊_ML 2T

K∞µ−1Bv1−Bv2L2_(0,T:X).

(3.4)

Proof. Lety(·, ϕ);(−∞, T ]→Xbe the function deﬁned by

y(t, ϕ):=

  

ϕ(t), −∞< t <0,

T (t)ϕ(0), t≥0. (3.5)

Denotey(t, ϕ)byy(t)with the continuous mapt→yt.

Next, for eachz∈C(0, T :X),z(0)=0, we denote by ˜z the function deﬁned by ˜

z(θ)=0, forθ≤0, and ˜z(t):=z(t), for 0≤t≤T.

So ifx(u)(t)satisﬁes (2.5), we can decompose it asx(u)(t)=z(u)(t)+y(t), for 0≤t≤T, which implies that xt(u)=z˜t(u)+yt, for 0≤t≤T and for each u∈ L2₍₀_{, T}_:_X)_{and that the function}_z(_·₎_satisﬁes

z(t)=S(t)F (0, φ)−Ft,z˜t(u)+yt− t

0

AS(t−s)Fs,z˜s(u)+ysds

+

t

0S(t−s)G

s,z˜s(u)+ys

ds+

t

0S(t−s)u(s)ds.

(3.6)

Thus for eachv1, v2∈L2(0, T:V ), it is clear that for 0≤t≤T , xt

Bv1

−xt

Bv2=z˜t

Bv1

+yt

−z˜t

Bv2

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=z˜t

Bv1

−z˜t

Bv2

≤Ft,z˜tBv1+yt−Ft,z˜tBv2+yt

+

t

0

AS(t−s)Fs,z˜sb

Bv1)+ys

−Fs,z˜s

Bv2

+ys

ds

+

t

0S(t−s)

Gs,z˜sBv1+ys−Gs,z˜sBv2+ysds

+

t

0S(t−s)

Bv1(s)−Bv2(s)

ds

≤(−A)−β·L1·K∞zBv1

−zt

Bv2_∞

+

_C αL1Tβ

β +ML2T

K∞zBv1

−zBv2_∞

+M√TBv1−Bv2L2(0,T:X).

(3.7)

By Gronwall’s inequality, we have

_x_· Bv1

−x·Bv2_∞

≤µ−1M√Texp CαL1 β T

β₊_ML 2T

K∞µ−1Bv1−Bv2L2(0,T:X).

(3.8)

Theorem _3.3. _{Under hypotheses (H}₁_{), (H}₂_{), (H}₃_{), (H}₄_{), (H}₅_{), and (P}₁_{), (P}₂_{), (P}₃_), KT(G)=X, that is, system (1.1) is approximately controllable.

Proof. Since byLemma 3.1,K_T(0)=X, it is suﬃcient to show thatK_T(0)⊂K_T(G). Letξ∈KT(0). Then for any given >0, there existsv∈L2(0, T:V )such that

ξ−h(T , ϕ)−ΦBv< 23,

h(T , ϕ)=S(T )ϕ(0)+F (0, ϕ)−FT , xT(Bv)

−

T

0

AS(T−s)Fs, xs(Bv)

ds. (3.9)

Assumev1∈L2(0, T :V ) is arbitrarily given. By hypothesis (P2), there exists some v2∈L2(0, T:V )such that

ΦBv−Gs, xsBv1−ΦBv2<

23. (3.10)

By (3.9) and (3.10), we obtain

ξ−h(T , ϕ)−ΦGs, xs

Bv1

−ΦBv2<

22. (3.11)

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and (P2) such that

ΦGs, xs

Bv2

−Gs, xs

Bv1

−ΦBw2<

23, (3.12)

and so by (P2) andLemma 3.2, _Bw₂_L₂_(0,T_:X)_≤_q₁_G

·, x·Bv2−G·, x·Bv1L2(0,T:X)

≤q1L2

√

TK∞·x·Bv2

−x·Bv1_∞

≤µ−1q1L2K∞MTexp Cα_βL1Tβ+ML2T

K∞µ−1

×Bv2−Bv1L2_(0,T_:X).

(3.13)

Thus we may deﬁnev3=v2−w2inL2(0, T:V ), which has the following property: ξ−h(T , ϕ)−ΦGs, xs

Bv2

−ΦBv3

=ξ−h(T , ϕ)−ΦGs, xs

Bv1

−ΦBv2+ΦBw2

−ΦGs, xsBv2−Gs, xsBv1<

1 22+

1 23

.

(3.14)

By induction, it is proved that there exists a sequencevninL2(0, T:V )such that _ξ₋_{h(T , ϕ)}₋_Φ_G

s, xs

Bvn

−ΦBvn+1< ₂12+···+

1 2n+1

, n=1,2, . . . , Bvn+1−BvnL2_(0,T_:X)

≤µ−1q1L2K∞MTexp Cα_βL1Tβ+ML2T

K∞µ−1

·Bvn−Bvn−1.

(3.15)

By hypothesis (P3), the sequence{Bvn:n=1,2, . . .}is a Cauchy sequence in the Banach

spaceL2₍₀_{, T}_:_X)_{, and there exists some}_u_in_L2₍₀_{, T}_:_X)_{such that lim}

n→∞Bvn=uin L2₍₀_{, T}_:_X)_{. Therefore, for any given}_>_{0, there exists some integer}_N

such that

_Φ_Bv_N

+1−ΦBvN<

2,

_ξ₋_{h(T , ϕ)}₋_Φ_G

s, xsBvN

−ΦBvN

≤ξ−h(T , ϕ)−ΦGs, xs

BvN

−ΦBvN+1+Φ

BvN+1

−ΦBvN

< 1 22+···+

1 2N

+1

2≤.

(3.16)

This means thatξ∈KT(G). Hence the nonlinear system (1.1) is approximately

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4. Example. We consider the boundary value problem

d dt

z(t, τ)+ t

−∞ π

0 b(s−t, η, τ)z(s, η)dη ds

= d2

dτ2z(t, τ)+ t

−∞a(s−t)z(s, τ) ds+Bv(t), 0≤t≤T , 0≤τ≤π ,

z(t,0)=z(t, π )=0,

z(θ, τ)=ϕ(θ, τ), θ≤0,0≤τ≤π .

(4.1)

To represent this problem as a Cauchy problem, we takeX=L2_([₀_{, π ])}_{and deﬁne} x(t):=z(t,·). LetA:X→Xbe deﬁned byAf (τ):=f(τ)with the domain

D(A):=f (·)∈L2[0, π ]:f(·)∈L2[0, π ], f (0)=f (π )=0. (4.2)

It is well known thatAgenerates a strongly continuous semigroupT (·)which is com-pact, analytic, and selfadjoint. Furthermore,Ahas discrete spectrum, the eigenvalues are−n2_,_n_∈_N_{, with corresponding normalized eigenvectors}_e

n(τ):=(2/π )1/2sin(nτ).

These eigenvectors satisfy the properties stated in [3]. Deﬁne an inﬁnite-dimensional spaceVby

V=

v|v=

∞

n=2

vnenwith ∞

n=2

vn2<+∞

. (4.3)

The norm inV is deﬁned byvV =( ∞

n=2vn2)1/2. Deﬁne a mappingB∈ᏸ(V→X)

as follows:

Bv=2v2e1+ ∞

n=2

vnen, forv= ∞

n=2

vnen∈V . (4.4)

Obviously,Bᏸ(V→X)≤

√

5.

Then the operatorB is well deﬁned byv(·,·)∈L2₍₍₀_{, T )}_×₍₀_{, π ))}_{; and by [}₈_{], we}

know thatBsatisﬁes hypotheses (H5), (P1), (P2), and (P3).

LetᏮdenote the spaceCr×L2(g;X)withr =0, as in [4]. To prove approximate

controllability of the problem (4.1), we assume that conditions (i)–(v) of [4] hold. Con-sequently, equation (4.1) can be formulated abstractly as

d dt

x(t)+Λ1

xt

=Ax(t)+Λ2

xt

+Bu(t), 0≤t≤ξ, x0=ϕ∈Ꮾ,

(4.5)

whereΛ1, Λ2are linear operators inᏮ. Using the assumptions stated in [4, pages

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Jong Yeoul Park: Department of Mathematics, Pusan National University, Pusan

609-735, Korea

E-mail address:[email protected]

Sang Nam Kang: Department of Mathematics, Pusan National University, Pusan