R E S E A R C H
Open Access
Controllability for nonlinear evolution
equations with monotone operators
Yong Han Kang
1, Jin-Mun Jeong
2*and Hyun-Hee Rho
2*Correspondence:
2Department of Applied
Mathematics, Pukyong National University, Busan, 608-737, Korea Full list of author information is available at the end of the article
Abstract
In this paper, we investigate the approximate controllability for nonlinear evolution equations with monotone operators and nonlinear controllers according to monotone operator theory. We also give the regularity for the nonlinear equation. Finally, an example, to which our main result can be applied, is given.
MSC: 35F25; 93C20
Keywords: nonlinear evolution equation; monotone operator; approximate controllability; regularity; hemicontinuous
1 Introduction
In this paper, we deal with the approximate controllability for the semilinear equation in a Hilbert spaceHas follows:
⎧ ⎨ ⎩
x(t) +Ax(t) +f(t,x(t),u(t)) +Bu(t) = , <t≤T,
x() =x.
(.)
In (.), the principal operator –Agenerates an analytic semigroupS(t). LetUbe a Hilbert space of control variables, and letBbe a linear (or nonlinear) operator fromUtoH, which is called a controller.
First, we consider the following initial value problem of a semilinear equation: ⎧
⎨ ⎩
x(t) +Ax(t) +f(t,x(t)) =h(t), <t≤T,
x() =x. (.)
IfA:D(A)⊂H→His an unbounded operator, Di Blasioet al.[] provedL-regularity for a retarded linear system in Hilbert spaces, and Jeong [] (also see []) considered the control problem for retarded linear systems withL-valued controller and more general Lipschitz continuity of nonlinear terms.
For the theory of monotone operators, there are many literature works; for example, see Lions [], Stampacchia [], Browder [], and the references cited therein. Kenmochi [] derived new results on monotone operator equations, and Ouchi [] proved the analytic-ity of solutions of semilinear parabolic differential equations with monotone nonlinearanalytic-ity. For the existence of solutions for a class of nonlinear evolution equations with monotone perturbations, one can refer to [–]. We refer to Pascali and Sburlan [], Morosanu []
to see the applications of nonlinear mapping of monotone type and nonlinear evolution equations. The classical solutions of (.) were obtained by Kato [] under the mono-tonicity condition on the nonlinear termf as an operator from [,T]×HtoH.
In the first part of this note, we apply results of [] to findL-regularity of solutions in the wider sense of (.) under the more general monotonicity of a nonlinear operatorf
fromR×VtoV∗, which is related to the results of Tanabe [, Theorem ..].
Next, we extend and develop control problems on this topic. In recent years, as for the controllability for semilinear differential equations with Lipschitz continuity of a nonlin-ear operatorf, Naito [] and [–] proved the approximate controllability under the range conditions of the controllerB. However, we can find few articles which extend the known general controllability problems to nonlinear evolution equation (.) with mono-tone operators and nonlinear controllers.
In this paper, based on the regularity for solutions of equation (.), we obtain the ap-proximate controllability for nonlinear evolution equation (.) with monotone operators and nonlinear controllers.
The paper is organized as follows. In Section , we explain several notations of this paper and state results about L-regularity for linear equations in the sense of [, , ]. In Section , we give the regularity for nonlinear equation (.). In Section , we obtain the approximate controllability for nonlinear evolution equation (.) with hemicontinuous monotone operators by using the theory of monotone operators. In the end, an example is provided to illustrate the application of the obtained results.
2 Preliminaries
IfH is identified with its dual space, we may writeV⊂H⊂V∗densely and the corre-sponding injections are continuous. The norm onV,HandV∗will be denoted by · ,| · | and · ∗, respectively. The duality pairing between the elementvofV∗and the element
vofVis denoted by (v,v), which is the ordinary inner product inHifv,v∈H. Forl∈V∗, we denote (l,v) by the valuel(v) oflatv∈V. The norm oflas element ofV∗
is given by
l∗=sup
v∈V |(l,v)|
v .
Therefore, we assume that V has a stronger topology thanH and, for brevity, we may regard that
u∗≤ |u| ≤ u, ∀u∈V. (.)
Leta(·,·) be a bounded sesquilinear form defined in V×V and satisfying Gårding’s inequality
Rea(u,u)≥ωu–ω|u|, (.)
whereω> andωis a real number. LetAbe an operator associated with this sesquilinear form:
Then –Ais a bounded linear operator fromV toV∗ by the Lax-Milgram theorem. The realization ofAinH, which is the restriction ofAto
D(A) ={u∈V:Au∈H},
is also denoted byA. It is well known thatAis positive definite and self-adjoint and gen-erates an analytic semigroupS(t) in bothHandV∗. From the following inequalities:
ωu≤Rea(u,u) +ω|u|≤C|Au||u|+ω|u|≤max{C,ω}uD(A)|u|,
where
uD(A)=
|Au|+|u|/
is the graph norm ofD(A), it follows that there exists a constantC> such that
u ≤Cu/D(A)|u|
/. (.)
Thus we have the following sequence:
D(A)⊂V⊂H⊂V∗⊂D(A)∗, (.)
where each space is dense in the next one, which is continuous injection.
Lemma . With notations(.), (.),we have
V,V∗/,=H,
D(A),H/,=V,
where(V,V∗)/,denotes the real interpolation space between V and V∗ (Section.of
[]or[]).
IfXis a Banach space,L(,T;X) is the collection of all strongly measurable square inte-grable functions from (,T) intoX, andW,(,T;X) is the set of all absolutely continuous functions on [,T] such that their derivative belong toL(,T;X).C([,T];X) will denote the set of all continuous functions from [,T] intoXwith the supremum norm. IfXand
Yare two Banach spaces,L(X,Y) is the collection of all bounded linear operators fromX
intoY, andL(X,X) is simply written asL(X). Here, we note that by using interpolation theory, we have
L(,T;V)∩W,,T;V∗⊂C[,T];H. (.)
First, we consider the following linear system: ⎧
⎨ ⎩
x(t) +Ax(t) =k(t),
By virtue of Theorem . of [] (or Theorem . of [, ]), we have the following result on the corresponding linear equation of (.).
Lemma . ()For x∈V = (D(A),H)/,(see Lemma.)and k∈L(,T;H),T > ,
there exists a unique solution x of(.)belonging to
L,T;D(A)∩W,(,T;H)⊂C[,T];V
and satisfying
xL(,T;D(A))∩W,(,T;H)≤C
x+kL(,T;H)
, (.)
where Cis a constant depending on T.
()Let x∈H and k∈L(,T;V∗),T> .Then there exists a unique solution x of(.)
belonging to
L(,T;V)∩W,,T;V∗⊂C[,T];H
and satisfying
xL(,T;V)∩W,(,T;V∗)≤C
|x|+kL(,T;V∗)
, (.)
where Cis a constant depending on T.
Throughout this paper, strong convergence is denoted by ‘→’ and weak convergence by ‘’.
Definition . LetXandY be Banach spaces andLbe a mapping fromXintoY. The
domainD(L) ofLis assumed to be convex.Lis called hemicontinuous ifL(( –λ)x+λx) for anyx,x∈D(L) is continuous in ≤λ≤ in weak topology ofY.
The linear operator is obviously hemicontinuous.
Definition . LetXandY be Banach spaces andLbe a single-valued mapping fromX
intoY.Lis called demicontinuous ifxn∈D(L) andxn→x∈D(L) imply thatLxnLx.
Definition . LetLbe a mapping from a Banach spaceXinto its conjugate spaceX∗.
Lis said to be pseudo-monotone if the following condition is satisfied. Ifxiis a directed family of points, contained inD(L), which converges weakly to an elementxofD(L) and iflim sup(Lxi,xi–x)≤, thenlim inf(Lxi,xi–y)≥(Lx,y) for ally∈D(L).
Definition . LetLbe a generally multi-valued mapping from a Hilbert spaceXinto itself. If
for allx,y∈D(L),xˆ∈Lxandyˆ∈Ly, thenLis called a monotone operator. SometimesLis also called a monotone operator if
(xˆ–ˆy,x–y)≤
holds instead of (.).
Definition . A real-valued continuous functionjis called a gauge function defined on [,∞) if it is strictly monotone increasing and satisfiesj() = andlimr→∞j(r) =∞.
LetXbe a Banach space andX∗its conjugate. For anyx∈X, we set
F(x) =x∗∈X∗:x∗,x=x∗∗x,x∗∗=jx .
The multi-valued operatorF:X→X∗is called the duality mapping ofXwith a gauge functionj.
Let us denote byΛthe operator determined by an inner product ((·,·)) onV: (Λx,y) = ((x,y)). Then it is immediate thatΛis a duality mapping fromVintoV∗with a gauge func-tionj(r) =r. It is also known that the duality mapping is monotone and hemicontinuous, and hence it is pseudo-monotone.
Lemma . We have briefly explained the theory of monotone operators(see[, Sec-tion .]).
() In Definition.above,lim(Lxi,xi–x) = is seen by takingx=y. () Hemicontinuous monotone mappings from a Banach spaceXintoX∗are
pseudo-monotone.
() LetXbe a reflexive Banach space,and let bothXandX∗be strictly convex.Further, letM⊂X×X∗be monotone andΛbe a duality mapping fromXintoX∗.If
R(M+Λ) =X∗,thenMis maximal monotone.
() LetXbe a reflexive Banach space andLbe a closed monotone linear operator fromX
andX∗.If the dual operatorL∗is monotone,thenLis maximal monotone. () LetXbe a reflexive Banach space,M⊂X×X∗be maximal monotone andLbe a
pseudo-monotone bounded mapping fromD(L) =XintoX∗.If there exists [x,f]∈Msuch that
lim
x→∞(Lx+f,x–x)/x=∞,
thenR(M+L) =X∗,that is,for everyf∈X∗,(M+L)xf has a solutionx∈D(M).
The following inequality is referred to as Young’s inequality.
Lemma .(Young’s inequality) Let a> ,b> and/p+ /q= ,where≤p<∞and
<q<∞.Then,for everyλ> ,one has
ab≤λ
pap
p + bq
Lemma . Let H be a Hilbert space and V be a reflexive Banach space.Suppose that V is a dense subspace of H and that V has a stronger topology than H.Therefore,V⊂H⊂V∗.
Let T> and X=Lp(,T;V)with≤p<∞.Then the operator L defined by ⎧
⎨ ⎩
D(L) ={x∈X:x∈X∗,x() = },
Lx=x for each x∈D(L),
is maximal monotone linear.
Proof SinceX∗=Lq(,T;V∗) with q=p/(p– ), noting thatx(t)∈V∗ and by Young’s
inequality, we have
x(t)= t
(d/ds)x(s)ds= t
Rex(s),x(s)ds
≤q t
x(s)q/q ds+ t
x(s)p/p ds,
so thatxbelongs toC([,T];H). Therefore, we find thatx() is well defined as an element ofH. It is easily shown thatLis a closed linear operator fromXintoX∗andD(L) is dense inX. Further, since
(Lx,x) =x,x=x(T)/≥,
we know thatLis monotone. It is also easily seen that the adjoint operator ofLis given by ⎧
⎨ ⎩
D(L∗) ={x∈X:x∈X∗,x(T) = },
L∗x= –x for eachx∈D(L∗).
Hence,L∗ is also monotone. Therefore, from () of Lemma ., it is concluded thatLis
maximal monotone linear.
3 Nonlinear equations
We consider the following initial value problem of a semilinear equation: ⎧
⎨ ⎩
x(t) +Ax(t) +f(t,x(t)) =h(t), <t≤T,
x() =x.
(.)
The following result is due to Kato [] (or [, Theorem ..]).
Lemma . Let f be a demicontinuous bounded mapping from[,T]×H into H.Assume that f(t,·)is monotone for each t∈[,T]:
f(t,x) –f(t,y),x–y> .
Assume further that–A is a generator of a contraction semigroup S(t) (t≥).Then,for any
(x,h)∈H×C([,T];H),there exists a solution x∈C([,T];H)of the integral equation
x(t) =S(t)x– t
corresponding to(.),and it is unique.Let xand xbe the solutions with initial values x
and x,respectively.Then the estimate
x(t) –x(t)≤ |x–x|
holds on≤t≤T.Hence,the mapping which carries the initial value xto the solution x
is a continuous mapping from H into C([,T];H).
Next, we apply Lemma . to find a solution in the wider sense of (.) under some-what different assumptions. Concerning the nonlinear mappingf, assume the following hypothesis.
Assumption (F) The mappingf is demicontinuous bounded from [,T]×Hinto V∗, andf(t,·) for eachtis monotone as a mapping fromV intoV∗.
The following theorem is a part of Theorem .. due to Tanabe [].
Theorem . Let Assumption(F)be satisfied,and let the assumptions on the principal operator A stated in Sectionbe satisfied.Assume that xis an arbitrary element of H and
h∈L(,T;V∗).Then there exists a solution x∈L(,T;V),satisfying x∈L(,T;V∗),of
⎧ ⎨ ⎩
x(t) +Ax(t) +f(t,x(t)) =h(t), <t≤T,
x() =x
(.)
and it is unique.Moreover,there exists a constant Csuch that
xL(,T;V)∩W,(,T;V∗)≤C
|x|+hL(,T;V∗)
,
where Cis a constant depending on T and the mapping
H×L,T;V∗(x,h)→x∈L(,T;V)∩W,
,T;V∗⊂C[,T];H
is Lipschitz continuous.
Proof As far as the existence of the solution is concerned, we may put x = . By Lemma ., the operator defined by
⎧ ⎨ ⎩
D(L) ={x∈L(,T;V) :x∈L(,T;V∗),x() = },
Lx=x, ∀x∈D(L),
is a maximal monotone linear operator from L(,T;V) intoL(,T;V∗). Let us write
(Ax)(t) =Ax(t) and (Fx)(t) =f(t,x(t)) for each x∈L(,T;V). Then bothAand F are monotone operators fromL(,T;V) intoL(,T;V∗) andD(L)⊂D(F). Since we assumed
x= , equation (.) is equivalent to
Note thatL+Fis monotone, if it is shown to be maximal monotone, assumption () of Lemma . is satisfied with M=L+F,L=Aandx= . Then we haveR(L+F+A) =
L(,T;V∗), which implies the existence of the solution. Thus, from now on, we prove the maximal monotonicity ofL+F. SinceΛdetermined by an inner product ((·,·)) on
V is a duality mapping fromVintoV∗, as seen under Definition ., to see the maximal monotonicity ofL+F, on account of () of Lemma ., it is enough to verify thatR(L+F+
Λ) =L(,T;V∗).
Lethbe an arbitrary element ofL(,T;V∗),h
n∈C([,T];H) andhn→hinL(,T;V∗). SinceΛis positive definite and self-adjoint in bothHandV∗, the domainΛ/coincides withVandH, respectively. Hence,In= (I+n–Λ/)–is a contraction operator in bothH andV, and it converges strongly toIasn→ ∞. It is also easy to see that (Inf,g) = (f,Ing) holds forf ∈V∗andg∈H. Let us definefn(t,x) =Inf(t,x). Then the mappingfnsatisfies the assumption forf in Lemma .. Hence, Lemma . can be applied to the initial value problem
⎧ ⎨ ⎩
xn(t) +Λxn(t) +fn(t,xn(t)) –hn(t) = ,
x() = . (.)
Let us denote the semigroup generated byΛbyT(t). Then it ensures the existence of a solutionx∈C([,T];H) of the equation
xn(t) + t
T(t–s)fn
s,xn(s)–hn(s)ds= . (.)
Multiplying byxn(t) on (.) and integrating over [,t], we have
xn(t)
+
t
xn(s)ds= t
hn(s) –fn
s,xn(s),xn(s)ds. (.)
By using Young’s inequality and the monotonicity off, the following holds:
xn(t)
+
t
xn(s)ds
= – t fn
s,xn(s)–fn(s, ),xn(s)ds
– t
fn(s, ),xn(s)
ds+ t
hn(s),xn(s) ds ≤ t
hn(s)∗ds+
t
fn(s, )∗ds+ t
xn(s)ds.
So, we obtain
xn(t)+ t
xn(s)ds≤C
t
hn(s)∗ds+ t
fn(s, )∗ds
,
as-sumexnxinL(,T;V). By lettingn→ ∞in (.), we have
x(t) + t
T(t–s)fs,x(s)–h(s)ds= .
Since
xn(t) –x(t) +Λxn(t) –x(t)
=hn(t) –h(t)–fn
t,xn(t)–ft,x(t)
=hn(t) –h(t)–fn
t,xn(t)–fn
t,x(t)–fn
t,x(t)–ft,x(t),
from multiplying byxn(t) –x(t) and the monotonicity offn, it follows that
xn(t) –x(t)
+ t
xn(s) –x(s)
ds
≤
t
hn(s) –h(s)∗ds
+
t
fn
s,x(s)–fs,x(s)∗ds+ t
xn(s) –x(s)
ds.
Thus, noting thathnandfn(t,x) converge strongly tohandf(t,x) inL(,T;V∗), respec-tively, we see thatxis a solution of the equation (L+F+Λ)x=h. Finally, to prove the uniqueness of the solution, suppose thatxandxare solutions with initial conditionsx andxand forcing termshandh, respectively. Then it is easy to see that there exists
C> such that
x(t) –x(t)+ t
x(s) –x(s)
ds≤C
|x–x|+ t
h(s) –h(s)∗ds
.
This completes the proof of Theorem ..
Remark . In a similar way to Theorem ., we also obtain the existence of solutions of (.) in the case wheref is a demicontinuous bounded mapping from [,T]×V intoH. Moreover, assume thatf(t,·) for eachtis monotone as a mapping fromD(A) intoHand
x∈V,h∈L(,T;H), then there exists a unique solutionxof (.) such that
x∈L,T;D(A)∩W,(,T;H)⊂C[,T];V.
Moreover, there exists a constantCsuch that
xL(,T;D(A))∩W,(,T;H)≤Cx+hL(,T;H),
whereCis a constant depending onTand the mapping
V×L(,T;H)(x,h)→x∈L,T;D(A)∩W,(,T;H)
4 Approximate controllability
In this section, we deal with the approximate controllability for the semilinear equation in
Has follows. ⎧ ⎨ ⎩
x(t) +Ax(t) +f(t,x(t),u(t)) +Bu(t) = , <t≤T,
x() =x. (.)
In (.), the principal operator –Agenerates an analytic semigroupS(t) as stated in Sec-tion . LetUbe a Hilbert space of control variables, and letBbe a bounded linear operator fromUtoH, which is called a controller. The mild solution of initial value problem (.) is the following form:
x(t;f,u) =S(t)x– t
S(t–s)fs,x(s),u(s)+Bu(s) ds.
Letf be a nonlinear mapping satisfying the following.
Assumption (F) The mapping f is demicontinuous bounded from [,T]× H ×U
intoV∗. Assume thatf(t,·,u) for each (t,u)∈[,T]×Uis monotone as a mapping fromV
intoV∗withf(t,·, ) = , andf(t,x,·) for each (t,x)∈[,T]×Vis monotone as a mapping fromUintoV∗.
For eachu∈L(,T;U), let us defineF(t,x(t)) =f(t,x(t),u(t)). Then from Theorem . it follows that solution (.) exists and is unique inL(,T;V)∩W,(,T;V∗). Letx(T;f,u) be a state value of system (.) at timeTcorresponding to the nonlinear termf and the controlu. We define the reachable sets for system (.) as follows:
RT(f) =
x(T;f,u) :u∈L(,T;U) ,
RT() =x(T; ,u) :u∈L(,T;U) .
Definition . System (.) is said to be approximately controllable at timeTif for every desired final statex∈Hand> , there exists a control functionu∈L(,T;U) such that the solutionx(T;f,u) of (.) satisfies|x(T;f,u) –x|<, that is,RT(f) =H, whereRT(f) is the closure ofRT(f) inH.
Definition . LetLbe a mapping from a Banach spaceXinto its conjugate spaceX∗.
T is called coercive if there existsu∈D(L) such that
lim
u∈D(L),u→∞
(Lu,u–u)
u =∞.
Remark .[, Theorem .] It is well known that ifXis a reflexive Banach space and
Lis monotone, everywhere defined and hemicontinuous fromD(L) =XintoX∗, thenLis maximal monotone. If in additionLis coercive monotone, thenR(L) =X∗.
oper-atorf. So,H=Uobviously. Consider the linear system given by ⎧
⎨ ⎩
y(t) +Ay(t) +u(t) = ,
y() =x
(.)
and the following semilinear control system: ⎧
⎨ ⎩
x(t) +Ax(t) +f(t,x(t),v(t)) +v(t) = ,
x() =x. (.)
Lemma . Let Assumption(F)be satisfied,and let y(t)be the solution of(.) corre-sponding to a control u.Then there exists v∈L(,T;H)such that
⎧ ⎨ ⎩
v(t) =u(t) –f(t,y(t),v(t)), <t≤T,
v() =u(). (.)
Proof Set
w(t) =v(t) –u(t), gt,w(t)=ft,y(t),w(t) +u(t).
Let (Gw)(t) =g(t,w(t)). Then equation (.) is equivalent to
(I+G)w= . (.)
It is easy to see thatGis monotone as an operator fromL(,T;H) toL(,T;V∗), and is a demicontinuous bounded mapping as an operator fromL(,T;H) intoL(,T;V∗).
Let the collection of all finite dimensional subspaces ofH be denoted byY, and when
Y ∈Y, let the orthogonal projection onY be denoted byPY. Foru∈L(,T;H), let us define (PYu)(t) =PYu(t); thusPY also denotes the orthogonal projection inL(,T;H). According to Assumption (F), we have that the operatorI+PYgis a coercive monotone operator fromYinto itself. In general, any demicontinuous operator is hemicontinuous. Therefore, by Remark ., we haveR(I+PYg) =Y, which (.) implies the existence of a solution to
wY(t) +PYg
t,wY(t)
= , wY() =uY(). (.)
Since
wY(t)
= –PYg
t,wY(t)
,wY(t)
= –PYg
t,wY(t)
–PYg(t, ),wY(t)
–PYg(t, ),wY(t)
≤g(t, )wY(t),
we have
Hence, the solution of (.) is bounded onC([,T];H). Letwbe an arbitrary element of
L(,T;H). Then we can takew
n∈Yn∈Ysatisfying (.) such thatwn→winL(,T;H). SinceGis monotone as an operator fromL(,T;H) toL(,T;V∗), and is a demicontin-uous bounded mapping, we haveGwnGwinL(,T;V∗). Hence, we obtain that for
v∈L(,T;V),
≤(I+PYG)v– (I+PYG)wn,v–wn
=(I+PYG)v,v–wn
asn→ ∞, so that
(I+PYG)v,v–w
≥. (.)
Ifvis replaced byw+n–vin (.), we have
≤w+n–v+PYG
w+n–v,n–v,
which, by the demicontinuity ofG, leads to
≤(w+Gw,v), ∀v∈L(,T;V)
in the limit asn→ ∞. Sincevis arbitrary, we obtainw+Gw= .
Remark . As seen in [], we know that ifXis a Hilbert space andG⊂X×Xis maximal monotone, thenR(I+G) =X. So, (.) is easily obtained if the operatorGin Theorem . is maximal monotone.
Theorem . Under Assumption(F)and B=I,we have
RT()⊂RT(f).
Therefore,if linear system(.)with f = is approximately controllable at time T,then so is semilinear system(.).
Proof Lety,xbe the solutions of (.) and (.), respectively. Letv(t) =u(t) –f(t,y(t),v(t)) in the sense of Lemma .. Then, since
x(t) +Ax(t) +ft,x(t),v(t)+v(t)
=x(t) +Ax(t) +ft,x(t),v(t)+u(t) –ft,y(t),v(t),
we have ⎧ ⎨ ⎩
x(t) –y(t) +A(x(t) –y(t)) +f(t,x(t),v(t)) –f(t,y(t),v(t)) = ,
x() –y() = .
Acting on both sides of the above equation, byx(t) –y(t), from the monotonicity off, it follows
d
dtx(t) –y(t)
+ωx(t) –y(t)
≤ωx(t) –y(t)
which is
x(t) –y(t)+ ω t
x(s) –y(s)ds≤ω t
x(s) –y(s)ds.
By using Gronwall’s inequality, we get x =y in C([,T];H). Noting that x(·),y(·) ∈
C([,T];H), every solution of the linear system with controluis also a solution of the semilinear system with controlv, that is, we have thatRT()⊂RT(f).
From now on, we consider the initial value problem for semilinear parabolic equation (.). LetUbe a Hilbert space, and let the controller operatorBbe a nonlinear operator fromUtoH.
Theorem . Let Assumption(F)and R(f)⊂R(B)be satisfied.Assume that the inverse mapping B–of the controller B exists and is monotone.Then the linear system
⎧ ⎨ ⎩
y(t) +Ay(t) +Bu(t) = ,
y() =x,
(.)
is approximately controllable at time T,so is nonlinear system(.).
Proof Letybe a solution of (.) corresponding to a controlu. Consider the following semilinear system:
⎧ ⎨ ⎩
x(t) +Ax(t) +f(t,x(t),v(t)) +Bu(t) –f(t,y(t),v(t)) = ,
x() =x.
(.)
Set
v(t) =u(t) –B–ft,y(t),v(t). (.)
We put
w(t) =v(t) –u(t), (Bu)(t) =Bu(t), (Gw)(t) =g
t,w(t)=ft,y(t),w(t) +u(t).
Equation (.) is equivalent to
I+B– Gw= . (.)
Here, similarly to the proof of Lemma ., we have that there exists an element v∈ L(,T;U) satisfying (.), that is,Bv(t) =Bu(t) –f(t,y(t),v(t)). In a similar way to the proof of Theorem ., we getx=y. Since system (.) is equivalent to (.), we conclude
thatRT()⊂RT(f).
Assumption (F) Assume thatf(t,x,·) for each (t,x)∈[,T]×V is maximal monotone as a mapping fromUintoV∗.
The following result is well known from semigroup properties.
Lemma . If p∈L(,T;H)and
t
S(t–s)p(s)ds= , ≤t≤T,
then p(t) = for almost all t∈[,T].
Theorem . Let Assumption(F)and R(f)⊂R(B)be satisfied.Assume that B is a hemi-continuous monotone mapping from V into V∗;moreover,if it is coercive,then linear system
(.)is approximately controllable at time T,so is semilinear system(.).
Proof LetξT∈D(A). We define the linear operatorSˆfromL(,T;H) toHby
ˆ
Sp= T
S(T–s)p(s)ds
forp∈L(,T;H). Asξ
T∈D(A), there existsp∈C(,T;H) such that
ˆ
Sp=ξT–S(T)x;
for instance, takep(s) = (ξT–sAξT) –S(s)x/T. By expressing (Bu)(t) =Bu(t) for allu∈
L(,T;V). By Remark ., sinceR(B) =V∗, there existsu
∈L(,T;V) such that
p=Bu.
Sincepis an arbitrary element ofL(,T;V∗),p
n∈C(,T;H) andpn→pinL(,T;V∗). This implies that linear system (.) is approximately controllable.
To prove the approximate controllability of (.), we will show thatD(A)⊂RT(f),i.e., for givenε> andξT∈D(A), there existsu∈L(,T;V) such that
ξT–x(T;f,u)<ε. (.)
Letx∈L(,T;V). Then we writeGu(t) =f(t,x(t),u(t)) for eachu∈L(,T;H). Then we rewrite (.) as
Sˆ(p+Gu+Bu)<ε.
Thus, in view of Lemma ., it is enough to verify that there exists an arbitrary element
uofL(,T;V) such that (G+B
)u= –p. By () of Lemma .,Bis pseudo-monotone and satisfies the condition () of Lemma .. Thus, we haveR(G+B) =V∗. Sincepis an arbitrary element ofL(,T;V∗),p
n∈C(,T;H) andpn→pinL(,T;V∗). This implies
Remark . We know that by Assumption (F) and (.),B+Gis monotone, hemi-continuous and coercive from U into V∗. Therefore, as seen in Remark ., we have
R(B+G) =L(,T;V∗), that is, system (.) is approximately controllable.
5 Example
Let be a bounded region inRnwith smooth boundary∂ . We define the following spaces:
H( ) =
u:u,∂u
∂xi ∈
L( ),i= , , . . . ,n
,
H( ) =
u:u,∂u
∂xi , ∂
u
∂xi∂xj
∈L( ),i,j= , , . . . ,n
,
where ∂∂xu
i and
∂u
∂xi∂xj are derivatives ofuin the distribution sense. The norm ofH
( ) is
defined by
u=
u(x)+ n
i=
∂u(x)
∂xi dx .
HenceH( ) is a Hilbert space.
H( ) =u:u∈H( ),u|∂ = = the closure ofC∞( ) inH( ).
The norm and inner product ofH
( ) are defined by
u=
n
i=
∂u(x)
∂xi
dx
=u, ((u,v)) =
n
i=
∂u(x)
∂xi ·
∂v(x)
∂xi
dx
for anyu,v∈H
( ). We put∇= (∂∂x, . . . ,
∂
∂xn). Define the operatorAby
D(A) = domain ofA
=u:u∈H( )∩H( ) =u:u∈H( ),u|∂ = ,
Au= –u for allu∈D(A).
The operatorAinL( ) is defined so that for anyv∈H
( ), there existsf ∈L( ) such that
(u,v)= (f,v).
Then, for anyu∈D(A),Au=f andAis a positive definite self-adjoint operator. LetH–( ) =H
( )∗be a dual space ofH( ). For anyl∈H–( ) andv∈H( ), the notation (l,v) denotes the valuelatv.
Letube fixed if we consider the functionalH
u,v∈H ( ),
((u,v)) = (Au,v),
that is,Au=l. The operatorAis a one-to-one mapping fromH( ) toH–( ). The relation of operatorsAandAsatisfy that
D(A) =u∈H( ),Au∈L( ) ,
Au=Au for anyu∈D(A).
From now on, bothAandAare denoted simply byA.
We introduce a simple example of the control operatorBwhich satisfies the condition in Theorem .. Consider the caseU=H, and define the intercept operator <α<Ton
L(,T;H) by
gαv(t) = ⎧ ⎨ ⎩
, ≤t≤α,
|v(t)|p, p≥,α≤t≤T, v∈L
(,T;H),
Bu(t) =gαv(t)v(t).
ThenBis a continuous monotone mapping such that there exists a constantβ>β> such that
β|u–u| ≥(Bu–Bu)≥β|u–u|, ∀u,u∈H.
Let
H=L( ), V=H( ), V∗=H–( ).
Letn≥. Then by Sobolev’s imbedding theorem, we have
V⊂Lp( )⊂L( )⊂Lq( )⊂V∗,
wherep= n/(n– ) andq=p/(p– ) = n/(n+ ). Assume thatfˆ(λ,μ) is a continuous and increasing function defined onRsuch that|ˆf(λ,·)|=O(|λ|(n+)/n) as|λ| → ∞. If we putf(t,x,u)(y) =f(t,x(y),u) =fˆ(x(y),u) for eachx∈H, thenfˆ(·,x,·)∈Lq( ), so that it is clear thatfˆis monotone as an operatorLp( )×UintoLq( ). To show thatfˆis a demi-continuous mapping fromHintoLq( ), letx
n→xinH. Since{xn}is bounded inH, so is{ˆf(xn)}inLq( ). Hence, there exists a subsequence{x
nj}such thatxnj(y)→x(y) almost
everywhere in and there exists an elementg∈Lq( ) such thatfˆ(x
nj,·)g. Sincefˆ(λ,·)
is a continuous function of real variablesλ,fˆ(xnj(y),·)→ ˆf(x(y),·) almost everywhere in
. Otherwise, one can find an appropriate convex combinationgk=
n≥kλknfˆn, where ˆ
fn(x) =fˆ(xnj,·), which is strongly convergent toginLp( ). This says thatgk(y)→ ˆf(x(y),·)
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
YHK drafted the manuscript and corrected the main results, JMJ carried out the main proof of this paper, and HHR participated in its design and coordination.
Author details
1Institute of Liberal Education, Catholic University of Daegu, Daegue, 712-702, Korea.2Department of Applied
Mathematics, Pukyong National University, Busan, 608-737, Korea.
Acknowledgements
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0007560).
Received: 7 August 2013 Accepted: 22 October 2013 Published:12 Nov 2013
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