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International Journal of Control
ISSN: 0020-7179 (Print) 1366-5820 (Online) Journal homepage: http://www.tandfonline.com/loi/tcon20
Control problems for semilinear retarded
integrodifferential equations by the Fredholm
theory
Yong Han Kang & Jin-Mun Jeong
To cite this article: Yong Han Kang & Jin-Mun Jeong (2017): Control problems for semilinear
retarded integrodifferential equations by the Fredholm theory, International Journal of Control, DOI:
10.1080/00207179.2017.1390260
To link to this article: http://dx.doi.org/10.1080/00207179.2017.1390260
Accepted author version posted online: 27
Oct 2017.
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Download by: [Chalmers University of Technology]
Date: 28 October 2017, At: 22:20
Publisher: Taylor & Francis
Downloaded by [Chalmers University of Technology] at 22:20 28 October 2017
Journal: International Journal of Control
DOI: https://doi.org/10.1080/00207179.2017.1390260
Control problems for semilinear retarded
integrodifferential equations by the
Fredholm theory
Yong Han Kang1 and Jin-Mun Jeong2
1
Institute of Liberal Education, Catholic University of Daegu
Kyeongsan 680-749, Korea
2
Department of Applied Mathematics, Pukyong National University
Busan 608-737, Korea
Abstract
In this paper, we deal with the approximate controllability for semilinear
retarded functional integrodifferential equations by using the Fredholm theory in Hilbert spaces. We no longer require the compactness of structural
operators to obtain the approximate controllability for the nonlinear differential system, but instead we use the theory of interpolation spaces and the
regularity of solutions of semilinear given equations with unbounded principal
operators. Finally, based on the properties of general degree theory in infinite dimensional spaces, we investigate the relation between the reachable set
of trajectories of the semilinear retarded functional integrodifferential system
and that of its corresponding linear system excluded by the nonlinear term.
Email: 1 yhkang2642@gmail.com, 2 jmjeong@pknu.ac.kr( Corresponding author)
1
Keywords: retarded differential equation; semilinear equation; Fredholm
theory; approximate controllability; reachable set.
AMS Classification: Primary 35B37; Secondary 93C20
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1
Introduction
This paper is concerned with the approximate controllability for the following semilinear retarded functional integrodifferential equation:

∂

x) + A(x, Dx )u(t, x) + A1 (x, Dx )u(t − h, x)
 ∂t u(t,
R0
(1.1)
+ −h a(s)A2 (x, Dx )u(t + s, x)ds

Rt

= 0 k(t − s)g(s, u(s, x(s))ds + (Bw)(t), (t, x) ∈ [0, T ] × Ω.
Here, Ω ⊂ Rn is a bounded domain with smooth boundary ∂Ω, A(x, Dx ) and
Aι (x, Dx )(ι = 1, 2) are second order linear differential operators with real coefficients, and A(x, Dx ) is an elliptic operator in Ω. The function a(s) is a real scalar
function on [−h, 0], where h > 0 is a delay time. The controller B is a bounded
linear operator from L2 (0, T ; U) to L2 (0, T ; H), where U is some Banach space. The
boundary condition attached to (1.1) is given by Dirichlet boundary condition
u|∂Ω = 0,
0 < t ≤ T,
(1.2)
and the initial condition is given by
u(0, x) = g 0 (x),
u(s, x) = g 1 (s, x)
− h ≤ s ≤ 0.
(1.3)
Precise assumptions on the nonlinear function g are given in the next section.
This kind of abstract formulations of many partial integrodifferential equations
arise in the mathematical description of the dynamical processes with heat flow in
material with memory, viscoelasticity, and many physical phenomena (See Fitzgibbon, 1980;Heard, 1981). Many authors have discussed the structural properties
for retarded systems(see Di Blasio, Kunisch, Sinestrari, 1984;Jeong, 1993;Manitius,
1980;Nakagiri, 1988;Tanabe, 1988), and references therein.
The numerical solution of FredholmVolterra integro-differential equations for periodic boundary value problems has discussed in (Abu Arqub and M. Al-Smadi,
2004;Abu Arqub and Rashaideh, 2017; Momani, Abu Arqub, Hayat, and Al-Sulami;
2014). To obtain sufficient conditions for the approximate controllability, many
authors frequently used either the range condition of controller with a compact
2
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semigroup introduced in Naito (1987) or the so called resolvent condition for using a fixed point theorem introduced in (Bashirov and Mahmudov, 1999;Fu, 2004).
Recently, approximate controllability for semilinear control systems with a range
condition of the control action operator can be founded in (Jeong, Kwun and
Park, 1999;Mahmudov, 2006;Radhakrishnan and Balachandran, 2012;Sukavanam
and Tomar, 2007;Wang, 2009). In particular, with conditions the range condition
of controller with a compact semigroup, Wang (2009) established the approximate
controllability for System (1,1) assuming
||f (·, u)||L2(0,T ;H)
:= γ
||u||L2(0,T ;H)
|||u||L2 (0,T ;H) →∞
lim sup
(1.4)
is sufficiently small, where
f (t, x) =
Z
0
t
k(t − s)g(s, x(s))ds
for a k belonging to L2 (0, T ).
In this paper, we will show the approximate controllability for the system (1,1)
even if γ 6= 1 of (1.4) by using so called Fredholm theory: (λT − S)(u) = f is
solvable, where T and S are nonlinear operators defined on a Banach space X
with value in a Banach space Y . In our case, the operator T acts as the identity
operator while S is completely continuous. Instead of the uniform boundedness
of the nonlinear terms and the compactness of the principal operators studied by
most parts of previous results, we make the natural assumption that the embedding
D(A0 ) := −D(A(x, Dx ) ⊂ V is compact. Then the embedding L2 (0, T ; D(A0 )) ∩
W 1,2 (0, T ; H) ⊂ L2 (0, T ; V ) is compact in view of Aubin (1963), and we show that
the mapping which maps w to the mild solution of (1.1) is a compact operator from
L2 (0, T ; U) to L2 (0, T ; V ).
The paper is organized as follows. In Section 2, we present the regularity and
variations of constant formula of solutions for the system (1.1), and introduce basic
properties of the Fredholm theory by using general degree theory in infinite dimensional spaces in Section 3. In Section 4, we will obtain the equivalent relations
between the reachable set of the semilinear retarded integrodifferential equation and
that of its corresponding linear system. Finally we give a simple example to which
our main result can be applied.
2
Retarded control equations
Let H and V be two complex Hilbert spaces such that V is a dense subspace of H.
The norm of H(resp. V ) is denoted by | · | ( resp. || · ||). Assume that the injection
3
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of V into H is continuous. The antidual of V is denoted by V ∗ , and the norm of
V ∗ by || · ||∗. Identifying H with its antidual, we may consider that H is embedded
in V ∗ . Hence we have V ⊂ H ⊂ V ∗ densely and continuously. 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.
We realize the operator A(x, Dx ), Aι (x, Dx ), ι = 1, 2, in Hilbert spaces by
A0 v = −A(x, Dx )v,
Aι v = −Aι (x, Dx )v, ι = 1, 2,
v∈V
(2.1)
in the distribution sense.
Let b(·, ·) be a sesquilinear form defined in V × V , that is, for each u, v ∈ V there
corresponds a complex number a(u, v) which is linear in u and antilinear in v:
b(u1 + u2 , v) = b(u1 , v) + b(u2 , v),
b(u, v1 + v2 ) = b(u, v1 ) + b(u, v2 ),
b(λu, v) = λb(u, v),
b(u, λv) = λb(u, v).
We assume that b(·, ·) is bounded and satisfies Gårding’s inequality
Re b(v, v) ≥ c0 ||v||2 − c1 |v|2 ,
c0 > 0,
c1 ≥ 0.
(2.2)
The operator A0 defined as (2.1) may be associated with the sesquilinear form −b(·, ·)
as follows:
(A0 v1 , v2 ) = −b(v1 , v2 ), v1 , v2 ∈ V.
The following result is known as the Lax-Milgram theorem.
Lemma 2.1. Let X be a Hilbert space. Assume that b(u, v) is sesquilnear form
defined on X × X satisfying (2.2). Then if f ∈ X ∗ , then there uniquely exists an
element u ∈ X such that f (v) = b(u, v) for every v ∈ X.
We can apply Lemma 2.1 to show that if f ∈ V ∗ then there exists a u ∈ V
such that (f, u) = b(u, v) for all v ∈ V , i.e., f = A0 u and hence, A0 is a bounded
linear operator from V to V ∗ . Moreover, it is well known that A0 and generates
an analytic semigroup S(t)(t ≥ 0) in both of H and V ∗ (see Tanabe, 1979), and its
realization in H which is the restriction of A0 to
D(A0 ) = {v ∈ V ; A0 v ∈ H}
is also denoted by A0 .
From the following inequalities
c0 ||v||2 ≤ Re b(v, v) + ω2 |v|2 ≤ |A0 u| |u| + c2 |u|2 ≤ max{1, c2 }||u||D(A0) |u|,
4
where
||v||D(A0) = (|Av|2 + |v|2)1/2
is the graph norm of D(A0 ), it follows that there exists a constant C0 > 0 such that
1/2
||v|| ≤ C0 ||v||D(A0) |v|1/2 .
(2.3)
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Thus, we have the following sequence
D(A0 ) ⊂ V ⊂ H ⊂ V ∗ ⊂ D(A0 )∗ ,
(2.4)
where each space is dense in the next one, which is continuous injection.
Lemma 2.2. With the notations (2.3), (2.4), we have
(V, V ∗ )1/2,2 = H and (D(A0 ), 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 Butzer and Berens, 1967;Triebel, 1977).
The mixed problem (1.1) can be formulated abstractly as

R0
d

 dt u(t) = A0Ru(t) + A1 u(t − h) + −h a(s)A2 u(t + s)ds
t
+ 0 k(t − s)g(s, u(t, x(s))ds + (Bw)(t), 0 ≤ t ≤ T,


u(0) = g 0, u(s) = g 1 (s), −h ≤ s ≤ 0.
(2.5)
The operators A1 and A2 are bounded linear operators from V to V ∗ such that their
restrictions to D(A0 ) are bounded linear operators from D(A0 ) equipped with the
graph norm of A0 to H. The function a(·) is assumed to be real valued and belongs
to L2 (−h, 0).
We assume the following hypotheses on the nonlinear mappings g in (2.5):
Assumption (F). g : [0, T ] × V −→ H is a nonlinear mapping such that
(i) t 7→ g(t, x) is measurable;
(iii) x 7→ g(t, x) is odd mapping (g(·, x) = −g(·, −x));
(iii) there exists a constant L such that
|g(t, x) − g(t, y)| ≤ L||x − y||,
x, y ∈ V.
We assume that g(t, 0) = 0 for the sake of simplicity.
5
For x ∈ L2 (0, T ; V ) we set
f (t, x) =
Z
0
t
k(t − s)g(s, x(s))ds,
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where k belongs to L2 (0, T ).
Lemma 2.3. Let u ∈ L2 (0, T ; V ), T > 0. Then f (·, u) ∈ L2 (0, T ; H),
√
|f (t, u1(t)) − f (t, u2 (t))| ≤ L t||k||L2(0,t) ||u1 − u2 ||C([0,t];V )
and
√
||f (·, u)||L2(0,T ;H) ≤ L||k||L2(0,T ) T ||u||L2(0,T ;V ) .
Moreover if u1 , u2 ∈ L2 (0, T ; V ), then
√
||f (·, u1) − f (·, u2)||L2 (0,T ;H) ≤ L||k||L2(0,T ) T ||u1 − u2 ||L2 (0,T ;V ) .
Proof. By using Hölder inequality, we obtain
Z t
|f (t, u1 (t)) − f (t, u2 (t))| ≤
|k(t − s){g(s, u1(s)) − g(s, u2(s))}|ds
0
≤
Z
t
2
0
|k(t − s)| ds
1/2 Z
t
2
0
2
L ||u1(s) − u2 (s)|| ds
√
≤ L t||k||L2(0,t) ||u1 − u2 ||C([0,t];V ) .
1/2
From Assumption (F) and using the Hölder inequality, it is easily seen that
Z T Z t
2
||f (·, u)||L2(0,T ;H) ≤
|
k(t − s)g(s, u(s))ds|2dt
0
0
Z TZ t
2
L2 ||u(s)||2dsdt
≤ ||k||L2 (0,T )
0
0
≤ L2 ||k||2L2 (0,T ) T ||u||2L2(0,T ;V ) .
Finally, if u1 , u2 ∈ L2 (0, T ; V ), then
Z T Z t
||f (·, u1) − f (·, u2)||L2 (0,T ;H) =
|
k(t − s) g(s, u1(s)) − g(s, u2(s)) ds|2 dt
0
0
√
≤ L||k||L2 (0,T ) T ||u1 − u2 ||L2 (0,T ;V ) ,
which shows the second paragraph.
6
Let the solution spaces W0 (T ) and W1 (T ) of strong solutions be defined by
W0 (T ) = L2 (0, T ; D(A)) ∩ W 1,2 (0, T ; H),
W1 (T ) = L2 (0, T ; V ) ∩ W 1,2 (0, T ; V ∗ ).
Here, we note that by using interpolation theory, we have
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W0 (T ) ⊂ C([0, T ]; V ),
W1 (T ) ⊂ C([0, T ]; H).
By virtue of Theorem 3.1 of Jeong, Kwun and Park (1999), we have the following
result on the corresponding linear equation of (2.5).
Proposition 2.1. Suppose that the assumptions stated above are satisfied. Then
the following properties hold:
1)(linear case) Let f (t, x) ≡ f (t) ∈ L2 (0, T ; H) as considered a forcing term, and
let the controller B be a bounded linear operator from L2 (0, T ; U) to L2 (0, T ; H).
Then for (g 0, g 1 ) ∈ V × L2 (−h, 0; D(A0 )) and w ∈ L2 (0, T ; U), T > 0, there exists
a unique solution u of the linear equation (2.5) belonging to
W0 (T ) ⊂ C([0, T ]; V ),
and satisfying
||u||W0(T ) ≤ C1 (||g 0|| + ||g 1||L2 (−h,0;D(A0 )) + ||f ||L2(0,T ;H) + ||Bw||L2(0,T ;H) ),
(2.6)
where C1 is a constant depending on T .
2) Let the controller B be a bounded linear operator from L2 (0, T ; U) to L2 (0, T ; H),
and let Assumption (F) be satisfied. Then for (g 0 , g 1) ∈ V × L2 (−h, 0; D(A0 )) and
w ∈ L2 (0, T ; U), there exists a unique solution u of the semilinear equation (2.5)
belonging to
W0 (T ) ⊂ C([0, T ]; V ),
and satisfying
||u||W0(T ) ≤ C1 (1 + ||g 0|| + ||g 1||L2 (−h,0;D(A0 )) + ||Bw||L2(0,T ;H) ),
(2.7)
where C1 is a constant depending on T .
3) Let the controller B be a bounded linear operator from L2 (0, T ; U) to L2 (0, T ; V ∗ )
and let Assumption (F) be satisfied. Then for (g 0, g 1 ) ∈ H × L2 (−h, 0; V ) and
w ∈ L2 (0, T ; U), there exists a unique solution u of the semilinear equation (2.5)
belonging to
W1 (T ) ⊂ C([0, T ]; H),
and satisfying
||u||W1(T ) ≤ C1 (1 + |g 0| + ||g 1||L2 (−h,0;V ) + ||Bw||L2(0,T ;V ∗ ) ),
where C1 is a constant depending on T .
7
(2.8)
Given u ∈ L2 (0, T ; V ), we extend it to the space L2 (−h, T ; V ) by setting u(s) =
g (s) for s ∈ (−h, 0).
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1
Theorem 2.1. Suppose that Assumption (F) is satisfied and the controller B is
bounded from L2 (0, T ; U) to L2 (0, T ; H). If (g 0, g 1 ) ∈ V × L2 (−h, 0; D(A0 )) and
w ∈ L2 (0, T ; U), then u ∈ W0 (T ), and the mapping (g 0 , g 1 , w) 7→ u ∈ W0 (T ) is
continuous.
Proof. For (g 0, g 1 ) ∈ V × L2 (−h, 0; D(A0 )) and w ∈ L2 (0, T ; U), by Proposition 2.1,
u belongs to W0 (T ). Let (gi0 , gi1, wi )∈V ×L2 (−h, 0; D(A0 )) ×L2 (0, T ; H), and ui be
the solution of (2.5) with (gi0 , gi1 , wi ) in place of (g 0 , g 1 , w) for i = 1, 2. Then in view
of (2.6) and Lemma 2.3 we have
||u1 − u2 ||W0 (T )) ≤ C1 {||g10 − g20 ||
(2.9)
1
1
+ ||g1 − g2 ||L2 (−h,0:D(A0 )) + ||f (·, u1) − f (·, u2)||L2 (0,T ;H)
+ ||Bw1 − Bw2 ||L2 (0,T ;H) }
≤ C1 {||g10 − g20 || + ||g11 − g21 ||L2 (−h,0:D(A0 )) + ||Bw1 − Bw2 ||L2 (0,T ;H)
√
+ L||k||L2(0,T ) T ||u1 − u2 ||L2 (0,T :V ) }.
Since
u1 (t) − u2 (t) =
we get
||u1 − u2 ||L2 (0,T ;H) ≤
g10
√
−
g20
+
Z
0
t
(u̇1 (s) − u̇2(s))ds,
T
T |g01 − g20 | + √ ||u1 − u2 ||W 1,2(0,T ;H) .
2
Hence, arguing as in (2.9) we get
1/2
1/2
||u1 − u2 ||L2 (0,T ;V ) ≤ C0 ||u1 − u2 ||L2 (0,T ;D(A0 )) ||u1 − u2 ||L2 (0,T ;H)
(2.10)
1/2
≤ C0 ||u1 − u2||L2 (0,T ;D(A0 ))
T
1/2
× {T 1/4 |g10 − g20 |1/2 + ( √ )1/2 ||u1 − u2 ||W 1,2 (0,T ;H) }
2
T
1/2
≤ C0 T 1/4 |g10 − g20 |1/2 ||u1 − u2 ||L2 (0,T ;D(A0 )) + C0 ( √ )1/2 ||u1 − u2 ||W0 (T )
2
T
≤ 2−7/4 C0 |g10 − g20 | + 2C0 ( √ )1/2 ||u1 − u2 ||W0 (T ) .
2
8
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Combining (2.9) and (2.10) we obtain
||u1 − u2 ||W0 (T ) ≤C1 {||g10 − g20 ||X + ||g11 − g21 ||L2 (−h,0:D(A0 ))
+ ||Bw1 − Bw2 ||L2 (0,T ;H) }
√
+ 2−7/4 C0 C1 L||k||L2 (0,T ) T |g10 − g20|
√
T
+ 2C0 C1 ( √ )1/2 L||k||L2 (0,T ) T ||u1 − u2 ||W0 (T ) .
2
(2.11)
Suppose that (gn0 , gn1 , wn ) → (g 0 , g 1 , w) in V ×L2 (−h, 0; D(A0 ))×L2 (0, T ; H), and
let un and u be the solutions (2.5) with (gn0 , gn1 , wn ) and (g 0, g 1 , w) respectively. Let
0 < T1 ≤ T be such that
p
√
2C0 C1 (T1 / 2)1/2 L||k||L2 (0,T ) T1 < 1.
Then by virtue of (2.11) with T replaced by T1 , we see that un → u in W0 (T1 ).
This implies that (un (T1 ), (un )T1 ) 7→ (u(T1 ), uT1 ) in V ×L2 (−h, 0; D(A0 )). Hence the
same argument shows that un → u in
L2 (T1 , min{2T1 , T }; D(A0 )) ∩ W 1,2 (T1 , min{2T1 , T }; H).
Repeating this process we conclude that un → u in W0 (T ).
3
Basic Fredholm theory
Let us introduce the theory of the degree for completely continuous perturbations of
the identity operator, which is the infinite dimensional version of Borsuk’s theorem.
Let 0 ∈ D be a bounded open set in a Banach space X, D its closure and ∂D its
boundary. The number d[I −T ; D, 0] is the degree of the mapping I −T with respect
to the set D and the point 0 (see Fučik, Nečas, Souček and Souček, 1973;Lloid, 1978).
Theorem 3.1. (Borsuk’s theorem) Let D be a bounded open symmetric set in a
Banach space X, 0 ∈ D. Suppose that T : D → X be odd completely continuous
operator satisfying T (x) 6= x for x ∈ ∂D. Then d[I − T ; D, 0] is odd integer. That
is, there exists at least one point x0 ∈ D such that (I − T )(x0 ) = 0.
Definition 3.1. Let T be a mapping defined by on a Banach space X with value in
a real Banach space Y . The mapping T is said to be a (K, L, α)-homeomorphism of
X onto Y if
(i) T is a homeomorphism of X onto Y ;
9
(ii) there exist real numbers K > 0, L > 0, and α > 0 such that
L||x||αX ≤ ||T (x)||Y ≤ K||x||αX ,
∀x ∈ X.
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Lemma 3.1. Let T be an odd (K, L, α)-homeomorphism of X onto Y and S : X →
Y a continuous operator satisfying
||S(x)||Y
= N ∈ R+ .
α
||x||
||x||X →∞
X
lim sup
N N
, L ] ∪ {0},
Then for |λ| ∈
/ [K
lim
||x||X →∞
||λT (x) − S(x)||Y = ∞.
Proof. Suppose that there exist a constant M > 0 and a sequence {xn } ⊂ X such
that
||λT (xn ) − S(xn )||Y ≤ M
as xn → ∞. From this it follows that
λT (xn )
S(xn )
−
→ 0.
α
||xn ||X
||xn ||αX
Hence, we have
lim sup
n→∞
|λ|||T (xn )||Y
= N,
||xn ||αX
N N
, L ].
and so, |λ|K ≥ N ≥ |λ|L. It is a contradiction with |λ| ∈
/ [K
Proposition 3.1. Let T be an odd (K, L, α)-homeomorphism of X onto Y and
S : X → Y an odd completely continuous operator. Suppose that for λ 6= 0,
lim
||x||X →∞
||λT (x) − S(x)||Y = ∞.
(3.1)
Then λT − S maps X onto Y .
Proof. We follow the proof Theorem 1.1 in Chapter II of Fučik, Nečas, Souček and
Souček (1973). Suppose that there exists y ∈ Y such that λT (x) = y. Then from
(3.1) it follows that ST −1 : Y → Y is an odd completely continuous operator and
y
||y − ST −1 ( )||Y = ∞.
||y||Y →∞
λ
lim
10
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Let y0 ∈ Y . There exists r > 0 such that
y
||y − ST −1 ( )||Y > ||y0 ||Y ≥ 0
λ
for each y ∈ Y satisfying ||y||Y = r. Let Yr = {y ∈ Y : ||y||Y < r} be a open ball.
Then by view of Theorem 3.1, we have d[y − ST −1 ( λy ); Yr , 0] is an odd number. For
each y ∈ Y satisfying ||y||Y = r and t ∈ [0, 1], there is
y
y
||y − ST −1( ) − ty0 ||Y ≥ ||y − ST −1 ( )||Y − ||y0||Y > 0
λ
λ
and hence, by the homotopic property of degree, we have
y
y
d[y − ST −1 ( ); Yr , y0 ] = d[y − ST −1( ); Yr , 0] 6= 0.
λ
λ
Hence, by the existence theory of the Leray-Schauder degree, there exists a y1 ∈ Yr
such that
y1
y1 − ST −1( ) = y0 .
λ
We can choose x0 ∈ X satisfying λT (x0 ) = y1 , and so, λT (x0 ) − S(x0 ) = y0 . Thus,
it implies that λT − S is a mapping of X onto Y .
Combining Lemma 3.1. and Proposition 3,1, we have the following results.
Corollary 3.1. Let T be an odd (K, L, α)-homeomorphism of X onto Y and S :
X → Y an odd completely continuous operator satisfying
||S(x)||Y
= N ∈ R+ .
α
||x||
||x||X →∞
X
lim sup
N N
, L ] ∪ {0}, the operator λT − S maps X onto Y .
Then for |λ| ∈
/ [K
4
Controllability of semilinear retarded control
equations
Let U is a Banach space of control variables and let the controller B be a linear
bounded operator from L2 (0, T ; U) to L2 (0, T ; H). In this section, we concern with
the approximate controllability of the following retarded semilinear control system
(2.5) with initial data (g 0 , g 1) = (0, 0):
(
R0
d
u(t) = A0 u(t) + A1 u(t − h) + −h a(s)A2 u(t + s)ds + f (t, u) + (Bw)(t),
dt
u(0) = 0, u(s) = 0 − h ≤ s < 0.
(4.1)
11
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The solution of (4.1) is denoted by u(t; g, w). Let S(·) be the analytic semigroup
generated by A0 , and let W (·) be the fundamental solution of the linear equation
associated with (4.1) in the sense of Nakagiri (1988);Tanabe (1992), which is the
operator valued function satisfying
(
Rt
R0
W (t) = S(t) + 0 S(t − s){A1 W (s − h) + −h a(τ )A2 W (s + τ )dτ }ds, t > 0
W (0) = I,
W (t) = 0 − h ≤ t < 0.
Then
u(t; g, w) =
Z
t
0
W (t − s){f (s, u(·, g, w)) + (Bw)(s)}ds,
and in view of (2.8)
||u(·; g, w)||L2(0,T ;V )∩W 1,2 (0,T ;V ∗ ) ≤ C1 (1 + ||B||||w||L2(0,T ;U ) ).
(4.2)
In order to obtain approximate controllability for the system (4.1), we need to impose
the following conditions.
Assumption (A). The embedding D(A0 ) ⊂ V is completely continuous.
Assumption (F1). g : [0, T ] × H −→ H is a nonlinear mapping such that
(i) t 7→ g(t, x) is measurable;
(iii) x 7→ g(t, x) is odd mapping g(·, x) = −g(·, −x));
(iii) there exists a constant L such that
|g(t, x) − g(t, y)| ≤ L|x − y|,
x, y ∈ H.
We assume that g(t, 0) = 0 for the sake of simplicity.
Here, we remark that since Assumption(F) is more general than (F1), the regular
property of solutions of (4.1) mentioned in Section 2 can be available.
Theorem 4.1. For w ∈ L2 (0, T ; U), let uw be the solution of equation (4.1), and
let Assumptions (A) and (F1) be satisfied. Then the mapping w 7→ uw is compact
from L2 (0, T ; U) to L2 (0, T ; V ).
Proof. If w ∈ L2 (0, T ; U), then in view of (2.8) in Proposition 2.1
||uw ||W1 (T ) ≤ C1 (1 + ||B|| ||w||L2(0,T ;U ) ).
12
(4.3)
Since uw ∈ L2 (0, T ; V ), we have f (·, uw ) ∈ L2 (0, T ; H). Consequently
uw ∈ L2 (0, T ; D(A0 ) ∩ W 1,2 (0, T ; H).
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Hence, with aid of (2.6) of Proposition 2.1, Lemma 2.3, and (4.3),
||uw ||L2 (0,T ;D(A0 )∩W 1,2 (0,T ;H)
≤ C1 (||f (·, uw ) + Bw||L2(0,T ;H) )
√
≤ C1 {L||k||L2 (0,T ) T ||u||L2(−h,T −h;V ) + ||Bw||L2(0,T ;H) }
√
≤ C1 [C1 L||k||L2 (0,T ) T {(1 + ||B|| ||u||L2(0,T ;U ) )} + ||B|| ||w||L2(0,T ;U ) ].
(4.4)
Hence, if w is bounded in L2 (0, T ; H), then so is uw in L2 (0, T ; D(A0))∩W 1,2 (0, T ; H).
Since D(A0 ) is compactly embedded in V by assumption, the embedding
L2 (0, T ; D(A0) ∩ W 1,2 (0, T ; V ) ⊂ L2 (0, T ; V )
is completely continuous in view of Theorem 2 of Aubin (1963), and by Theorem 2.1,
the mapping w 7→ uw is completely continuous from L2 (0, T ; U) to L2 (0, T ; V ).
We define the operator F by
F (w) = f (·, uw ),
∀w ∈ L2 (0, T ; U).
(4.5)
By virtue of Assumption(F) in Section 2, we obtain the following results.
Corollary 4.1. Let Assumptions (A) and (F1) be satisfied. Then F defined by (4.5)
is a completely continuous mapping from L2 (0, T ; U) to L2 (0, T ; H).
Let us define the reachable sets for the system (4.1) as follows:
RT (g) = {u(T ; g, w) : w ∈ L2 (0, T ; U)},
RT (0) = {u(T ; 0, w) : w ∈ L2 (0, T ; U)}.
Definition 4.1. If RT (g) = H, where RT (g) is the closure of RT (g) in H, then the
system (4.1) is called approximately controllable at time T .
By the same argument as in the proof of Lemma 2.3 in the sense of Assumption
(F1), we know that
√
||f (·, u)||L2(0,T ;H) ≤ L||k||L2 (0,T ) T ||u||L2(0,T ;H) .
Hence, we have
√
||f (·, u)||L2(0,T ;H)
:= γ < L||k||L2 (0,T ) T .
||u||L2(0,T ;H)
|||u||L2 (0,T ;H) →∞
lim sup
13
(4.6)
Theorem 4.2. Assume that (A) and (F1) are satisfied and γ ∈ R+ − {1}, where γ
is the constant in (4.6). Then we have
RT (0) ⊂ RT (g).
Therefore, if the linear system (4.1) with g = 0 is approximately controllable, then
so is the nonlinear system (4.1).
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Proof. Let
η=
Z
0
T
W (T − s)Bv(s)ds ∈ RT (0).
We are going to show that there exists w such that
η = x(T ; g, w).
Let
2
N = {p ∈ L (0, T ; H) :
Z
T
0
W (T − s)p(s)ds = 0}
and denote by N ⊥ be the orthogonal complement of N in L2 (0, T ; H). We denote
the range of the operator B by HB . For y ∈ N ⊥ , let P y be the unique minimum
norm element of {y + N} ∩ H B . Then the proof of Lemma 1 of Naito (1987) can
be applied to show that P is a linear and continuous operator from N ⊥ to H B . Let
Ỹ = L2 (0, T ; H)/N be the quotient space and the norm of a coset ỹ = y + N ∈ Ỹ
is defined of ||ỹ|| = inf{|y + g| : g ∈ N}. We define by Q the isometric isomorphism
from Ỹ onto N ⊥ , that is, Qỹ is the minimum norm element in ỹ = {y + g : g ∈ N}.
Let
F ỹ = F (P Qỹ) + N
for ỹ ∈ Ỹ . Then by Lemma 2.3 and Theorem 4.1, F is a completely continuous
mapping from Ỹ to itself. Moreover, by (4.6), it holds
||F (y)||Ỹ
= γ < ∞.
||y||Ỹ
||y||Ỹ →∞
lim sup
(4.7)
Since the identity operator IỸ on Ỹ is an odd (1, 1, 1)-homeomorphism and, by
Assumption (F1) and (4.7), −F : Ỹ → Ỹ an odd completely continuous operator
satisfying
lim ||λI(y) + F (y)||Y = ∞,
||y||Ỹ →∞
where λ 6= γ , noting that γ ∈ R+ − {1} and by Corollary 3.1, we have that λI + F
maps Ỹ onto Ỹ . Hence, if we set z = Bv, there exists w̃ ∈ Ỹ such that
z̃ = w̃ + F w̃.
14
(4.8)
Put w = Qw̃ and wB = P Qw̃. Then we have that wB = P w and w−wB = w−P w ∈
N. Hence,
z̃ = F (wB ) + w + N = F (wB ) + wB + N.
(4.9)
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For g ∈ L2 (0, T ; H) let ûg be the solution of equation with B = I



Then
d
u(t)
dt
= A0 u(t) + A1 u(t − h)
R0
+ −h a(s)A2 u(t + s)ds + f (t, x) + g(t),


u(0) = 0, u(s) = 0 − h ≤ s < 0.
ûg (t) =
Z
0
(4.10)
t
W (t − s){f (s, ûg (·)) + g(s)}ds,
and by (2.8),
||ûg ||L2 (0,T ;V )∩W 1,2 (0,T ;V ∗ ) ≤ C2 (1 + ||g||L2(0,T ;H) ).
Since ûg ∈ L2 (0, T ; V ), f (·, ûg ) ∈ L2 (0, T ; H). Consequently, ûg ∈ L2 (0, T ; D(A0)) ∩
W 1,2 (0, T ; H). Thus, from (4.9) it follows that
η=
Z
T
W (T − s)(F (wB )(s) + wB (s))ds
0
=
Z
0
T
W (T − s)(f (s, ûwB ) + wB (s))ds.
Since wB ∈ H B , there exists a sequence {vn } ∈ L2 (0, T ; U) such that Bvn 7→ wB in
L2 (0, T ; H). Then by the last part of Theorem 2.1 we have that u(·; g, vn) 7→ ûuB
in L2 (0, T ; D(A0)) ∩ W 1,2 (0, T ; H), and hence u(T ; g, vn) 7→ ûuB (T ) = η in V . Thus
V
V
we conclude η ∈ RT (g) , where RT (g) is the closure of RT (g) in V , which says
η ∈ RT (g).
Remark 4.1. In general, to obtain sufficient conditions for approximate controllability of semilinear control systems, many authors frequently used the following range
condition of controller with a compact semigroup introduced in Naito (1987):
Assumption (B). For each p ∈ L2 (0, T ; X) there exists a function q ∈ X B :
S̃p = S̃q, where
Z T
S̃p =
S(T − s)p(s)ds,
0
where S(t)(t ≥ 0) is an analytic semigroup generated by A0 .
15
In particular, in Wang (2009) established the approximate controllability with
local Lipschitz continuity for nonlinear term of System (4,1) assuming Assumption
(B) with the compactness of S(t)(t ≥ 0) and
||f (·, u)||L2(0,T ;H)
:= γ
||u||L2(0,T ;H)
|||u||L2 (0,T ;H) →∞
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lim sup
is sufficiently small.
In this paper, to prove approximate controllability of (4.1) for γ ∈ R+ − {1}
more generally, we no longer require the compactness of S(t)(t ≥ 0), Assumption
(B) and the uniform bundedness of the nonlinear term, but instead we need the
natural assumption that the embedding D(A0 ) ⊂ V is compact.
5
Examples
Let Ω be a bounded domain in Rn with smooth boundary ∂Ω. Let aij be a realvalued and smooth function for each i, j = 1, · · · , n. Assume that 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
n
X
i,j=1
aij (x)ξi ξj ≥ c0 |ξ|2
(5.1)
for
ξ. Let bi ∈ L∞ (Ω) and c ∈ L∞ (Ω). Put βi =
Pn all x ∈ Ω̄ and all real vectors
∞
i=1 ∂aij /∂xj + bi , then βi ∈ L (Ω). Consider an elliptic differential operator of
second order in (1.1) as
n
n
X
X
∂
∂
∂
(ai,j (x)
)+
bi (x)
+ c(x).
A(x, Dx ) = −
∂xj
∂xi
∂xi
i=1
i,j=1
For each u, v ∈ H1 (Ω), we put
)
Z (X
n
n
X
∂u ∂v
∂u
b(u, v) =
aij
+
βi
v̄ + cuv̄ dx.
∂xi ∂xj
∂x
i
Ω
i,j=1
i=1
Since {aij } is real symmetric, by (5.1) the inequality
n
X
i,j=1
aij (x)ζi ζ̄j ≥ c0 |ζ|2
16
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holds for all complex vectors ζ = (ζ1 , · · · , ζn ). On the other hand, by this hypothesis, there exists a certain K such that |βi (x)| ≤ K and |c(x)| ≤ K hold almost
everywhere. Hence,
Z
Z X
Z
n n X
∂u 2
∂u Re b(u, u) ≥ c0
|u|2dx
∂xi dx − K
∂xi |u|dx − K
Ω
Ω i=1
Ω
i=1
!
2
2
Z X
Z X
n
n ∂u ∂u 1
ε
2
≥c0
∂xi + 2ǫ |u| dx
∂xi dx − K
2
Ω i=1
Ω i=1
Z
−K
|u|2dx
Ω
2
Z
n Z ǫ X ∂u nK
= c0 − K
|u|2dx.
∂xi dx − 2ε + K
2
Ω
i=1 Ω
By choosing ǫ = c0 K −1 , we obtain
2
Z
n Z c0 X ∂u nK 2
Re b(u, u) ≥
dx −
+K
|u|2dx
2 i=1 Ω ∂xi
2c0
Ω
c0
c0
nK 2
kuk2 .
= kuk21 −
+K +
2
2c0
2
Therefore, it follows that, for any closed subspace V of H1 (Ω) containing H01 (Ω),
the quadratic form b(u, v) satisfies (2.2), where H = L2 (Ω). Let A0 be the operator
associated with this sesquilinear form:
(A0 u, v) = b(u, v),
u, v ∈ V.
Then −A0 is a bounded linear operator from V to V ∗ by the Lax-Milgram Theorem.
The realization of A0 in H which is the restriction of A0 to
D(A0 ) = {u : u ∈ H 2 (Ω) ∩ H01 (Ω)} = {u : u ∈ H 2 (Ω), u|∂Ω = 0},
A0 u = −A(x, Dx )u for any u ∈ D(A)
is also denoted by A0 . as the result known as Sobolev’s imbedding theorem, the
embedding D(A0 ) ⊂ V is compact. Let Aι (x, Dx )(ι = 1, 2) be second order linear
differential operators with real coefficients. We realize the operator Aι (x, Dx ), ι =
1, 2, in Hilbert spaces by
Aι v = −Aι (x, Dx )v, ι = 1, 2,
17
v∈V
in the distribution sense.
We set
′
g(x(t, y)) = g1 (|x(t, y)|2)x(t, y),
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where the function g1 given is real valued belong to C 1 ([0, ∞)) which satisfies the
conditions:
(i) g1 (0) = 0, g1 (r) ≥ 0 for r > 0,
′
(ii) |g1(r) ≤ c for c > 0.
If we present
f (t, u(t, x)) =
Z
0
and
t
k(t − s)g(s, u(s, x(s))ds,
||f (·, u)||L2(0,T ;H)
6= 1.
||u||L2(0,T ;H)
|||u||L2(0,T ;H) →∞
lim sup
then Assumption (F1) is satisfied. Hence, from Theorem 4.2, we have
RT (0) ⊂ RT (g),
that is, if the linear system (1.1) with g = 0 is approximately controllable, then so
is the nonlinear system (1.1).
6
Conclusion
In this paper, authors establish the approximate controllability for a class of semilinear integro-differential functional control equations, which is provided under general
sufficient conditions on the system operator, controller and nonlinear terms. To
obtain sufficient conditions for the approximate controllability, many authors frequently used either the range condition of controller with a compact semigroup or
the so called resolvent condition for using a fixed point theorem.
This paper wants to use a different method than the previous one. The approach
followed here is the results similar to Fredholm alternative for nonlinear operators
under restrictive assumption, which is on the solution of nonlinear operator equations λT (x) − S(x) = y in dependence on the real number λ, where T and S are
nonlinear operators defined a Banach space X with values in a Banach space Y .
In order to obtain the approximate controllability for a class of semilinear integrodifferential functional control equations, it is necessary to suppose that T acts as the
18
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identity operator while S related to the nonlinear term of control equations is completely continuous. The approach developed in this paper can be extended to more
general sufficient assumptions such as the growth condition on nonlinear terms.
Future works will concentrate on extending the developed our results to both
nonlinear neutral evolution systems with nonlocal conditions and impulsive control
with delays, as well as semilinear retarded stochastic systems
Acknowledgement This research was supported by Basic Science Research Program through the National research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2015R1D1A1A09059030).
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