SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Approximate controllability and regularity for nonlinear differential equations

Jin-Mun Jeong1*, Jin-Ran Kim2 and Eun-Young Ju1

Author Affiliations

1 Department of Applied Mathematics, Pukyong National University, Busan 608-737, Korea

2 Department of Mathematics, Dong-A University Saha-Gu, Busan 604-714, Korea

For all author emails, please log on.

Advances in Difference Equations 2011, 2011:27  doi:10.1186/1687-1847-2011-27

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2011/1/27


Received:24 March 2011
Accepted:23 August 2011
Published:23 August 2011

© 2011 Jeong et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this article, we deal with the existence, uniqueness, and a variation of solutions of the nonlinear control system with nonlinear monotone hemicontinuous and coercive operator. Moreover, the approximate controllability for the given nonlinear control system is studied.

Keywords:
nonlinear differential equation; regularity; reachable set; degree theory; approximately controllable

1 Introduction

Let H and V be two real separable Hilbert spaces such that V is a dense subspace of H. We are interested in the following nonlinear differential control system on H:

x ( t ) + A x ( t ) = g ( t , x t , 0 t k ( t , s , x s ) d s ) + ( B u ) ( t ) , 0 < t , x ( 0 ) = ϕ 0 , x ( s ) = ϕ 1 ( s ) - h s 0 , (SE)

where the nonlinear term, which is a Lipschitz continuous operator, is a semilinear version of the quasi-linear form. The principal operator A is assumed to be a single valued, monotone operator, which is hemicontinuous and coercive from V to V*. Here, V* stands for the dual space of V . Let U be a Banach space of control variables. The controller B is a linear-bounded operator from a Banach space L2(0, T; U) to L2(0, T; H) for any T > 0. Let the nonlinear mapping k be Lipschitz continuous from ℝ × [- h, 0] × V into H. If the right-hand side of the equation (SE) belongs to L2(0, T; V* ), then it is well known as the quasi-autonomous differential equation(see Theorem 2.6 of Chapter III in [1]).

The problem of existence for solutions of semilinear evolution equations in Banach spaces has been established by several authors [1-3]. We refer to [2,4,5] to see the existence of solutions for a class of nonlinear evolution equations with monotone perturbations

First, we begin with the existence, and a variational constant formula for solutions of the equation (SE) on L2(0, T; V ) ∩ W1,2(0, T; V* ), which is also applicable to optimal control problem. We prove the existence and uniqueness for solution of the equation by converting the problem into a fixed point problem. Thereafter, based on the regularity results for solutions of (SE), we intend to establish the approximate controllability for (SE). The controllability results for linear control systems have been proved by many authors, and several authors have extended these concepts to infinite dimensional semilinear system (see [6-8]). In recent years, as for the controllability for semilinear differential equations, Carrasco and Lebia [9] discussed sufficient conditions for approximate controllability of parabolic equations with delay, and Naito [10] and other authors [6-8,11] proved the approximate controllability under the range conditions of the controller B.

The previous results on the approximate controllability of a semilinear control system have been proved as a particular case of sufficient conditions for the approximate solvability of semilinear equations, assuming either that the semigroup generated by A is a compact operator or that the corresponding linear system (SE) when g ≡ 0 is approximately controllable. However, Triggian [12] proved that the abstract linear system is never exactly controllable in an infinite dimensional space when the semigroup generated by A is compact. Thus, we will establish the approximate controllability under more general conditions on the nonlinear term and the controller.

Our aim in this article is to establish the approximate controllability for (SE) under a stronger assumption that {y : y(t) = (Bu)(t), u L2(0, T; U)} is dense subspace of L2(0, T, H), which is reasonable and widely used in case of the nonlinear system. We show that the input to solution (control to state) map is compact by using the fact that L2(0, T; V ) ∩ W1,2(0, T; V* ) furnished with the usual topology is compactly embedded in L2(0, T, H), provided that the injection V H is compact.

In the last section, we give a simple example to which the range conditions of the controller can be applied.

2 Nonlinear functional equations

Let H and V be two real Hilbert spaces. Assume and V is dense subspace in H and the injection of V into H is continuous. If H is identified with its dual space, then we may write V H V* densely, and the corresponding injections are continuous. The norm on V (resp. H) will be denoted by || · || (resp. |· |). 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. For the sake of simplicity, we may consider

u * u u , u V

where || · ||* is the norm of the element of V*. If an operator A is bounded linear from V to V* and generates an analytic semigroup, then it is easily seen that

H = { x V * : 0 T A e t A x * 2 d t < } ,

for the time T > 0. Therefore, in terms of the intermediate theory we can see that

( V , V * ) 1 2 , 2 = H

where ( V , V * ) 1 2 , 2 denotes the real interpolation space between V and V*.

We note that a nonlinear operator A is said to be hemicontinuous on V if

w - lim t 0 A ( x + t y ) = A x

for every x, y V where "w - lim" indicates the weak convergence on V*. Let A : V V* be given a single-valued, monotone operator and hemicontinuous from V to V* such that

(A1) A(0) = 0, (Au - Av, u - v) ≥ ω1 ||u - v||2 - ω2 |u - v|2 ,

(A2) ||Au||* ω3(||u|| + 1)

for every u, v V where ω2 ∈ ℝ and ω1, ω3 are some positive constants.

Here, we note that if 0 ≠ A(0), then we need the following assumption:

( A u , u ) ω 1 u 2 - ω 2 u 2

for every u V . It is also known that A is maximal monotone, and R(A) = V* where R(A) denotes the range of A.

Let the controller B is a bounded linear operator from a Banach space L2(0, T; U) to L2(0, T; H) where U is a Banach space.

For each t ∈ [0, T], we define xt : [ -h, 0] → H as

x t ( s ) = x ( t + s ) , - h s 0 .

We will set

Π = L 2 ( - h , 0 ; V ) and + = [ 0 , ) .

Let ℒ and ℬ be the Lebesgue σ-field on [0, ∞) and the Borel σ-field on [- h, 0] respectively. Let k : ℝ+ × ℝ+ × ∏ → H be a nonlinear mapping satisfying the following:

(K1) For any x. ∈ ∏ the mapping k(·, ·, x.) is strongly ℒ × ℬ -measurable;

(K2) There exist positive constants K0 , and K1 such that

k ( t , s , x . ) - k ( t , s , y . ) K 1 x . - y . Π , k ( t , s , 0 ) K 0

for all (t, s) ∈ ℝ+ × [ -h, 0] and x., y. ∈ ∏.

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

(G1) For any x ∈ ∏, y H the mapping g(·, x., y) is strongly ℒ -measurable;

(G2) There exist positive constants L0, L1, and L2 such that

g ( t , x . , y ) - g ( t , x ^ . , ŷ ) L 1 x . - x ^ . Π + L 2 y - ŷ , g ( t , 0 , 0 ) L 0

for all t ∈ ℝ+, x, x ^ . Π , and y, ŷ H .

Remark 2.1. The above operator g is the semilinear case of the nonlinear part of quasi-linear equations considered by Yong and Pan [13].

For x L2(-h, T; V ), T > 0 we set

G ( t , x ) = g ( t , x t , 0 t k ( t , s , x s ) d s ) .

Here, as in [13], we consider the Borel measurable corrections of x(·).

Lemma 2.1. Let x L2(- h, T; V ). Then, the mapping t xt belongs to C([0, T ]; ∏) and

x . L 2 ( 0 . T ; Π ) T x L 2 ( - h , T ; V ) . (2.1)

Proof. It is easy to verify the first paragraph and (2.1) is a consequence of the estimate:

x . L 2 ( 0 . T ; Π ) 2 0 T x t Π 2 d t 0 T - h 0 x ( t + s ) 2 d s d t 0 T d t - h T x ( s ) 2 d s T x L 2 ( - h , T ; V ) 2 .

Lemma 2.2. Let x L2(- h, T; V ), T > 0. Then, G, x) ∈ L2(0, T; H) and

G ( , x ) L 2 ( 0 , T ; H ) L 0 T + L 2 K 0 T 3 2 3 + ( L 1 T + L 2 K 1 T 3 2 2 ) x L 2 ( - h , T ; V ) . (2.2)

Moreover, if x1, x2 L2(- h, T; V ), then

G ( , x 1 ) - G ( , x 2 ) L 2 ( 0 , T ; H ) ( L 1 T + L 2 K 1 T 3 2 2 ) x 1 - x 2 L 2 ( - h , T ; V ) . (2.3)

Proof. It follows from (K2) and (2.1) that

0 k ( , s , x s ) d s L 2 ( 0 , T ; H ) 0 k ( , s , 0 ) d s L 2 ( 0 , T ; H ) + 0 ( k ( , s , x s ) - k ( , s , 0 ) ) d s L 2 ( 0 , T ; H ) K 0 T 3 2 3 + { 0 T 0 t K 1 x s Π d s 2 d t } 1 2 K 0 T 3 2 3 + { 0 T K 1 2 t 0 t x s Π 2 d s d t } 1 2 K 0 T 3 2 3 + K 1 T 2 x L 2 ( 0 , T ; Π ) K 0 T 3 2 3 + K 1 T 3 2 2 x L 2 ( - h , T ; V )

and hence, from (G2), (2.1), and the above inequality, it is easily seen that

G ( , x ) L 2 ( 0 , T ; H ) G ( , 0 ) + G ( , x ) - G ( , 0 ) L 0 T + L 1 x L 2 ( 0 , T ; Π ) + L 2 0 k ( , s , x s ) d s L 2 ( 0 , T ; H ) L 0 T + L 1 T x L 2 ( - h , T ; V ) + L 2 ( K 0 T 3 2 3 + K 1 T 3 2 2 x L 2 ( - h , T ; V ) ) .

Similarly, we can prove (2.3).

Let us consider the quasi-autonomous differential equation

x ( t ) + A x ( t ) = f ( t ) , 0 < t T , x ( 0 ) = ϕ 0 (E)

where A satisfies the hypotheses mentioned above. The following result is from Theorem 2.6 of Chapter III in [1].

Proposition 2.1. Let Φ 0 H and f L2(0, T; V* ). Then, there exists a unique solution x of (E) belonging to

C ( [ 0 , T ] ; H ) L 2 ( 0 , T ; V ) W 1 , 2 ( 0 , T ; V * )

and satisfying

x ( t ) 2 + 0 t x ( s ) 2 d s C 1 ( ϕ 0 2 + 0 t f ( s ) * 2 d s ) , (2.4)

0 t d x ( s ) d s * 2 d t C 1 ( ϕ 0 2 + 0 t f ( s ) * 2 d s ) (2.5)

where C1 is a constant.

Acting on both sides of (E) by x(t), we have

1 2 d d t x ( t ) 2 + ω 1 x ( t ) 2 ω 2 x ( t ) 2 + ( f ( t ) , x ( t ) ) .

As is seen Theorem 2.6 in [1], integrating from 0 to t, we can determine the constant C1 in Proposition 2.1.

We establish the following result on the solvability of the equation (SE).

Theorem 2.1. Let A and the nonlinear mapping g be given satisfying the assumptions mentioned above. Then, for any (Φ0, Φ1) ∈ H × L2(- h, 0; V ) and f L2(0, T; V*), T > 0, the following nonlinear equation

x ( t ) + A x ( t ) = G ( t , x ) + f ( t ) , 0 < t T , x ( 0 ) = ϕ 0 , x ( s ) = ϕ 1 ( s ) , - h s 0 (2.6)

has a unique solution x belonging to

L 2 ( - h , T ; V ) W 1 , 2 ( 0 , T ; V * ) C ( [ 0 , T ] ; H )

and satisfying that there exists a constant C2 such that

x L 2 ( - h , T ; V ) W 1 , 2 ( 0 , T ; V * ) C 2 ( 1 + ϕ 0 + ϕ 1 L 2 ( - h , 0 ; V ) + f L 2 ( 0 , T ; V * ) ) . (2.7)

Proof. Let y L2(0, T; V ). Then, we extend it to the interval (-h, 0) by setting y(s) = Φ1(s) for s ∈ (-h, 0), and hence, G, y(·)) ∈ L2(0, T; H) from Lemma 2.2. Thus, by virtue of Proposition 2.1, we know that the problem

x ( t ) + A x ( t ) = G ( t , y ) + f ( t ) , 0 < t , x ( 0 ) = ϕ 0 , x ( s ) = ϕ 1 ( s ) - h s 0 (2.8)

has a unique solution xy L2(0, T; V ) ∩ W1,2(0, T; V* ) corresponding to y. Let us fix T0 > 0 so that

ω 1 - 1 e ω 2 T 0 ( L 1 T 0 + L 2 K 1 T 0 3 2 2 ) < 1 . (2.9)

Let xi, i = 1, 2, be the solution of (2.8) corresponding to yi. Multiplying by x1(t) - x2(t), we have that

( 1 ( t ) - 2 ( t ) , x 1 ( t ) - x 2 ( t ) ) + ( A x 1 ( t ) - A x 2 ( t ) , x 1 ( t ) - x 2 ( t ) ) = ( G ( t , y 1 ) - G ( t , y 2 ) , x 1 ( t ) - x 2 ( t ) ) ,

and hence it follows that

1 2 d d t x 1 ( t ) x 2 ( t ) 2 + ω 1 x 1 ( t ) x 2 ( t ) 2 ω 2 x 1 ( t ) x 2 ( t ) 2 + G ( t , y 1 ) ) G ( t , y 2 ) * x 1 ( t ) x 2 ( t ) .

From Lemma 2.2 and integrating over [0,t], it follows

1 2 x 1 ( t ) - x 2 ( t ) 2 + ω 1 0 t x 1 ( s ) - x 2 ( s ) 2 d s 1 2 c 0 t G ( s , y 1 ) - G ( s , y 2 ) * 2 d s + c 2 0 t x 1 ( s ) - x 2 ( s ) 2 d s + ω 2 0 t x 1 ( s ) d s - x 2 ( s ) d s 2 d s ,

where c is a positive constant satisfying 2ω1 - c > 0. Here, we used that

a b a p p + b q q , p - 1 + q - 1 = 1 ( 1 < p < )

for any pair of nonnegative numbers a and b. Thus, from (2.3) it follows that

x 1 ( t ) - x 2 ( t ) 2 + ( 2 ω 1 - c ) 0 t x 1 ( s ) d s - x 2 ( s ) d s 2 d s c - 1 ( L 1 T 0 + L 2 K 1 T 0 3 2 2 ) 2 0 t y 1 ( s ) - y 2 ( s ) 2 d s + 2 ω 2 0 t x 1 ( s ) - x 2 ( s ) 2 d s .

By using Gronwall's inequality, we get

x 1 ( T 0 ) - x 2 ( T 0 ) 2 + ( 2 ω 1 - c ) 0 T 0 x 1 ( s ) - x 2 ( s ) 2 d s c - 1 ( L 1 T 0 + L 2 K 1 T 0 3 2 2 ) 2 e 2 ω 2 T 0 0 T 0 y 1 ( s ) - y 2 ( s ) 2 d s .

Taking c = ω1, it holds that

x 1 x 2 L 2 ( 0 , T 0 ; V ) ω 1 1 e ω 2 T 0 ( L 1 T 0 + L 2 K 1 T 0 3 / 2 / 2 ) y 1 y 2 L 2 ( 0 , T 0 ; V ) .

Hence, we have proved that y x is a strictly contraction from L2(0, T0; V ) to itself if the condition (2.9) is satisfied. It shows that the equation (2.6) has a unique solution in [0, T0].

From now on, we derive the norm estimates of solution of the equation (2.6). Let y be the solution of

y ( t ) + A y ( t ) = f ( t ) , 0 < t T 0 , y ( 0 ) = ϕ 0 . (2.10)

Then,

d d t ( x ( t ) - y ( t ) ) + ( A x ( t ) - A y ( t ) ) = G ( t , x ) ,

by multiplying by x(t) - y(t) and using the assumption (A1), we obtain

1 2 d d t x ( t ) - y ( t ) 2 + ω 1 x ( t ) - y ( t ) 2 ω 2 x ( t ) - y ( t ) 2 + G ( t , x ) * x ( t ) - y ( t ) .

By integrating over [0, t] and using Gronwall's inequality, we have

x y L 2 ( 0 , T 0 ; V ) ω 1 1 e ω 2 T 0 G ( η , x ) L 2 ( 0 , T 0 ; V * ) ω 1 1 e ω 2 T 0 { L 0 T 0 + L 2 K 0 T 0 3 / 2 / 3 + ( L 1 T 0 + L 2 K 1 T 0 3 / 2 / 2 ) ( x L 2 ( 0 , T 0 ; V ) + ϕ 1 L 2 ( h , 0 ; V ) } ,

and hence, putting

N = ω 1 - 1 e ω 2 T 0 and L = L 1 T 0 + L 2 K 1 T 0 3 2 2 ,

it holds

x L 2 ( 0 , T 0 ; V ) N 1 N L ( L 0 T 0 + L 2 K 0 T 0 3 / 2 / 3 ) + 1 1 N L y L 2 ( 0 , T 0 ; V ) + N L 1 N L ϕ 1 L 2 ( h , 0 ; V ) N 1 N L ( L 0 T 0 + L 2 K 0 T 0 3 / 2 / 3 ) + C 1 1 N L ( ϕ 0 + f L 2 ( 0 , T 0 ; V * ) ) + N L 1 N L ϕ 1 L 2 ( h , 0 ; V ) C 2 ( 1 + ϕ 0 + ϕ 1 L 2 ( h , 0 ; V ) + f L 2 ( 0 , T 0 ; V * ) ) (2.11)

for some positive constant C2. Since the condition (2.9) is independent of initial values, the solution of (2.6) can be extended to the internal [0, nT0] for natural number n, i.e., for the initial value (x(nT0), xnT0 ) in the interval [nT0, (n + 1)T0], as analogous estimate (2.11) holds for the solution in [0, (n + 1)T0].

Theorem 2.2. If (Φ0, Φ1) ∈ H × L2(-h, 0, V )) and f L2(0, T; V* ), then x L2(-h, T; V ))∩ W1,2(0, T; V* ), and the mapping

H × L 2 ( h , 0 ; V ) × L 2 ( 0 , T ; V * ) ( ϕ 0 , ϕ 1 , f ) x L 2 ( h , T ; V ) ) W 1 , 2 ( 0 , T ; V * )

is continuous.

Proof. It is easy to show that if (Φ 0, Φ1) ∈ H × L2(-h, 0; V )) and f L2(0, T; V* ) for every T > 0, then x belongs to L2(- h, T; V )∩W1,2(0, T; V*). Let

( ϕ i 0 , ϕ i 1 , f i ) H × L 2 ( - h , 0 ; V ) × L 2 ( 0 , T 1 ; V * )

and xi be the solution of (2.6) with ( ϕ i 0 , ϕ i 1 , f i ) in place of ( ϕ 0 , ϕ 1 , f ) for i = 1,2.

Then, in view of Proposition 2.1 and Lemma 2.2, we have

1 2 d d t x 1 ( t ) - x 2 ( t ) 2 + ω 1 x 1 ( t ) - x 2 ( t ) 2 ω 2 x 1 ( t ) - x 2 ( t ) 2 + G ( t , x 1 ) - G ( t , x 2 ) * x 1 ( t ) - x 2 ( t ) + f 1 ( t ) - f 2 ( t ) * x 1 ( t ) - x 2 ( t ) (2.12)

If ω1 - c/2 > 0, we can choose a constant c1 > 0 so that

ω 1 - c 2 - c 1 2 > 0

and

f 1 ( t ) f 2 ( t ) ) * x 1 ( t ) x 2 ( t ) 1 2 c 1 f 1 ( t ) f 2 ( t ) * 2 + c 1 2 x 1 ( t ) x 2 ( t ) 2 .

Let T1 < T be such that

2 ω 1 - c - c 1 - c - 1 e 2 ω 2 T 1 ( L 1 T 1 + L 2 K 1 T 1 3 2 2 ) 2 > 0 .

Integrating on (2.12) over [0, T1] and as is seen in the first part of proof, it follows

( 2 ω 1 c c 1 ) x 1 x 2 L 2 ( 0 , T 1 ; V ) 2 e 2 ω 2 T 1 { ϕ 1 0 ϕ 2 0 2 + 1 c G ( t , x 1 ) G ( t , x 2 ) L 2 ( 0 , T 1 ; V * ) 2 + 1 c 1 f 1 f 2 L 2 ( 0 , T 1 ; V * ) 2 } e 2 ω 2 T 1 { ϕ 1 0 ϕ 2 0 2 + 1 c ( L 1 T 1 + L 2 K 1 T 1 3 / 2 / 2 ) 2 x 1 x 2 L 2 ( h , T 1 ; V ) 2 + 1 c 1 f 1 f 2 L 2 ( 0 , T 1 ; V * ) 2 } .

Putting that

N 1 = 2 ω 1 - c - c 1 - c - 1 e 2 ω 2 T 1 ( L 1 T 1 + L 2 K 1 T 1 3 2 2 ) 2

we have

x 1 - x 2 L 2 ( 0 , T 1 ; V ) e ω 2 T 1 N 1 1 2 ( ϕ 1 0 - ϕ 2 0 + 1 c 1 f 1 - f 2 L 2 ( 0 , T 1 ; V * ) ) + c - 1 2 e ω 2 T 1 ( L 1 T 1 + L 2 K 1 T 1 3 2 2 ) N 1 1 2 ϕ 1 1 - ϕ 2 1 L 2 ( - h , 0 ; V ) . (2.13)

Suppose that

( ϕ n 0 , ϕ n 1 , f n ) ( ϕ 0 , ϕ 1 , f ) in H × L 2 ( h , 0 ; V ) ) × L 2 ( 0 , T ; V * ) ,

and let xn and x be the solution (2.6) with ( ϕ n 0 , ϕ n 1 , f n ) and ( ϕ 0 , ϕ 1 , f ) respectively.

By virtue of (2.13) with T being replaced by T1, we see that

x n x ℯin L 2 ( h , T 1 ; V ) ) W 1 , 2 ( 0 , T 1 ; V * ) C ( [ 0 , T 1 ] ; H ) .

This implies that ( x n ( T 1 ) , ( x n ) T 1 ) ( x ( T 1 ) , x T 1 ) in H × L2 (-h, 0; V). Hence, the same argument shows that

x n x in L 2 ( T 1 , min { 2 T 1 , T } ; V ) W 1 , 2 ( T 1 , min { 2 T 1 , T } ; V * ) .

Repeating this process, we conclude that

x n x in L 2 ( - h , T ; V ) W 1 , 2 ( 0 , T ; V * ) .

Remark 2.2. For x L2(0, T; V ), we set

G ( t , x ) = 0 t k ( t - s ) g ( s , x ( s ) ) d s

where k belongs to L2(0, T) and g : [0, T] × V H be a nonlinear mapping satisfying

g ( t , x ) - g ( t , y ) L x - y

for a positive constant L. Let x ∈ L2(0, T; V ), T > 0. Then, G, x) ∈ L2(0, T; H) and

G ( , x ) L 2 ( 0 , T ; H ) L k L 2 ( 0 , T ) T x L 2 ( 0 , T ; V ) .

Moreover, if x1, x2 ∈ L2(0, T; V ), then

G ( , x 1 ) - G ( , x 2 ) L 2 ( 0 , T ; H ) L k T x 1 - x 2 L 2 ( 0 , T ; V ) .

Then, with the condition that

ω 1 - 1 e ω 2 T 0 L k T 0 < 1

in place of the condition (2.9), we can obtain the results of Theorem 2.1.

3 Approximate controllability

In what follows we assume that the embedding V H is compact, and A is a continuous operator from V to V* satisfying (A1) and (A2). For h ∈ L2(0, T; H) and let xh be the solution of the following equation with B = I:

x ( t ) + A x ( t ) = G ( t , x ) + h ( t ) , 0 < t , x ( 0 ) = 0 x ( s ) = 0 - h s 0 , (3.1)

where

G ( t , x ) = g ( t , x t , 0 t k ( t , s , x s ) d s ) .

We define the solution mapping S from L2(0, T; V*) to L2(0, T; V ) by

( S h ) ( t ) = x h ( t ) , h L 2 ( 0 , T ; V * ) . (3.2)

Let A and G be the Nemitsky operators corresponding to the maps A and G, which are defined by A ( x ) ( ) = A x ( ) and G ( h ) ( ) = G ( , x h ) , respectively. Then, since the solution x belongs to L2(-h, T; V ) ∩ W1,2(0, T; V*) ⊂ C([0, T]; H), it is represented by

x h ( t ) = 0 t ( ( I + G A S ) h ) ( s ) d s , (3.3)

and with aid of Lemma 2.2 and Proposition 2.1

S h L 2 ( 0 , T ; V ) W 1 , 2 ( 0 , T ; V * ) = x h C 1 G ( η , x h ) + h L 2 ( 0 , T ; V * ) C 1 { L 0 T + L 2 K 0 T 3 / 2 / 3 + ( L 1 T + L 2 K 1 T 3 / 2 / 2 ) x L 2 ( 0 , T ; V ) + h L 2 ( 0 , T ; V * ) } C 1 { L 0 T + L 2 K 0 T 3 / 2 / 3 + ( L 1 T + L 2 K 1 T 3 / 2 / 2 ) ( 1 + h L 2 ( 0 , T ; V * ) ) + h L 2 ( 0 , T ; V * ) } . (3.4)

Hence, if h is bounded in L2 (0, T; V*), then so is xh in L2(0, T; V)∩W1,2(0, T; V*). Since V is compactly embedded in H by assumption, the embedding L2(0, T; V) ∩ W1,2 (0, T; V*) ⊂ L2 (0, T; H) is compact in view of Theorem 2 of Aubin [14]. Hence, the mapping h Sh = xh is compact from L2(0, T; V*) to L2(0, T; H). Therefore, G is a compact mapping from L2(0, T; V*) to L2(0, T; H) and so is A S from L2(0, T; V*) to itself. The solution of (SE) is denoted by x(T; g, u) associated with the nonlinear term g and control u at time T.

Definition 3.1. The system (SE) is said to be approximately controllable at time T if Cl{x(T; g, u): u L2(0, T; U)} = V* where Cl denotes the closure in V*.

We assume

(T) 1 - ω 1 - 1 ω 3 e ω 2 T > 0

(B) Cl{y : y(t) = (Bu)(t), a.e. u L2(0, T; U)} = L2(0, T; U)}. Here Cl is the closure in L2(0, T; H).

Theorem 3.1. Let the assumptions (T) and (B) be satisfied. Then,

C l { ( I A S ) h : h L 2 ( 0 , T ; V * ) } = L 2 ( 0 , T ; V * ) . (3.5)

Therefore, the following nonlinear differential control system

d x ( t ) d t + A x ( t ) = ( B u ) ( t ) , 0 < t T , x ( 0 ) = x 0 (3.6)

is approximately controllable at time T.

Proof. Let z ∈ L2(0, T; V*) and r be a constant such that

z U r = { x L 2 ( 0 , T ; V * ) : x L 2 ( 0 , T ; V * ) < r } .

Take a constant d > 0 such that

( r + ω 3 + N 2 x 0 ) ( 1 N 2 ) 1 < d , (3.7)

where

N 2 = ω 1 - 1 ω 3 e ω 2 T .

Taking scalar product on both sides of (3.1) with G = 0 by x(t)

1 2 d d t x ( t ) 2 + ω 1 x ( t ) 2 ω 2 x ( t ) 2 + h ( t ) * x ( t ) ω 2 x ( t ) 2 + 1 2 c h ( t ) * 2 + c 2 x ( t ) 2

where c is a positive constant satisfying 2ω1 - c > 0. Integrating on [0, t], we get

1 2 x ( t ) 2 + ω 1 0 t x ( s ) 2 d s 1 2 x 0 2 + 1 2 c 0 t h ( s ) * 2 d s + c 2 0 t x ( s ) 2 d s + ω 2 0 t x ( s ) 2 d s ,

and hence,

x ( t ) 2 + ( 2 ω 1 - c ) 0 t x ( s ) 2 d s x 0 2 + 1 c 0 t h ( s ) * 2 d s + 2 ω 2 0 t x ( s ) 2 d s .

By using Gronwall's inequality, it follows that

x ( T ) 2 + ( 2 ω 1 - c ) 0 T x ( s ) 2 d s e 2 ω 2 T ( x 0 2 + 1 c 0 T h ( s ) * 2 d s ) ,

that is,

S h L 2 ( 0 , T ; V ) = x L 2 ( 0 , T ; V ) e ω 2 T ( 2 ω 1 - c ) - 1 2 ( x 0 + c - 1 2 h L 2 ( 0 , T ; V * ) ) . (3.8)

Let us consider the equation

z = ( I A S ) w . (3.9)

Let w be the solution of (3.9). Then z ∈ Ud and taking c = ω1, from (3.7), (3.8)

w L 2 ( 0 , T ; V * ) z L 2 ( 0 , T ; V * ) + A S w L 2 ( 0 , T ; V * ) r + ω 3 ( S w L 2 ( 0 , T ; V * ) + 1 ) r + ω 3 { ω 1 1 / 2 e ω 2 T ( x 0 + ω 1 1 / 2 w ) + 1 } ,

and hence

w ( r + ω 3 + N 2 x 0 ) ( 1 N 2 ) 1 < d

it follows that w ∂Ud where ∂Ud stands for the boundary of Ud. Thus, the homotopy property of topological degree theory there exists w ∈ L2(0, T; V*) such that the equation (3.9) holds. Based on the assumption (B), there exists a sequence {un} ∈ L2(0, T; U) such that Bun w in L2(0, T; V*). Then, by the last paragraph of Theorem 2.1, we have that x(·; g, un) ↦ xw in L2(0, T; V ) ∩ W1,2(0, T; V*) ⊂ C([0, T]; H). Hence, we have proved (3.5). Let y ∈ V*. Then, there exists an element u ∈ L2(0, T; U) such that

y T ( I A S ) B u L 2 ( 0 , T ; V * ) < ε T .

Thus

y x ( T ) * = y 0 T ( ( I A S ) B u ) ( s ) d s * 0 T y T ( ( I A S ) B u ) ( s ) * d s T y T ( I A S ) B u L 2 ( 0 , T ; V * ) < ε .

Therefore, the system (3.6) is approximately controllable at time T.

In order to investigate the controllability of the nonlinear control system, we need to impose the following condition.

(F) g is uniformly bounded: there exists a constant Mg such that

g ( t , x , y ) M g ,

for all x, y ∈ V.

By (F) it holds that

G ( , x ) L 2 ( 0 , T ; H ) M g T ,

and for every h ∈ L2(0, T; V*)

G ( h ) L 2 ( 0 , T ; H ) M g T (3.10)

Theorem 3.2. Let the assumptions (T), (B), and (F) be satisfied. Then, we have

C l { ( G + I A S ) h : h L 2 ( 0 , T ; V * ) } = L 2 ( 0 , T ; V * ) . (3.11)

Thus, the system (SE) is approximately controllable at time T.

Proof. Let Ur be the ball with radius r in L2(0, T; V*) and z ∈ Ur. To prove (3.11), we will also use the degree theory for the equation

z = λ ( G A S ) w + w , 0 λ 1 (3.12)

in open ball Ud where the constant d satisfies

( r + ω 3 + N 2 x 0 + M g T ) ( 1 N 2 ) 1 < d (3.13)

where the constant N2 is in Theorem 3.1. If w is the solution of (3.12), then z ∈ Ud and from Lemma 2.1

w L 2 ( 0 , T ; V * ) z + A S w + G w r + ω 3 ( S w + 1 ) + M g T ) r + ω 3 { ω 1 1 / 2 e ω 2 T ( x 0 + ω 1 1 / 2 w ) + 1 } + M g T ,

and hence

w ( r + ω 3 + N 2 x 0 + M g T ) ( 1 N 2 ) 1 < d

it follows that w ∂Ud. Hence, there exists w ∈ L2(0, T; V*) such that the equation (3.12) holds. Using the similar way to the last part of Theorem 3.1 and the assumption (B), there exists a sequence {un} ∈ L2(0, T; U) such that Bun w in L2(0, T; V*) and x(·, g, un) ↦xw in L2(0, T; V ) ∩ W1,2(0, T; V*) ⊂ C([0, T]; H). Thus, we have proved (3.11), and the system (1.1) is approximately controllable at time T.

4 Example

Let -A be an operator associated with a bounded sesquilinear form a(u, v) defined in V × V and satisfying Gårding inequality

R e a ( u , v ) c 0 u 2 - c 1 u 2 , c 0 > 0 , c 1 0

for any u ∈ V. It is known that A generates an analytic semigroup in both H and V*. By virtue of the Riesz-Schauder theorem, if the embedding V H is compact, then the operator A has discrete spectrum:

σ ( A ) = { μ n : n = 1 , 2 , . . . }

which has no point of accumulation except possibly when μ = ∞. Let μn be a pole of the resolvent of A of order kn and Pn the spectral projection associated with μn

P n = 1 2 π i Γ n ( μ - A ) - 1 d μ ,

where Γn is a small circle centered at μn such that it surrounds no point of σ(A) except μn. Then, the generalized eigenspace corresponding to μn is given by

H n = P n H = { P n u : u H } ,

and we have that from P n 2 = P n and Hn V ; it follows that

P n V = { P n u : u V } = H n .

Definition 4.1. The system of the generalized eigenspaces of A is complete in H if Cl {span{Hn : n = 1, 2,...}} = H where Cl denotes the closure in H.

We need the following hypotheses:

(B1) The system of the generalized eigenspaces of A is complete.

(B2) There exists a constant d > 0 such that

υ d B υ , υ L 2 ( 0 , T ; U ) .

We can see many examples which satisfy (B2)(cf. [8,11]).

Consider about the intercept controller B define d by

( B u ) ( t ) = n = 1 u n ( t ) , (4.1)

where

u n = 0 ,   0 t T n   P n u ( t ) ,   T n < t T .  

Hence, we see that u1(t) ≡ 0 and un(t) Im Pn.

First of all, for the meaning of the condition (B) in section 3, we need to show the existence of controller satisfying Cl{Bu : u ∈ L2(0, T; U)} ≠ L2(0, T; H). In fact, by completion of the generalized eigenspaces of A, we may write that f ( t ) = n = 1 P n f ( t ) for ∈L2(0, T; H). Let us choose fL2(0, T; H) satisfying

0 T P 1 f ( t ) 2 d t > 0 .

Then, since

0 T f ( t ) - B u ( t ) 2 d t = 0 T n = 1 P n ( f ( t ) - B u ( t ) ) 2 d t 0 T P 1 ( f ( t ) - B u ( t ) ) 2 d t = 0 T P 1 f ( t ) 2 d t > 0 ,

the statement mentioned above is reasonable.

Let fL2(0, T; H) and α = T/(T - T/n). Then we know

f ( ) α K [ T , T n ] f ( α ( - T n ) ) in L 2 ( 0 , T ; H ) ,

where K[T,T/n] is the characteristic of [T,T/n]. Define

w ( s ) = n = 1 w n ( s ) , w n ( s ) = α K [ T , T n ] B - 1 P n f ( α ( s - T n ) ) .

Thus ( B w ) ( t ) = n = 1 P n f ( s ) , a .e . Since the system of the generalized eigenspaces of A is complete, it holds that for every f ∈ L2(0, T; H) and ∈ > 0

f ( ) - n = 1 P n f ( ) L 2 ( 0 , T ; H ) = f ( ) - B w L 2 ( 0 , T ; H ) < ε .

Thus, the intercept controller B define d by (4.1) satisfies the condition (B).

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

JMJ carried out the main proof of this manuscript, JRK drafted the manuscript and corrected the main theorems, EYJ conceived of the study, and participated in its design and coordination.

Acknowledgements

This study was supported by the Korea Research Foundation(KRF) grant funded by the Korea government (MOEHRD, Basic Research Promotion Fund) (KRF-351-C00102).

References

  1. Barbu, V: Nonlinear Semigroups and Differential Equations in Banach space. Noordho Leiden, Netherland (1976)

  2. Ahmed, NU, Xiang, X: Existence of solutions for a class of nonlinear evolution equations with nonmonotone perturbations. Nonlinear Anal Theory Methods Appl. 22(1), 81–89 (1994). Publisher Full Text OpenURL

  3. Pazy, A: Semigroup of Linear Operator and Applications to Partial Differential Equations. Springer-Verlag, New York (1983)

  4. Aizicovici, S, Papageorgiou, NS: Infinite dimensional parametric optimal control problems. Jpn J Ind Appl Math. 10, 307–332 (1993). Publisher Full Text OpenURL

  5. Hirano, N: Nonlinear evolution equations with nonmonotonic perturbations. Nonlinear Anal Theory Methods Appl. 13(6), 599–609 (1989). Publisher Full Text OpenURL

  6. Jeong, JM, Ju, JE, Lee, KY: Controllability for variational inequalities of parabolic type with nonlinear pertirbation. J Inequal Appl. 2010, 1–16 (2010) Article ID 768469

  7. Sukavanam, N, Tomar, Nutan Kumar: Approximate controllability of semilinear delay control system. Nonlinear Func Anal Appl. 12, 53–59 (2007)

  8. Zhou, HX: Approximate controllability for a class of semilinear abstract equations. SIAM J Control Optim. 21, 551–565 (1983). Publisher Full Text OpenURL

  9. Carrasco, A, Lebia, H: Approximate controllability of a system of parabolic equations with delay. J Math Anal Appl. 345, 845–853 (2008). Publisher Full Text OpenURL

  10. Naito, K: Controllability of semilinear control systems dominated by the linear part. SIAM J Control Optim. 25, 715–722 (1987). Publisher Full Text OpenURL

  11. Zhou, HX: Controllability properties of linear and semilinear abstract control systems. SIAM J Control Optim. 22, 405–422 (1984). Publisher Full Text OpenURL

  12. Triggiani, R: A note on the lack of exact controllability for mild solutions in Banach spaces. SIAM J Control Optim. 15, 407–411 (1977). Publisher Full Text OpenURL

  13. Yong, J, Pan, L: Quasi-linear parabolic partial differential equations with delays in the highest order partial derivatives. J Aust Math Soc. 54, 174–203 (1993). Publisher Full Text OpenURL

  14. Aubin, JP: Un thèoréme de compacité. C R Acad Sci. 256, 5042–5044 (1963)