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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

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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:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M1">View MathML</a>

(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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M2">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M3">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M4">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M5">View MathML</a> denotes the real interpolation space between V and V*.

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M6">View MathML</a>

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 onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M7">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M8">View MathML</a>

We will set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M9">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M10">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M11">View MathML</a>

for all t ∈ ℝ+, x, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M12">View MathML</a>, and y, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M13">View MathML</a>.

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M14">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M15">View MathML</a>

(2.1)

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M16">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M17">View MathML</a>

(2.2)

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M18">View MathML</a>

(2.3)

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M19">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M20">View MathML</a>

Similarly, we can prove (2.3).

Let us consider the quasi-autonomous differential equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M21">View MathML</a>

(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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M22">View MathML</a>

and satisfying

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M23">View MathML</a>

(2.4)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M24">View MathML</a>

(2.5)

where C1 is a constant.

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M25">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M26">View MathML</a>

(2.6)

has a unique solution x belonging to

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M27">View MathML</a>

and satisfying that there exists a constant C2 such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M28">View MathML</a>

(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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M29">View MathML</a>

(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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M30">View MathML</a>

(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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M31">View MathML</a>

and hence it follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M32">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M33">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M34">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M35">View MathML</a>

By using Gronwall's inequality, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M36">View MathML</a>

Taking c = ω1, it holds that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M37">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M38">View MathML</a>

(2.10)

Then,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M39">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M40">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M41">View MathML</a>

and hence, putting

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M42">View MathML</a>

it holds

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M43">View MathML</a>

(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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M44">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M45">View MathML</a>

and xi be the solution of (2.6) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M46">View MathML</a> in place of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M47">View MathML</a> for i = 1,2.

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M48">View MathML</a>

(2.12)

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M49">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M50">View MathML</a>

Let T1 < T be such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M51">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M52">View MathML</a>

Putting that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M53">View MathML</a>

we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M54">View MathML</a>

(2.13)

Suppose that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M55">View MathML</a>

and let xn and x be the solution (2.6) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M56">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M47">View MathML</a> respectively.

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M57">View MathML</a>

This implies that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M58">View MathML</a> in H × L2 (-h, 0; V). Hence, the same argument shows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M59">View MathML</a>

Repeating this process, we conclude that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M60">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M61">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M62">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M63">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M64">View MathML</a>

Then, with the condition that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M65">View MathML</a>

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:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M66">View MathML</a>

(3.1)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M67">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M68">View MathML</a>

(3.2)

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M111">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M112">View MathML</a> be the Nemitsky operators corresponding to the maps A and G, which are defined by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M113">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M114">View MathML</a>, 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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M69">View MathML</a>

(3.3)

and with aid of Lemma 2.2 and Proposition 2.1

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M70">View MathML</a>

(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, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M112">View MathML</a> is a compact mapping from L2(0, T; V*) to L2(0, T; H) and so is <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M115">View MathML</a> 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) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M71">View MathML</a>

(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,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M72">View MathML</a>

(3.5)

Therefore, the following nonlinear differential control system

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M73">View MathML</a>

(3.6)

is approximately controllable at time T.

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M74">View MathML</a>

Take a constant d > 0 such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M75">View MathML</a>

(3.7)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M76">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M77">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M78">View MathML</a>

and hence,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M79">View MathML</a>

By using Gronwall's inequality, it follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M80">View MathML</a>

that is,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M81">View MathML</a>

(3.8)

Let us consider the equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M82">View MathML</a>

(3.9)

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M83">View MathML</a>

and hence

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M84">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M85">View MathML</a>

Thus

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M86">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M87">View MathML</a>

for all x, y ∈ V.

By (F) it holds that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M88">View MathML</a>

and for every h ∈ L2(0, T; V*)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M89">View MathML</a>

(3.10)

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M90">View MathML</a>

(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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M91">View MathML</a>

(3.12)

in open ball Ud where the constant d satisfies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M92">View MathML</a>

(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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M93">View MathML</a>

and hence

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M94">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M95">View MathML</a>

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:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M96">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M97">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M98">View MathML</a>

and we have that from <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M99">View MathML</a> and Hn V ; it follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M100">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M101">View MathML</a>

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

Consider about the intercept controller B define d by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M102">View MathML</a>

(4.1)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M103">View MathML</a>

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 <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M104">View MathML</a> for ∈L2(0, T; H). Let us choose fL2(0, T; H) satisfying

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M105">View MathML</a>

Then, since

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M106">View MathML</a>

the statement mentioned above is reasonable.

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M107">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M108">View MathML</a>

Thus <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M109">View MathML</a> Since the system of the generalized eigenspaces of A is complete, it holds that for every f ∈ L2(0, T; H) and ∈ > 0

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/27/mathml/M110">View MathML</a>

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).

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