Controllability of a backward fractional semilinear differential equation

Applied Mathematics and Computation 242 (2014) 168–178

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Controllability of a backward fractional semilinear differential equation Gisèle Mophou ⇑ Laboratoire CEREGMIA, Université des Antilles et de la Guyane, Campus Fouillole, 97159 Pointe-à-Pitre, Guadeloupe Laboratoire MAINEGE, Université Ouaga 3S, 06 BP 10347 Ouagadougou 06, Burkina Faso

a r t i c l e

i n f o

Keywords: Fractional Caputo derivatives Fractional differential equations Mild solution Approximate controllability Optimal controls Regularization of Tikhonov type

a b s t r a c t In this paper we study the approximate controllability of a fractional semilinear differential equation involving the right fractional Caputo derivative. More precisely, we construct by means of Tikhonov type regularization method, the controllability operator. Then under certain condition on this operator, we obtain that the associate backward fractional linear system can be steered to an arbitrary small neighborhood of the state at initial time. This allows us to prove the approximate controllability of the semilinear system. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction In this paper, we study the approximate controllability of a system governed by the following fractional evolution equation with right fractional Caputo derivative in a Hilbert space X:



DaC yðtÞ ¼ A yðtÞ þ f ðt; yðtÞÞ þ B v ðtÞ;

0 < t 6 T;

yðTÞ ¼ y1 ;

ð1Þ

where T > 0; 0 < a < 1; DaC is the right fractional Caputo derivative of order a and the function f is an appropriate defined on ½0; T  X. The control v 2 L2 ðð0; TÞ; YÞ and B 2 LðY; XÞ. The operator A is the adjoint of A, and A : DðAÞ ! X is the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators fRðtÞ; t P 0g. This means that there exists M > 1 such that

sup kjRðtÞkj 6 M:

ð2Þ

t2½0;T

There is much literature on approximate controllability of differential or partial differential equations in finite dimensional as in infinite dimensional. We refer for instance to [7,3–5,8–12,22,23,6,24–26] and the reference therein. To obtain this quality property of the control, many approaches are developed and among them, the Tikhonov type regularization method. This method which uses the notion of adjoint state allows to obtain approximate controllability which is well adapted to partial differential equations with entire derivative (see for instance Mahmudov et al. [3,4] and the reference therein). This means in the limited case: a ¼ 1. In this case Eq. (1) becomes

⇑ Address: Laboratoire CEREGMIA, Université des Antilles et de la Guyane, Campus Fouillole, 97159 Pointe-à-Pitre, Guadeloupe. E-mail address: [email protected] http://dx.doi.org/10.1016/j.amc.2014.05.042 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

G. Mophou / Applied Mathematics and Computation 242 (2014) 168–178



y0 ðtÞ ¼ A yðtÞ þ f ðt; yðtÞÞ þ B v ðtÞ;

169

0 < t 6 T;

yðTÞ ¼ y1 ; 0

where y ðtÞ is first derivative of y with respect to t. It is well known that this latter problem is approximately controllable on ½0; T if the set

Rð0; y1 ; v Þ ¼ fyð0; y1 ; v Þ : v 2 L2 ðð0; TÞ; YÞg satisfies Rð0; y1 ; v Þ ¼ X. Actually, in this limited case, the approximately controllability is achieved because the controllability operator is symmetric. This is due to the fact that the mild solution is given with the same compact semigroup or the same compact operator and also because the adjoint of dtd , the first derivative with respect to the time, is its opposite,  dtd . In the case of fractional differential systems, mild solutions expressed with density probabilities are sometimes given with two operators (see [5,11,21]). Consequently, the operator of controllability may not be positive if one wants to steer the state of the system at given time to an arbitrary small neighborhood of this state. Motivated by these observations and the fact that mild solutions of fractional differential systems involving left fractional Riemann Liouville derivative of order 0 < a < 1 are expressed with the same operator [18] on the one hand, and the fact that the adjoint of right fractional Caputo derivative of order 0 < a < 1 is the left fractional Riemann Liouville derivative of same order (see Lemma 2.7 below), we study in this paper the approximate controllability of semilinear fractional differential Eq. (1). We prove that it is a backward semilinear fractional differential of a semilinear fractional differential involving left fractional Caputo derivative. Then by means of Tikhonov type regularization method, we construct an operator of controllability, which is symmetric, linear and bounded. Finally we obtain under certain condition on this operator, the approximate controllability of the system (1). The paper is organized as follows. In Section 2, we give some preliminary results. Section 3 is devoted to the study of the approximate controllability of the linear fractional differential system associate to (1). In particular we prove in this section that the adjoint of the right fractional Caputo derivative of order 0 < a < 1 is the left fractional Riemann Liouville derivative of same order. In Section 4, we prove under appropriate conditions on the nonlinear function f, the existence of mild solutions to system (1) and then, we show under some conditions on the operator of controllability and the function f that this system is approximately controllable. An example is given to illustrate our results in Section 5. 2. Preliminaries Throughout this paper, we denote by X; Y two separable Hilbert spaces with inner products h; i and h; iY respectively, and the corresponding norms k  kX and k  kY . Also, we denote by LðX; YÞ the space of bounded linear operators from X into Y endowed with the norm of operators, by A the adjoint of the operator A. The identity operator is denoted by I. Cð½0; T; XÞ is the space of all X-valued continuous functions on ½0; T with the norm kuk1 ¼ supfkuðtÞk; t 2 ½0; Tg; Lp ð½0; T; XÞ the space of R 1=p T X-valued Bochner integrable functions on ½0; T with the norm kf kLp ð½0;T;XÞ ¼ 0 kf ðtÞkp dt , where 1 6 p < 1 and L1 ð½0; T; XÞ the space of X-valued essentially bounded functions on ½0; T with the norm kf kL1 ð½0;T;XÞ ¼ ess supt2½0;T kf ðtÞk. Now, let us recall some basic definitions and results on fractional differentiation and integration. Definition 2.1 ([20,27]). The fractional order integral of the function f 2 L1 ð½0; T; XÞ of order a 2 Rþ is defined by

Ia f ðtÞ ¼

1 CðaÞ

Z

t

ðt  sÞa1 f ðsÞds;

0

where C is the Gamma function. Definition 2.2 ([20,16]). The left Riemann–Liouville fractional order derivative of order a 2 ð0; 1Þ of a function f 2 L1 ð½0; T; XÞ given on the interval ½0; T is defined by

DaRL f ðtÞ ¼

1 d Cð1  aÞ dt

Z

t

ðt  sÞa f ðsÞds:

0

Definition 2.3 ([27,14]). The left Caputo fractional order derivative of order a 2 ð0; 1Þ of a function f 2 L1 ð½0; T; XÞ given on the interval ½0; T is defined by

Da f ðtÞ ¼

1 Cð1  aÞ

Z

t

ðt  sÞa f ð1Þ ðsÞds:

0

Definition 2.4 ([27,14]). The right Caputo fractional order derivative of order a 2 ð0; 1Þ of a function f 2 L1 ð½0; T; XÞ given on the interval ½0; T is defined by

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G. Mophou / Applied Mathematics and Computation 242 (2014) 168–178

DaC f ðtÞ ¼

1 Cð1  aÞ

Z

t

ðs  tÞa f 0 ðsÞds:

T

Since f takes values in Banach space X, the integrals which appear in above four definitions are taken in Bochner’s sense. We have the following result which proves that Ia f is Hölder continuous. Lemma 2.5 ([20,19]). Let 0 < b < 1; X be a Banach space and f 2 Cð½0; T; XÞ. Then for all t1 ; t2 2 ½0; T,

b  kf kL1 ðð0;TÞ;XÞ I f ðt 1 Þ  Ib f ðt 2 Þ 6 jt  t2 jb : X Cðb þ 1Þ 1 Remark 2.6. From this lemma, we deduce that Ia f 2 L2 ðð0; TÞ; XÞ since Ia f 2 Cð½0; T; XÞ. Thus, we have this formula of integration by parts. Lemma 2.7. Let 0 < a < 1. Let u 2 L2 ðð0; TÞ; XÞ be such that DaRL u 2 L2 ðð0; TÞ; XÞ. Let also y 2 Cð½0; T; XÞ be such that DaC y 2 L2 ðð0; TÞ; XÞ. Then

Z

0

T



DaRL uðtÞ; yðtÞ



Z D E D E þ 1a 1a dt ¼ yðTÞ; I u ðTÞ  yð0Þ; I u ð0 Þ dx þ X X

X

0

T





uðtÞ; DaC yðtÞ X dt:

Remark 2.8. From Lemma 2.7, we deduce that the adjoint of the right Caputo fractional derivative of order 0 < a < 1 is the Riemann Liouville fractional derivative of order 0 < a < 1. Now, let Ua be the so-called Mainardi function[15]:

Ua ðzÞ ¼ We set

Sa ðtÞ ¼

þ1 X

ðzÞn : n!Cðan þ 1  aÞ n¼0

Z

1

Ua ðhÞRðht a Þdh and Pa ðtÞ ¼

Z

0

1

ahUa ðhÞRðta hÞdh:

ð3Þ

0

Then we have the following results Lemma 2.9 [28]. Let 0 < a < 1. Let also Sa ðtÞ and Pa ðtÞ be the operators defined by (3). Then (i) kSa ðtÞxk 6 Mkxk;

kPa ðtÞxk 6 M Cðaaþ1Þ kxk for all x 2 X and t P 0.

(ii) The operators ðSa ðtÞÞtP0 and ðPa ðtÞÞtP0 are strongly continuous. (iii) The operators ðSa ðtÞÞt>0 and ðPa ðtÞÞt>0 are compact if the semigroup ðRðtÞÞt>0 is compact. As the fractional semilinear differential Eq. (1) expressed with adjoint operators, we recall this result of Schauder [1] that will be useful in the following. Theorem 2.10. Let E and F be two Banach spaces. Denote respectively by E0 and F 0 , the dual of E and F. Then the operator T : E ! F is a compact if and only if T  : F 0 ! E0 is compact. So, denote by fR ðtÞ; t P 0g the adjoint of semigroup operator fRðtÞ; t P 0g and set

Sa ðtÞ ¼

Z

0

1

Ua ðhÞR ðht a Þdh and Pa ðtÞ ¼

Z

1

ahUa ðhÞR ðta hÞdh:

ð4Þ

0

Then fSa ðtÞ; t P 0g and fPa ðtÞ; t P 0g are respectively the adjoint of fSa ðtÞ; t P 0g and fPa ðtÞ; t P 0g. Moreover, in view of Theorem 2.10, Lemma 2.9 and (2), we have the following results. Lemma 2.11. Assume that the semigroup ðR ðtÞÞt>0 is compact. Let 0 < a < 1. Let also Sa ðtÞ and Pa ðtÞ be the operators defined by (4). Then (i) kSa ðtÞxk 6 Mkxk;

kPa ðtÞxk 6 M Cðaaþ1Þ kxk for all x 2 X and t P 0.



(ii) The operators ðSa ðtÞÞtP0 and ðPa ðtÞÞtP0 are strongly continuous. (iii) The operators ðSa ðtÞÞt>0 and ðPa ðtÞÞt>0 are compact. 3. Approximate controllability of the linear fractional system In this section, we study the approximate controllability of the backward fractional linear differential equation associate to (1). So, consider first the following fractional evolution equation with the right fractional Caputo derivative in a Hilbert space X:

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G. Mophou / Applied Mathematics and Computation 242 (2014) 168–178



DaC zðtÞ  A zðtÞ ¼ f ðtÞ; t 2 ½0; T zðTÞ

ð5Þ

z1 ;

¼

where f is continuous on ½0; T, the operator A is the adjoint of A; 0 < a < 1; z1 2 X and DaC is right fractional Caputo derivative of order a. Proposition 3.1. Let 0 < a < 1. Let also Sa ðtÞ and Pa ðtÞ be defined by (4). Then the function z 2 Cð½0; T; XÞ defined by

Z

zðtÞ ¼ Sa ðT  tÞz1 þ

Tt

0

ðT  t  sÞa1 Pa ðT  t  sÞf ðT  sÞds;

06t6T

ð6Þ

is the unique solution to the fractional evolution Eq. (5). Proof. Since the function f is continuous on ½0; T, using Lemma 2.9, we have that if z is given by (6) then z 2 Cð½0; T; XÞ. Now, let’s prove that (5) is a backward fractional equation defined with left fractional Caputo derivative. Set as in [19,17]

T T zðtÞ ¼ zðT  tÞ;

t 2 ½0; T:

ð7Þ

After calculations, we have that

DaC T T zðtÞ ¼ Da T T zðtÞ; where Da is the left fractional Caputo derivative of order a. Now, making the change of variable t ! T  t in (5), we obtain



Da T T zðtÞ  A T T zðtÞ ¼ T T f ðtÞ; T  t 2 ½0; T pð0Þ

¼

p0 :

Proceeding as in [13] with the Laplace transform, we deduce that

zðtÞ ¼ Sa ðT  tÞz1 þ

Z

T

t

ðs  tÞa1 Pa ðs  tÞf ðsÞds;

0 6 t 6 T:



Now, we are concerned with the approximate controllability of the linear fractional differential equation associate to (1) in a Hilbert space X:



DaC yðtÞ ¼ A yðtÞ þ f ðtÞ þ B v ðtÞ;

0 < t 6 T;

ð8Þ

yðTÞ ¼ y1 ;

where y1 2 X; T > 0; 0 < a < 1; DaC is the right Caputo fractional derivative of order a and the function f is continuous on ½0; T. The control v 2 L2 ðð0; TÞ; YÞ and B 2 LðY; XÞ. The operator A is the adjoint of A, and A : DðAÞ ! X is the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators fRðtÞ; t P 0g. Proceeding as for Eq. (5), we have that the solution yð:Þ ¼ yðv Þð:Þ of (8) satisfies

yðtÞ ¼ Sa ðT  tÞy1 þ

Z t

T

ðs  tÞa1 Pa ðs  tÞ½f ðsÞ þ B v ðsÞds;

Proposition 3.2. Let 1=2 < a < 1 Q : L2 ðð0; TÞ; YÞ ! Cð½0; T; XÞ by:

Q v ðtÞ ¼

Z t

T

and

Pa ðtÞ

ðs  tÞa1 Pa ðs  tÞB v ðsÞds;

be

defined

0 6 t 6 T:

as

in

(4).

8v 2 L2 ðð0; TÞ; YÞ t 2 ½0; T:

ð9Þ

Let

also

r > 0; B 2 LðY; XÞ

and

define

ð10Þ

Then fQ v : kv kL2 ðð0;TÞ;YÞ 6 rg is relatively compact in Cð½0; T; XÞ. Proof. Since the proof is straightforward, we omitted it.

h

From Propositions 3.2 and 3.1, the solution of (8), yð:Þ ¼ yðv Þð:Þ given by (9) belongs to Cð½0; T; XÞ. Thus, yð0Þ exists and belongs to X. Now, set

Rð0Þ ¼ fyðv Þð0Þ : v 2 L2 ðð0; TÞ; YÞg:

ð11Þ

Definition 3.3. A backward fractional system is said to be approximately controllable on ½0; T if Rð0Þ ¼ X. To prove the approximate controllability of system (8), we use the Tikhonov type regularization method. So, for any k > 0, we define the functional J by:

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G. Mophou / Applied Mathematics and Computation 242 (2014) 168–178

Jðv Þ ¼ kyð0Þ  hk2X þ k

Z

T

0

kv ðtÞk2Y dt

ð12Þ

where yð:Þ ¼ yðv Þð:Þ solution to the evolution system (5) and h 2 X. Note that the functional J is well defined since yð0Þ exists and belongs to X. We then consider the linear regulator problem:

Jðv Þ:

inf

v 2L2 ðð0;TÞ;YÞ

ð13Þ

Theorem 3.4. Let 1=2 < a < 1. Let Sa ðtÞ; Pa ðtÞ; Sa ðtÞ and Pa ðtÞ be respectively defined by (3) and (4). Let also B 2 LðY; XÞ. For a given h 2 X and k > 0, there exists a unique optimal control u 2 L2 ðð0; TÞ; YÞ such that (13) holds. Proof. The proof is straightforward. So we omit it

h

From now on, we define the operator KT from X to X by

KT ¼

Z

T



B sa1 Pa ðsÞ

 

B sa1 Pa ðsÞ ds:

ð14Þ

0

Then it is clear that the controllability operator KT is linear, bounded and positive. Consequently, the resolvent ðkI þ KT Þ1 is well defined for all k > 0. From now on, we set

Rðk; KT Þ ¼ ðkI þ KT Þ1 :

ð15Þ

Proposition 3.5. Let 1=2 < a < 1. Let Sa ðtÞ; Pa ðtÞ; Sa ðtÞ and Pa ðtÞ be respectively defined by (3) and (4). Let also B 2 LðY; XÞ and B be the adjoint of B . For a given h 2 X and k > 0, let u be the optimal control of (13). Then

uðtÞ ¼ k1 BpðtÞ ¼ k1 t a1 BPa ðtÞ½yð0Þ  h a:e on ½0; T;

ð16Þ

Z yð0Þ  h ¼ kRðk; KT Þ Sa ðTÞy1 þ

ð17Þ

0

T

 sa1 Pa ðsÞf ðsÞds  h ;

where Rðk; KT Þ is given by (15). Proof. We express the Euler–Lagrange optimality conditions which characterizes u:

lim

Jðu þ cwÞ  JðuÞ

c!0

¼ 0 8w 2 L2 ðð0; TÞ; YÞ;

c

ð18Þ

where w is the control associate to the state z solution to



DaC zðtÞ ¼ A zðtÞ þ B wðtÞ; zðTÞ

t 20; T;

ð19Þ

¼ 0:

After calculation, relation (18) gives

hyð0Þ  h; zð0ÞiX þ

Z

T

0

hkuðtÞ; wðtÞiY dt ¼ 0 8w 2 L2 ðð0; TÞ; YÞ;

ð20Þ

where w is the control associate to the state z solution to (19). To interpret (20), we consider the adjoint state p solution of

(

DaRL pðtÞ

¼ ApðtÞ;

t 20; T;

ð21Þ

I1a pð0þ Þ ¼ yð0Þ  h;

where I1a pðtÞ is the fractional integral of order 1  a of the function p and I1a pð0þ Þ ¼ limt!0þ I1a pðtÞ. Using the Laplace transform, one can prove as in Proposition 3.3 [18], that (21) has a unique p 2 Cðð0; T; XÞ that satisfies:

pðtÞ ¼ ta1 Pa ðtÞ½yð0Þ  h;

t 2 ð0; T:

ð22Þ

Then, multiplying the first equation in (19) by p and integrating by parts over ð0; TÞ, it follows by using Lemma 2.7 that

Z 0

T

hzðtÞ; DaRL pðtÞiX dt þ hzð0Þ; I1a pð0þ ÞiX ¼

which according (21) gives

hzð0Þ; yð0Þ  hiX ¼

Z

T 0

hwðtÞ; BpðtÞiY dt:

Z 0

T

hzðtÞ; ApðtÞiX dt þ

Z 0

T

hwðtÞ; BpðtÞiY dt;

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G. Mophou / Applied Mathematics and Computation 242 (2014) 168–178

Combining this latter identity with (20), we obtain that

Z 0

T

hwðtÞ; BpðtÞiY dt þ

Z

T

0

hkuðtÞ; wðtÞiY dt ¼ 0 8w 2 L2 ðð0; TÞ; YÞ:

Consequently,

uðtÞ ¼ k1 BpðtÞ ¼ k1 t a1 BPa ðtÞ½yð0Þ  h a:e on ½0; T: Now, since the solution of (8), y is given by

yðtÞ ¼ Sa ðT  tÞy1 þ

Z

T

t

ðs  tÞa1 Pa ðs  tÞf ðsÞds þ

Z t

T

ðs  tÞa1 Pa ðs  tÞB uðsÞds;

8t 2 ½0; T

and belongs to Cð½0; T; XÞ, using the expression of the optimal control u, we obtain that

yð0Þ ¼ Sa ðTÞy1 þ

Z

T

0

sa1 Pa ðsÞf ðsÞds  k1 KT ½yð0Þ  h;

with KT given by relation (14). Hence, it follows that

Z yð0Þ  h ¼ kRðk; KT Þ Sa ðTÞy1 þ

T

0

 sa1 Pa ðsÞf ðsÞds  h :



Proposition 3.6. Let 1=2 < a < 1; k > 0 and Rðk; KT Þ be defined by (15). Then the control system (8) is approximately controllable if, and only if kRðk; KT Þ converges to zero as k ! 0 in strong operator topology. Proof. Using the solution of the regulator problem (13) and (17), one obtains the results by proceeding exactly as for the proof of Theorem 2 in [3]. h

4. Approximate controllability of the semilinear fractional system In this section, we study the approximate controllability of (1). Actually, we will prove under suitable assumptions on the function f that the approximate controllability of the linear system (8) implies the approximate controllability of the nonlinear system (1). Assume that  ðH1 Þ: The function f : ½0; T  X ! X is continuous and there exists a constant p 2 ½0; a and a function l2 2 L1=p ð0; TÞ such that

kf ðt; xÞk 6 l2 ðtÞ;

for all t 2 ½0; T;

x 2 X:

 ðH2 Þ : f : ½0; T  X ! X is continuous and uniformly bounded, and there exists

kf ðt; xÞk 6 l;

for all t 2 ½0; T;

l > 0 such that

x 2 X:

Accordingly to Proposition 3.1, we give the following definition of mild solution to (1). Definition 4.1. A function yð:Þ ¼ yðv Þð:Þ 2 Cð½0; T; XÞ is said to be a mild solution of (1) if for any v 2 L2 ðð0; TÞ; YÞ, the integral equation

yðtÞ ¼ Sa ðT  tÞy1 þ

Z t

T

ðs  tÞa1 Pa ðs  tÞf ðs; yðsÞÞds þ

Z t

T

ðs  tÞa1 Pa ðs  tÞB v ðsÞds;

06t6T

ð23Þ

is satisfied. Lemma 4.2. Let 1=2 < a < 1. Let Sa ðtÞ and Pa ðtÞ be defined by (4). Let also B 2 LðX; YÞ. Assume that assumption (H1 ) holds. For any h 2 X and k > 0, let

v ðtÞ ¼ k1 ta1 BPa ðtÞ½yð0Þ  h a:e on ½0; T;

ð24Þ

Z yð0Þ  h ¼ kRðk; KT Þ Sa ðTÞy1 þ

ð25Þ

0

T

 sa1 Pa ðsÞf ðs; yðsÞÞds  h ;

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G. Mophou / Applied Mathematics and Computation 242 (2014) 168–178

where Rðk; KT Þ is given by (15). Then,

kyð0Þ  hkX 6 H

ð26Þ

kv kL2 ðð0;TÞ;YÞ 6 k1 kBkLðX;YÞ with

H ¼ Mky1 kX þ

Ma

Cða þ 1Þ

H

ð27Þ



Makl2 kL1=p ð0;TÞ 1  p 1p ap T þ khkX : Cða þ 1Þ ap

Proof. Using ðH1 Þ and Lemma 2.11, one obtains the estimates.

ð28Þ

h

We recall the following result of Schauder and Tikhonov. Theorem 4.3 [2]. Let K be a closed, bounded, convex subset of a Banach space X. If T : K ! K is compact, then T has a fixed point in K. Theorem 4.4. Let 1=2 < a < 1.Let Pa ðtÞ be defined by (3). Let Sa ðtÞ and Pa ðtÞ be defined by (4). Let also B 2 LðY; XÞ and B be the adjoint of B . Assume that ðH1 Þ holds. Then for any h 2 X and k > 0, the fractional semilinear control system (1) has a mild solution on ½0; T. Proof. For any k > 0, we define the operator Pk : CðI; XÞ ! CðI; XÞ by

Pk yðtÞ ¼ Sa ðT  tÞy1 þ

Z

T

t

ðs  tÞa1 Pa ðs  tÞ½f ðs; yðsÞÞ þ B v ðsÞds

with the control v given by (24). n o We will prove that for any k > 0, there exists c ¼ cðkÞ > 0 such that Pk y : supt2½0;T kykX 6 c is relatively compact in Cð½0; T; XÞ. Observing that Pk yðtÞ can be decomposed as Pk yðtÞ ¼ FyðtÞ þ WyðtÞ where the operators F : Cð½0; T; XÞ ! Cð½0; T; XÞ and W : Cð½0; T; XÞ ! Cð½0; T; XÞ are respectively defined by:

FyðtÞ ¼

Z t

T

ðs  tÞa1 Pa ðs  tÞf ðs; yðsÞÞds

and

WyðtÞ ¼ Sa ðT  tÞy1 þ

Z

T t

ðs  tÞa1 Pa ðs  tÞB v ðsÞds

n o n o it suffices to prove Fy : supt2½0;T kykX 6 c1 and Wy : supt2½0;T kykX 6 c2 are relatively compact in Cð½0; T; XÞ to obtain that n o Pk y : supt2½0;T kykX 6 c is relatively compact in Cð½0; T; XÞ with c ¼ c1 þ c2 . Using Lemma 2.11 and (27), we have

kWyðtÞkX 6 Mky1 kX þ

MakB kLðY;XÞ kv kL2 ðð0;TÞ;YÞ a1=2 k1 M 2 a2 kBkLðX;YÞ kB kLðY;XÞ 2a1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 6 Mky1 kX þ T H ¼ c2 : Cða þ 1Þ 2a  1 C2 ða þ 1Þð2a  1Þ

Consequently, writing WyðtÞ ¼ Sa ðT  tÞy1 þ Q v ðtÞ where Q is the operator defined in Proposition 3.2 and using the fact n o that the control v is bounded in L2 ðð0; T; YÞ since (27) holds, we deduce from Proposition 3.2 that Q v : kv kL2 ðð0;TÞ;YÞ 6 c2 is relatively compact in Cð½0; T; XÞ. Hence Sa ðtÞ being a compact operator for t > 0, one can easily prove that n o n o Wy : supt2½0;T kykX 6 c2 is relatively compact in Cð½0; T; XÞ. Thus, it remains to prove that Fy : supt2½0;T kykX 6 c1 is relatively compact in Cð½0; T; XÞ to complete the proof of Theorem 4.4. To this end, we proceed in four steps. Step 1. F is well defined. We prove that Fy 2 Cð½0; T; XÞ for all y 2 Cð½0; T; XÞ. Let 0 6 t 1 < t 2 6 T. Using Lemma 2.11, we have

kFyðt 1 Þ  Fyðt2 Þk 6



1p 1=1p 1p aMkl k 1=p aMkl2 kL1=p ð0;TÞ Z T  2 L ð0;TÞ 1  p  a1  ðs  t1 Þa1  ds þ ðt 2  t 1 Þap : ðs  t2 Þ Cða þ 1Þ Cða þ 1Þ ap t1 þ

Z

T

t2

js  t 2 ja1 kjP ðs  t1 Þ  P ðs  t 2 Þkj kB v ðsÞkX ds:

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Thus, using Lemma 2.11-(ii), we can conclude that Fy 2 Cð½0; T; XÞ. Step 2. F maps bounded sets into bounded sets in Cð½0; T; XÞ. Using ðH1 Þ and Lemma 2.11, we have for any y 2 Bc1

kðFyÞðtÞkX 6

Z

T

ðs  tÞa1 kPa f ðs; yðsÞÞkX ds 6

t

M akl2 kL1=p ð0;TÞ Z

Cð1 þ aÞ

t

T

1p

M akl2 kL1=p ð0;TÞ 1  p 1p ap a1 ðs  tÞ1p ds 6 T : Cð1 þ aÞ ap

Hence, we deduce that

supkðFyÞkX 6 t2½0;T



M akl2 kL1=p ð0;TÞ 1  p 1p ap T ¼ c1 : Cð1 þ aÞ ap

Step 3. fFy : supt2½0;T kykX 6 c1 g is equicontinuous. Let 0 6 t 1 < t 2 6 T, we have

kFyðt 1 Þ  Fyðt 2 ÞkX 6 I1 þ I2 þ I3 ; where

Z

I1 ¼

T

t2

Z

I2 ¼

T

t2

Z

I3 ¼

t2

t1

    a1   Pa ðs  t2 Þ  Pa ðs  t 1 Þ f ðs; yðsÞÞds; ðs  t 2 Þ  h i   a1  ðs  t2 Þa1 Pa ðs  t1 Þf ðs; yðsÞÞds;  ðs  t1 Þ     a1 ðs  t1 Þ Pa ðs  t1 Þf ðs; yðsÞÞds:

Using the continuity of Pa ðtÞ (Lemma 2.11) and the fact that Using again Lemma 2.11 and ðH1 Þ,

RT t2

ðs  t2 Þa1 f ðs; yðsÞÞds < 1, we conclude that limt1 !t2 I1 ¼ 0.

I2 6

1=1p 1p aMkl2 kL1=p ð0;TÞ Z T   a1  ðs  t 1 Þa1  ds ðs  t 2 Þ Cða þ 1Þ t2

I3 6

1p aMkl2 kL1=p ð0;TÞ Z t2 aMkl2 kL1=p ð0;TÞ 1  p 1p a1 ðs  t1 Þ1p ds ¼ ðt 2  t 1 Þap : Cða þ 1Þ Cða þ 1Þ ap t1

and

Consequently

lim I2 ¼ 0 and

lim I3 ¼ 0:

t 1 !t 2

t 1 !t 2

Thus fFy : supt2½0;T kykX 6 c1 g is equicontinuous. Step 4. We prove that the set fFy : supt2½0;T kykX 6 c1 g is precompact in X for every t 2 ½0; T. It is easy to see the set fFyðTÞ : supt2½0;T kykX 6 c1 g is precompact in X. Let t 2 ½0; TÞ. For each h 2 ð0; T  tÞ and  > 0, we define the operator F h; by

F h; yðtÞ ¼

Z

T

ðs  tÞa1

tþh

a

¼ Rðh Þ

Z

Z

1

ahUa ðhÞR ððs  tÞa hÞf ðs; yðsÞÞdhds

 T

ðs  tÞa1

Z

1

ahUa ðhÞR ððs  tÞa h  ha Þf ðs; yðsÞÞdhds:



tþh

Observing on the one hand that

Z   

T

ðs  tÞa1

tþh

6

Z 

1

 

ahUa ðhÞR ððs  tÞa h  ha Þf ðs; yðsÞÞdhds 6 aMkl2 kL1=p ð0;TÞ

1p

aMkl2 kL1=p ð0;TÞ 1  p Cða þ 1Þ ap

Z

X

T

tþh

1p Z a1 ðs  tÞ1p ds 

1

hUa ðhÞdh

T ap ;

since ðH1 Þ and Lemma 2.11 hold, and on the other hand that, the operators R ðtÞ; t > 0 are compact on X, we deduce that the sets fðF h; ÞyðtÞ : supt2½0;T kykX 6 cg are relatively compact in X. Moreover, using (4), we deduce that

kFyðtÞ  F h; yðtÞkX 6

Z t

þ

T

ðs  tÞa1

Z  0

Z

tþh

ðs  tÞ t

a1





ahUa ðhÞR ððs  tÞa hÞf ðs; yðsÞÞX dhds Z 

1





ahUa ðhÞR ððt  sÞa hÞf ðs; yðsÞÞX dhds:

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G. Mophou / Applied Mathematics and Computation 242 (2014) 168–178

R1 Then using (2) and the fact that  hUa ðhÞdhds ¼ 1, we obtain

kFyðtÞ  F h; yðtÞkX 6 aMkl2 kL1=p ð0;TÞ Consequently, for all



1p

1p Z  1p 1p ap T ap hUa ðhÞdh þ aMkl2 kL1=p ð0;TÞ h : ap ap 0

 > 0,

kFyðtÞ  F h; yðtÞkX 6

aMkl2 kL1=p ð0;TÞ 1  p 1p ap h : Cða þ 1Þ ap

Therefore, the set fðFyÞðtÞ : supt2½0;T kykX 6 c1 g is relatively compact in X for all t 2 ½0; TÞ. Hence, we have that the set fðFyÞðtÞ : supt2½0;T kykX 6 cg is relatively compact in X for all t 2 ½0; T since it is compact at t ¼ T. We thus have proved that fFy : supt2½0;T kykX 6 c1 g is relatively compact in Cð½0; T; XÞ and since fWy : supt2½0;T kykX 6 c2 g is relatively compact in Cð½0; T; XÞ, we have the relatively compactness of fPk yÞ : supt2½0;T kykX 6 cg in Cð½0; T; XÞ. Therefore by Arzela–Ascoli’s theorem, Pk is compact and by Schauder–Tikhonov fixed point theorem, Pk has a fixed point y 2 Bc which is a mild solution of (1) on ½0; T. h To prove the approximate controllability of the semilinear fractional system (1), we need the following results. Lemma 4.5. Let 1=2 < a < 1 and Pa ðtÞ be defined as in (4). Let also r > 0, define U : L2 ðð0; TÞ; XÞ ! Cð½0; T; XÞ by:

UwðtÞ ¼

Z

T t

ðs  tÞa1 Pa ðs  tÞwðsÞds 8w 2 L2 ðð0; TÞ; XÞ t 2 ½0; T:

ð29Þ

Then fUw : kwkL2 ðð0;TÞ;XÞ 6 rg is relative compact in Cð½0; T; XÞ. Proof. On proceeds exactly as for the proof of Proposition 3.2.

h

Theorem 4.6. Let 1=2 < a < 1. Let Pa ðtÞ be as in (3). Let Sa ðtÞ and Pa ðtÞ be as in (4). Let also B 2 LðY; XÞ and B be the adjoint of B . Assume that ðH1 Þ and ðH2 Þ hold. Assume also that the linear system (8) is approximately controllable on ½0; T. Then the semilinear fractional system (1) is approximately controllable on ½0; T. ^ be the fixed point of Pk in Bc . Then as a fixed point of Pk ; y ^k is a mild solution of (1) under the control Proof. Let y

v^ ¼ k1 ta1 BPa ðtÞkRðk; KT Þy^k;T and satisfies

^k ð0Þ  h ¼ kRðk; KT Þy ^k;T ; y RT



^k ðsÞÞds  h sa1 Pa ðsÞf ðs; y 0

^k;T ¼ Sa ðTÞy1 þ where y Now, in view of ðH2 Þ, we have

Z t

T

ð30Þ and Rðk; KT Þ is given by (15).

^k ðsÞÞk2X 6 l2 T: kf ðs; y

^k ð:ÞÞ is bounded in L2 ð0; TÞ; XÞ. Consequently we can extract a subsequence of ðf ð:; y ^k ð:ÞÞ still denoted This means that ðf ð:; y ^k ð:ÞÞ such that ðf ð:; y

^k ð:ÞÞ * gð:Þ weakly in L2 ð0; TÞ; XÞ: f ð:; y Set

f ¼ Sa ðTÞy1 þ

Z

T

0

sa1 Pa ðsÞgðsÞds  h:

Then

Z  ^k;T  fkX ¼  ky 

0

T

 Z    ^ðsÞÞ  gðsÞÞds sa1 Pa ðsÞðf ðs; y  6 sup X

t2½0;T

Therefore, Lemma 4.5 allows to say that

Z  lim sup   k!0 t2½0;T

t

T

  ^ðsÞÞ  gðsÞÞds ðs  tÞa1 Pa ðs  tÞðf ðs; y  ¼ 0: X

t

T

  ^ðsÞÞ  gðsÞÞds ðs  tÞa1 Pa ðs  tÞðf ðs; y  : X

G. Mophou / Applied Mathematics and Computation 242 (2014) 168–178

177

This implies that

^k;T  fkX ¼ 0: limky

ð31Þ

k!0

Using (30), we have

^k ð0Þ  hkX 6 kkRðk; KT Þðy ^k;T  fÞkX þ kkRðk; KT ÞðfÞkX 6 ky ^k;T  fkX þ kkRðk; KT ÞðfÞkX ; ky which according to (31) and Proposition 3.6 implies that

^k ð0Þ  hkX ¼ 0: limky k!0

This complete the proof of Theorem 4.6.

h

5. Example We consider a control system governed by the fractional partial differential equation:

@2 yðt; zÞ þ f ðt; yðtÞÞ þ v ðt; xÞ; @x2 yðt; 0Þ ¼ yðt; pÞ ¼ 0;

DaC yðt; xÞ ¼

ðt; xÞ 2 ð0; 1Þ  ð0; pÞ; ð32Þ

yðT; xÞ ¼ y1 ðxÞ; where DaC is the right fractional Caputo derivative of order 0 < a < 1; f is a given continuous function. Let us take X ¼ L2 ð0; pÞ and define the operator A by Az ¼ z00 with

DðAÞ ¼ fzð:Þ 2 L2 ð0; pÞ; z; z0 absolutely continuous; z00 ð:Þ 2 L2 ð0; pÞ; zð0Þ ¼ zðpÞ ¼ 0g: Then

Az ¼

1 X

n2 hz; en ien ;

z 2 DðAÞ;

n¼1

pffiffiffiffiffiffiffiffiffi where en ðwÞ ¼ 2=p sinðnwÞ; 0 6 w 6 p; n ¼ 1; 2; 3; . . . . It is clear that A generates a compact analytic semigroup RðtÞ; t > 0 in X which is given by

RðtÞz ¼

1 X 2 en t hz; en ien ;

z 2 X:

n¼1

Put yðtÞ ¼ yðt; :Þ. That is yðtÞðxÞ ¼ yðt; xÞ; ðt; xÞ 2 ð0; 1Þ  ð0; pÞ. Define the function f : ½0; 1  X ! X by f ðt; yðtÞÞðxÞ ¼ f ðt; yðt; xÞÞ, the bounded linear operator B : L2 ð0; pÞ ! X by B v ðtÞðxÞ ¼ v ðt; xÞ and take a ¼ 3=4 . Then, the system (32) can be written is the abstract form (1). Let us prove that the linear system corresponding to (32) is approximately controllable. Actually, for any z 2 X, the operator of controllability is defined as follows.

hKT z; ziX ¼

Z

T



0

2 sa1 kPa ðsÞÞzk2X ds;

where the operator Pa : X ! X is given by 1 Z X Pa ðtÞz ¼ a n¼1

1

hUa ðhÞen

2 ta h

dhhz; en ien ;

z 2 X:

0

So, observing that for any z 2 X; Pa ðtÞz ¼ 0; 0 6 t 6 T, implies z ¼ 0, we deduce that the linear system corresponding to (32) is approximately controllable (see. Remark 10 in [29]). 1 Now, take f ðt; yðt; xÞÞ ¼ t1=3 sin yðtÞ. Then assumptions ðH1 Þ and ðH2 Þ are satisfied and it follows from Theorem 4.6 that the backward fractional control system (32) is approximately controllable on [0, 1]. Acknowledgment We like to thank the referee for his/her careful reading and valuable suggestions. References [1] H. Brezis, Analyse fonctionnelle. Thórie et applications, Collection Mathématiques Appliquées pour la Matrise, Masson, Paris, 1983. [2] M. Bohner, A. Peterson, Dynamic equations on time scales, An Introduction with Applications, Birkhauser, Boston, 2001. [3] A.E. Bashirov, N.I. Mahmudov, On concepts of controllability for deterministic and stochastic systems, SIAM J. Control Optim. 37 (6) (1999) 1808–1821.

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