null controllability of nonlocal hilfer fractional ...

Miskolc Mathematical Notes Vol. 18 (2017), No. 2, pp. 1073–1083

HU e-ISSN 1787-2413 DOI: 10.18514/MMN.2017.2396

NULL CONTROLLABILITY OF NONLOCAL HILFER FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATIONS JINRONG WANG AND HAMDY M. AHMED Received 08 September, 2017 Abstract. In this paper, we study exact null controllability of Hilfer fractional semilinear stochastic differential equations in Hilbert spaces. By using fractional calculus and fixed point approach, sufficient conditions of exact null controllability for such fractional systems are established. An example is given to show the application of our results. 2010 Mathematics Subject Classification: 26A33; 93B05 Keywords: null controllability, stochastic differential equation, Hilfer fractional derivative

1. I NTRODUCTION The stochastic differential equations arise in many mathematical models [5,14,18, 20]. The problem of controllability of nonlinear stochastic or deterministic system has been discussed in [3, 4, 6, 8, 9, 16, 19]. Recently, basic theory of differential equations involving Caputo and RiemannLiouville fractional derivatives can be found in [1, 2, 13, 21–24, 26–30] and the references cited therein. Beside Caputo and Riemann-Liouville fractional derivatives, there exists a new definition of fractional derivative introduced by Hilfer, which generalized the concept of Riemann-Liouville derivative and has many application in physics, for more details, see [10–12, 25]. In this paper, we investigate the exact null controllability of Hilfer fractional semilinear stochastic differential equation of the form 8 ; ˆ < D0C x.t / D Ax.t / C Bu.t / / CF .t; x.t // C G.t; x.t // d!.t ; t 2 J D Œ0; b; (1.1) dt ˆ .1 /.1 / : I x.0/ C h.x/ D x0 ; 0C ;

where D0C is the Hilfer fractional derivative, 0 1; 12 < < 1; A is the infinitesimal generator of strongly continuous semigroup of bounded linear operators This work is supported by National Natural Science Foundation of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), and Unite Foundation of Guizhou Province ([2015]7640), and Graduate ZDKC ([2015]003). c 2017 Miskolc University Press

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JINRONG WANG AND HAMDY M. AHMED

S.t /; t 0; on a separable Hilbert space H with inner product h:; :i and norm k : k. There exists a M 1 such that sup t0 kS.t /k M: The control function u./ is given in L2 .J; U /, the Hilbert space of admissible control functions with U as a separable Hilbert space. The symbol B stands for a bounded linear operator from U into H . Here ! is an H -valued Wiener process associated with a positive, nuclear covariance operator Q; F is an H -valued map and G is a L.K; H /-valued map both defined on J H (where K is a real separable Hilbert space with norn k kK and L.K; H / is the space of all bounded, linear operators from K to H; we write simply L.H / if H D K) and h W C.J; H / ! H: 2. P RELIMINARIES In this section, some definitions and results are given which will be used throughout this paper. Definition 1 (see [15, 17]). The fractional integral operator of order > 0 for a function f can be defined as Z t 1 f .s/ ds; t > 0 I f .t / D ./ 0 .t s/1 where

./ is the Gamma function.

Definition 2 (see [11]). The Hilfer fractional derivative of order 0 1 and 0 < < 1 for a function f is defined by ;

.1 /

D0C f .t / D I0C

d .1 I dt 0C

/.1 /

f .t /:

Let .˝; ; P / be a complete probability space furnished with complete family of right continuous increasing sub -algebras f t W t 2 J g satisfying t : An H valued random variable is an - measurable function x.t / W ˝ ! H and a collection of random variables D fx.t; !/ W ˝ ! H jt 2 J g is called a stochastic process. Usually we suppress the dependence on ! 2 ˝ and write x.t / instead of x.t; !/ and x.t / W J ! H in the place of . Let ˇn .t / .n D 1; 2; :::/ be a sequence of real valued one-dimensional standard Brownian motions mutually independent over .˝; ; P /. Set 1 p X n ˇn .t /en ; t 0; !.t / D nD1

where n ; .n D 1; 2; :::/ are nonnegative real numbers and fen g .n D 1; 2; :::/ is a complete orthonormal basis in Let Q 2 L.K; K/ be an operator defined by Qen D PK: 1 n en with finite T r.Q/ D nD1 n < 1, (Tr denotes the trace of the operator). Then the above K-valued stochastic process !.t / is called Q-Wiener process. We assume that t D f!.s/ W 0 s t g is the -algebra generated by !:

HILFER FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATIONS

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For 2 L.K; H / we define k k2Q D T r.Q / D

1 X

k

p

n en k2 :

nD1

k2Q
0g is continuous in the uniform operator topology. (ii) For any fixed t > 0; S; .t / and P .t / are linear and bounded operators, and kP .t /xk

M t 1 M t . 1/. 1/ kxk; kS; .t /xk kxk: ./ ..1 / C /

(iii) fP .t / W t > 0g and fS; .t / W t > 0g are strongly continuous.

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To study the exact null controllability of (1.1) we consider the fractional linear system ( ; / D0C y.t / D Ay.t / C Bu.t / C F .t / C G.t / d!.t ; t 2 J D Œ0; b; dt (2.1) .1 /.1 / I0C y.0/ D y0 ; associated with the system (1.1). Define the operator Lb0 u D

b

Z

s/Bu.s/ds W L2 .J; U / ! H;

P .b 0

where Lb0 u has a bounded inverse operator .L0 / 1 with values in L2 .J; U /=ker.Lb0 /; and Z b b P .b s/F .s/ds N0 .y; F; G/ D S; .b/y C 0

b

Z C

P .b 0

s/G.s/d!.s/ W H L2 .J; U / ! H:

Definition 3. The system (2.1) is said to be exactly null controllable on J if Im Lb0 Im N0b : By [7], the system (2.1) is exactly null controllable if there exists > 0 such that k.Lb0 / yk2 k.N0b / yk2 for all y 2 H . Lemma 2 (see [16]). Suppose that the linear system (2.1) is exactly null controllable on J: Then the linear operator W D .L0 /

1

is bounded and the control " u.t / D .L0 /

1

S; .b/y0 C

N0b W H L2 .J; H / ! L2 .J; U /

Z

b

Z P .b

0

s/F .s/ds C

#

b

P .b

s/G.s/d!.s/

0

D W .y0 ; F; G/ transfers the system (2.1) from y0 to 0; where L0 is the restriction of Lb0 to Œker Lb0 ? ; F 2 L2 .J; H / and G 2 L2 .J; L.K; H //:


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3. E XACT NULL CONTROLLABILITY In this section, we formulate sufficient conditions for exact null controllability for the system (1.1). First, we give the definitions of mild solution and exact null controllability for it. Definition 4. We say x 2 C.J; L2 .˝; H // is a mild solution to (1.1) if it satisfies that Z t x.t / D S; .t /Œx0 h.0/ C P .t s/ŒF .s; x.s// C Bu.s/ds 0 Z t C P .t s/G.s; x.s//d!.s/; t 2 J: 0

Definition 5. The system (1.1) is said to be exact null controllable on the interval J if there exists a stochastic control u 2 L2 .J; U / such that the solution x of the system (1.1) satisfies x.b/ D 0: To prove the main result, we need the following hypotheses: .H1/ The fractional linear system (2.1) is exactly null controllable on J: .H 2/ The function F W J H ! H is locally Lipschitz continuous, for all t 2 J; x; x1 ; x2 2 H; there exist constant c1 > 0; such that kF .t; x2 /

F .t; x1 /k2 c1 kx2

x1 k2 ;

kF .t; x/k2 c1 .1 C kxk2 /:

.H 3/ The function G W J H ! L.K; H / is locally Lipschitz continuous, for all t 2 J; x; x1 ; x2 2 H; there exist constant c2 > 0; such that G.t; x1 /k2Q c2 kx2

kG.t; x2 /

x1 k2 ; kG.t; x/k2Q c2 .1 C kxk2 /:

.H 4/ The function h W C.J; H / ! H is continuous, for any x; x1 ; x2 2 C.J; H /; there exist constant c3 > 0; such that kh.x2 / Set %2 WD 1 C

h.x1 /k2 c3 kx2

4M 2 c3 /C/ 4M 2 b 2 1 kW k2 kBk2 . .2 1/ 2 ./

%1

WD

2 ..1

x1 k2 ; kh.x/k2 c3 .1 C kxk2 /: C

M 2 b 1C2. 1/ .c1 .2 1/ 2 ./

C c2 T r.Q//

and

Theorem 1. If the hypotheses .H1/-.H 4/ are satisfied, then the system (1.1) is exactly null controllable on J provided that % WD %1 %2 < 1:

(3.1)

Proof. For an arbitrary x define the operator ˚ on Y as follows .˚ x/.t / D S; .t /Œx0 h.x/ Z t C P .t s/ŒF .s; x.s// 0

(3.2) BW .x0

h.x/; F; G/ds

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JINRONG WANG AND HAMDY M. AHMED t

Z C

P .t 0

s/G.s; x.s//d!.s/; t 2 J;

where u.t / D W .x0

h.x/; F; G/.t /

D .L0 / 1 fS; .b/Œx0 h.x/ Z b Z C P .b s/F .s; x.s//ds C 0

b

P .b

s/G.s; x.s//d!.s/g:

0

It will be shown that the operator ˚ from Y into itself has a fixed point. Step 1. The control u./ D W .x0 h.x/; F; G/ is bounded on Y: Indeed, kuk2Y D sup t 2.1

/.1 /

sup t 2.1

/.1 /

Ekuk2

(3.3)

t 2J

EkW .x0

h.x/; F; G/.s/k2

t 2J

M2 ŒEkx0 k2 C c3 .1 C Ekxk2 / 2 ..1 / C / M 2 b 1C2. 1/ 2 C .1 C Ekxk /.c1 C c2 T r.Q// : .2 1/ 2 ./ kW k

2

Step 2. We show that ˚ maps Y into itself. From (3.2) and (3.3) for t 2 J; we have k.˚ x/.t /k2Y D sup t 2.1

/.1 /

t 2J

4 sup t 2.1

/.1

Ek.˚ x/.t /k2 / EkS; .t /Œx0

h.x/k2

t 2J

Z t

CE

P .t 0

C 4 sup t 2.1

2

s/F .s; x.s//ds

/.1 /

t2J

2 Z t

E

P .t s/ŒBW .x0 h.x/; F; G/.s/ds 0

2

Z t

C E P .t s/G.s; x.s//d!.s/

0 4M 2 ŒEkx0 k2 C c3 .1 C Ekxk2 / 2 ..1 / C /


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4M 2 b 1C2. 1/ 2 C .1 C Ekxk /.c1 C c2 T r.Q// .2 1/ 2 ./ M 2 kBk2 kW k2 kb 2 1 1C < 1: .2 1/ 2 ./ Therefore ˚ maps Y into itself. Step 3. We prove .˚ x/.t / is continuous on J for any x 2 Y: Let 0 < t b and > 0 be sufficiently small, then, .˚ x/.t /k2Y

k.˚ x/.t C / D sup t

2.1 /.1 /

(3.4)

Ek.˚ x/.t C /

.˚ x/.t /k

2

t2J

4 sup t 2.1

/.1 /

t 2J

Ek.S; .t C / C 4 sup t 2.1

S; .t //Œx0

h.x/k2

/.1 /

t 2J

Z t C

E P .t C s/ŒBW .x0 h.x/; F; G/.s/ds

0

2 Z t

P .t s/ŒBW .x0 h.x/; F; G/.s/ds

0

C 4 sup t 2.1

/.1 /

t 2J

Z

E

t C

Z P .t C

0

C 4 sup t 2.1

s/F .s; x.s//ds

t

P .t 0

2

s/F .s; x.s//ds

/.1 /

t 2J

Z

E

t C

0

Z P .t C

s/G.s; x.s//d!.s/

t

P .t 0

2

s/G.s; x.s//d!.s/

:

Clearly, from Lemma 1, .H 2/ and .H 3/, the right hand side of (3.4) tends to zero as ! 0: Hence, .˚ x/.t / is continuous on J: Step 4. We show that .˚ x/.t / is a contraction on Y: Let x1 ; x2 2 Y; for any t 2 .0; b be fixed, then k.˚ x2 /.t / D sup t 2.1

.˚ x1 /.t /k2Y /.1 /

Ek.˚ x2 /.t /

.˚ x1 /.t /k2

t2J

4 sup t 2.1 t 2J

/.1 /

Ek.S; .t /Œh.x2 /

h.x1 /k2

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C 4 sup t t 2J

BW .x0 C 4 sup t

Z t

E P .t s/ŒBW .x0

0

2

h.x1 /; F; G/.s/ds

2.1 /.1 /

t 2J

C 4 sup t

Z t

E P .t

0

Z t

/ E P .t

2.1 /.1 /

2.1 /.1

h.x2 /; F; G/.s/

s/ŒF .s; x2 .s//

2

F .s; x1 .s//ds

s/ŒG.s; x2 .s//

2

G.s; x1 .s//d!.s/

0

t 2J

%Ekx2

x1 k2 :

Hence, ˚ is a contraction in Y via (3.1). From the Banach fixed point theorem, ˚ has a unique fixed point. Therefore the system (1.1) is exact null controllable on J:

4. A N EXAMPLE Consider the following Hilfer fractional stochastic partial differential system 8 ; 32 @2 ˆ ˆ D ˆ 0C x.t; ´/ D @´2 x.t; ´/ C u.t; ´/ ˆ < / Cf .t; x.t; ´// C g.t; x.t; ´// d!.t ; t 2 J; 0 < ´ < 1; dt ˆ x.t; 0/ D x.t; 1/ D 0; t 2 J; ˆ ˆ ˆ Pp : 13 .1 / I0C .x.0; ´// C i D1 ki x.ti ; ´/ D x0 .´/; 0 ´ 1;

(4.1)

where p is a positive integer, 0 < t0 < t1 < ::: < tp < b and !.t / is Wiener process, u 2 L2 .0; b/; and H D L2 .Œ0; 1/: Let f W R R ! R and g W R R ! R are continuous and global Lipschitz continuous in the second variable. Also, let A W H ! H @y @2 be defined by Ay D @´ 2 y with domain D.A/ D fy 2 H W y; @´ are absolutely con2

tinuous, and @@´y2 2 H; y.0/ D y.1/ D 0g: 2 2 It is known that A is self-adjoin and has the eigenvalues p n D n ; n 2 N; with the corresponding normalized eigenvectors en .´/ D 2 sin.n´/: Furthermore, A generates an analytic compact semigroup of bounded linear operator S.t /; t 0; on a separable Hilbert space H which is given by S.t /y D

1 X nD1

.yn ; en /en D

1 X

2e

n2 2 t

nD1

If u 2 L2 .J; H /; then B D I; B D I:

Z sin.n´/ 0

1

sin.n/y./d ; y 2 H:


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Now we consider 8 ; 23 ˆ @2 ˆ D y.t; ´/ D @´ ˆ 2 y.t; ´/ C u.t; ´/ 0C ˆ < Cf .t; ´/ C g.t; ´/d!.t /; t 2 J; 0 < ´ < 1; y.t; 0/ D y.t; 1/ D 0; t 2 J; ˆ ˆ ˆ ˆ : I 13 .1 / .y.0; ´// D y .´/; 0 ´ 1; 0 0C

(4.2)

The system (4.2) is exact null controllability if there is a > 0; such that " # Z b Z b kB P .b s/yk2 ds kS; .b/yk2 C kP .b s/yk2 ds ; 0

0

or equivalently Z b kP .b 0

" 2

2

s/yk ds kS; .b/yk C

Z 0

#

b

kP .b

2

s/yk ds :

If f D 0 and g D 0 in (4.2), then the fractional linear system is exactly null controllable if Z b kP .b s/yk2 ds bkS; .b/yk2 : 0

Therefore, Z b kP .b 0

" Z b b 2 2 kS; .b/yk C kP .b s/yk ds 1Cb 0

# s/yk2 ds :

Hence, the linear fractional system (4.2) is exactly null controllable on Œ0; b. So the hypothesis .H1 / is satisfied. We define F W J H ! H; G W J H ! L.K; H / and h W C.J; PpH / ! H as follows: F .t; x/ D f .t; x.t; ´//; G.t; x/ D g.t; x.t; ´// and h.x/ D i D1 ki x.ti ; ´/ Then h./ satisfies .H 4/. By choosing the constants ki ; i D 1; 2; :::; p; M; c1 ; c2 and c3 such that % < 1: Hence, all the hypotheses of Theorem 1 are satisfied, so the Hilfer fractional stochastic partial differential system (4.1) is exact null controllable on Œ0; b: R EFERENCES [1] R. Agarwal, S. Hristova, and D. O’Regan, “A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations,” Fract. Calc. Appl. Anal, vol. 19, no. 2, pp. 290–318, 2016, doi: 10.1515/fca-2016-0017. [2] R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Appl. Math, vol. 109, no. 3, pp. 973–1033, 2010, doi: 10.1007/s10440-008-9356-6. [3] H. M. Ahmed, “Controllability of fractional stochastic delay equations,” Lobachevskii J. Math, vol. 30, no. 3, pp. 195–202, 2009, doi: 10.1134/S19950802090300019.

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Authors’ addresses JinRong Wang Guizhou University, Department of Mathematics, Guiyang 550025, China E-mail address: [email protected] Hamdy M. Ahmed Higher Institute of Engineering, El-Shorouk Academy, El-Shorouk City, Cairo, Egypt E-mail address: hamdy [email protected]