255–271 DOI: 10.18514/MMN.2018.2486 APPROXIMATE CONTROLLABILITY OF IMPULSIVE NON-LOCAL NON-LINEAR FRACTIONAL DYNAMICAL SYSTEMS AND OPTIMAL CONTROL SARRA GUECHI, AMAR DEBBOUCHE, AND DELFIM F

Vol. 19 (2018), No. 1, pp. 255–271 DOI: 10.18514/MMN.2018.2486

APPROXIMATE CONTROLLABILITY OF IMPULSIVE NON-LOCAL NON-LINEAR FRACTIONAL DYNAMICAL

SYSTEMS AND OPTIMAL CONTROL

SARRA GUECHI, AMAR DEBBOUCHE, AND DELFIM F. M. TORRES Received 24 December, 2017

Abstract. We establish existence, approximate controllability and optimal control of a class of impulsive non-local non-linear fractional dynamical systems in Banach spaces. We use frac- tional calculus, sectorial operators and Krasnoselskii fixed point theorems for the main results.

Approximate controllability results are discussed with respect to the inhomogeneous non-linear part. Moreover, we prove existence results of optimal triplets of corresponding fractional control systems with Bolza cost functionals.

2010Mathematics Subject Classification: 26A33; 45J05; 49J15; 93B05

Keywords: fractional nonlinear equations, approximate controllability,q-resolvent families, op- timal control, nonlocal and impulsive conditions

1. INTRODUCTION

We are concerned with an impulsive non-local non-linear fractional control dy- namical system of form

_C

D^q_tx.t /DAx.t /Cf .t; x.t /; .H x/.t //CBu.t /; t 2.0; bn ft1; t2; : : : ; tmg; x.0/Cg.x/Dx02X; 4x.ti/DIi.x.t_i //CDv.t_i /; iD1; 2; : : : ; m;

(1.1) where^CD^q_t is the Caputo fractional derivative of order0 < q < 1, the statex./takes its values in a Banach spaceX with normk k, andx02X. LetAWD.A/X!X be a sectorial operator of type.M; ; q; / onX, H WIIX !X represents a Volterra-type operator such that.H x/.t /DRt

0h.t; s; x.s//ds, the control functions u./andv./are given inL².I; U /,U is a Banach space,BandDare bounded linear operators fromU intoX. Here, one hasIDŒ0; b,0Dt0< t1< < tm< tmC1Db, Ii WX !X are impulsive functions that characterize the jump of the solutions at impulse points ti, the non-linear term f WIXX !X, the non-local function gWP C.I; X /!X, withP C defined later, 4x.ti/Dx.t_i^C/ x.t_i /, wherex.t_i^C/ andx.t_i /are the right and left limits ofxat the pointti, respectively.

c 2018 Miskolc University Press

Derivatives and integrals of arbitrary order, the main objects of Fractional Calcu- lus (FC), have kept the interest of many scientists in recent years, since they provide an excellent tool to describe hereditary properties of various materials and processes.

During the past decades, FC and its applications have gained a lot of importance, due to successful results in modelling several complex phenomena in numerous seem- ingly diverse and widespread fields of science and engineering, such as heat con- duction, diffusion, propagation of waves, radiative transfer, kinetic theory of gases, diffraction problems and water waves, radiation, continuum mechanics, geophysics, electricity and magnetism, as well as in mathematical economics, communication theory, population genetics, queuing theory and medicine. For details on the theory and applications of FC see [9]. For recent developments in non-local and impulsive fractional differential problems see [1,2,8,10] and references therein.

The problem of controllability is one of the most important qualitative aspects of dynamical systems in control theory. It consists to show the existence of a control function that steers the solution of the system from its initial state to a final state, where the initial and final states may vary over the entire space. This concept plays a major role in finite-dimensional control theory, so that it is natural to try to gen- eralize it to infinite dimensions [14]. Moreover, exact controllability for semi-linear fractional order systems, when the non-linear term is independent of the control func- tion, is proved by assuming that the controllability operator has an induced inverse on a quotient space. However, if the semi-group associated with the system is com- pact, then the controllability operator is also compact and hence the induced inverse does not exist because the state space is infinite dimensional [17]. Thus, the concept of exact controllability is too strong and has limited applicability, while approximate controllability is a weaker concept completely adequate in applications.

On the other hand, control systems are often based on the principle of feedback, where the signal to be controlled is compared to a desired reference, and the discrep- ancy is used to compute a corrective control action. Fractional optimal control of a distributed system is an optimal control problem for which the system dynamics is defined with fractional differential equations. Recently, attention has been paid to prove existence, approximate controllability and/or optimal control for different classes of fractional differential equations [4–7].

In [11], optimal control of non-instantaneous impulsive differential equations is studied. Qin et al. investigate approximate controllability and optimal control of fractional dynamical systems of order1 < q < 2in Banach spaces [13]. Debbouche and Antonov established approximate controllability of semi-linear Hilfer fractional differential inclusions with impulsive control inclusion conditions in Banach spaces [3]. Motivated by the above works, here we construct an impulsive non-local non- linear fractional control dynamical system and prove new sufficient conditions to treat the questions of approximate controllability and optimal control.

The paper is organized as follows. In Section2, we recall some facts from frac- tional calculus, q-resolvent families, and useful versions of fixed point techniques that are used for obtaining our main results. In Section3, we form appropriate suf- ficient conditions and prove existence results for the fractional control system (1.1).

In Section4, we investigate the question of approximate controllability. We end with Section5, where we obtain optimal controls corresponding to fractional control sys- tems with a Bolza cost functional.

2. PRELIMINARIES

Here we present some preliminaries from fractional calculus [9], operator theory [12] and fixed point techniques [1], which are used throughout the work to obtain the desired results.

Definition 1. The left-sided Riemann–Liouville fractional integral of order˛ > 0, with lower limita, for a functionf WŒa;C1/!R, is defined as

I_a^˛_Cf .t /D 1 .˛/

Z t a

.t s/^{˛ 1}f .s/ds;

provided the right side is point-wise defined on Œa;C1/, where ./ is the Euler gamma function. IfaD0, then we can writeI₀^˛_Cf .t /D.g˛f /.t /, where

g˛.t /WD 1

.˛/t^{˛ 1}; t > 0;

0; t0;

anddenotes convolution of functions. Moreover, lim

˛!0g˛.t /Dı.t /, withıthe delta Dirac function.

Definition 2. The left-sided Riemann–Liouville fractional derivative of order˛ >

0,n 1˛ < n,n2N, for a functionf WŒa;C1/!R, is defined by

LD^˛_aCf .t /D 1 .n ˛/

dⁿ dtⁿ

Z t a

f .s/

.t s/^{˛C1 n}ds; t > a;

where functionf has absolutely continuous derivatives up to ordern 1.

Definition 3. The left-sided Caputo fractional derivative of order˛ > 0,n 1 <

˛ < n,n2N, for a functionf WŒa;C1/!R, is defined by

CD_a^˛_Cf .t /D 1 .n ˛/

Z t a

f^.n/.s/

.t s/^{˛C1 n}dsDI_a^{n ˛}_C f^.n/.t /; t > a;

where functionf has absolutely continuous derivatives up to ordern 1.

Throughout the paper, byP C.I; X / we denote the space of X-valued bounded functions onI with the uniform normkxk^{P C} Dsupfkx.t /k; t2Igsuch thatx.t_i^C/ exists for anyiD0; : : : ; mandx.t /is continuous on.ti; tiC1,i D0; : : : ; m,t0D0 andtmC1Db.

Definition 4(See [16]). LetAWDX!X be a closed and linear operator. We say thatAissectorialof type.M; ; q; /, if there exists2R,0 < <₂ andM > 0 such that theq-resolvent ofAexists outside the sector

CS D fC^qW2C;jArg. ^q/j< g and

k.^qI A/ ¹k M

j^q j; ^q…CS:

Remark1. IfAis a sectorial operator of type.M; ; q; /, then it is not difficult to see thatAis the infinitesimal generator of aq-resolvent familyTq.t /_t0in a Banach space, whereTq.t /D_{2 i}¹ R

ce^tR.^q; A/d .

Definition 5 (Motivated by [3,16]). A state functionx 2P C.I; X / is called a mild solution of (1.1) if it satisfies the following integral equations:

x.t /DSq.t /.x0 g.x//C Z t

0

Tq.t s/.f .s; x.s/; .H x/.s//CBu.s//ds ift2Œ0; t1, and

x.t /DSq.t ti/Œx.t_i /CIi.x.t_i //CDv.t_i /

C Z t

t_i

Tq.t s/Œf .s; x.s/; .H x/.s//CBu.s/ds ift2.ti; tiC1,iD1; : : : ; m, where

Sq.t /D 1 2 i

Z

c

e^{tq 1}R.^q; A/d and Tq.t /D 1 2 i

Z

c

e^tR.^q; A/d withc being a suitable path such that^q…CS for2c.

Letxt_k.x.0/;4x.t_{k 1}/Iu; v/,kD1; : : : ; mC1, be the state value of (1.1) at time tk, corresponding to the non-local initial valuex.0/, the impulsive values4x.tk 1/D x.t_{k 1}^C / x.t_{k 1}/and the controlsuandv. For everyx.0/and4x.tk 1/2X, we introduce the set

R.t_k; x.0/;4x.t_{k 1}//D˚

xt_k.x.0/;4x.t_{k 1}/Iu; v/Wu./; v./2L².I; U / ; which is called thereachable setof system (1.1) at timet_k (ifkDmC1, thent_k is the terminal time). Its closure inX is denoted byR.t_k; x.0/;4x.t_{k 1}//.

Definition 6. The impulsive control system (1.1) is said to be approximately con- trollable on I ifR.t_k; x.0/;4x.t_{k 1}//DX, that is, given an arbitrary > 0, it is possible to steer from the pointsx.0/and4x.t_{k 1}/at timet_k all points in the state spaceX within a distance.

Consider the linear impulsive fractional control system 8

<

:

CD^q_tx.t /DAx.t /CBu.t /;

x.0/Dx02X;

4x.ti/DDv.t_i /; iD1; : : : ; m:

(2.1) Approximate controllability for the linear impulsive control fractional system (2.1) is a natural generalization of the notion of approximate controllability of a linear first- order control system (q D1 andti DDD0, i D1; 2; : : : ; m, i.e.,t2Œtm; tmC1D Œ0; b). The controllability operators associated with (2.1) are

tk

tk 1;1D Z tk

t_{k 1}

Tq.t_k s/BBT_q.t_k s/ds; kD1; : : : ; mC1;

tk

t_{k 1};2DSq.t_k t_{k 1}/DDS_q.t_k t_{k 1}/; kD2; : : : ; mC1;

(2.2) where T_q./, S_q./, B and D denote the adjoints ofTq./, Sq./, B andD, re- spectively. Moreover, for > 0, we consider the relevant operator

R.; _t^tk

k 1;i/D

IC _t^t_k^k _1;i 1

; i D1; 2: (2.3)

It is easy to verify that _t^t_k^k _1;1and _t^t_k^k _1;2are linear bounded operators.

Lemma 1 (See [3]). The linear impulsive control fractional system(2.1) is ap- proximately controllable onI if and only ifR.; _t^tk

k 1;i/!0as!0^C,iD1; 2, in the strong operator topology.

Lemma 2 (Krasnoselskii theorem [15]). LetX be a Banach space andE be a bounded, closed, and convex subset ofX. LetQ1; Q2be maps ofEintoX such that Q1xCQ2y2E for everyx; y2E. IfQ1is a contraction andQ2is compact and continuous, then equationQ1xCQ2xDxhas a solution onE.

3. EXISTENCE OF A MILD SOLUTION

We prove existence for system (1.1). Define K_iDsup_t2IRti

ti 1m.t; s/ds <1, i D1; : : : ; mC1. For anyr > 0, let˝r WD fx2P C.I; X /jkxk rg. We make the following assumptions:

(H1) The operatorsSq.t /t0andTq.t /t0, generated byA, are bounded and com- pact, such that sup_t2IkSq.t /k M and sup_t2IkTq.t /k M.

(H2) The non-linearityf WIXX !X is continuous and compact; there ex- ist functionsi 2L¹.I;R^C/,i D1; 2; 3, and positive constants˛1and˛2

such thatkf .t; x; y/k 1.t /C2.t /kxk C3.t /kykandkf .t; x; H x/

f .t; y; Hy/k D˛1kx yk C˛2kH x Hyk.

(H3) FunctiongWP C.I; X /!Xis completely continuous and there exists a pos- itive constantˇsuch thatkg.x/ g.y/k ˇkx yk,x; y2X.

(H4) Associated withhWX !X, there existsm.t; s/2P C.;R^C/such that kh.t; s; x.s//k m.t; s/kxk for each .t; s/2 andx; y 2X, where D f.t; s/2R²jtis,ttiC1,iD0; : : : ; mg.

(H5) For everyx1; x2; x2X andt2.ti; tiC1,iD1; : : : ; m,Ii are continuous and compact and there exist positive constantsdi,ei such that

kIi.x1.t_i // Ii.x2.t_i //k di sup

t2.t_i;t_iC1kx1.t / x2.t /k andkIi.x.t_i //k ei sup

t2.t_i;t_iC1kx.t /k.

Theorem 1. Letx02X. If conditions (H1)–(H5) hold, then the impulsive non- local fractional control system (1.1) has a fixed point on I provided Mˇ < 1 and M.1Cdi/ < 1, i D1; : : : ; m, that is, (1.1) has at least one mild solution on t 2 Œ0; bn ft1; : : : ; tmg.

Proof. Define the operatorsQ1andQ2on˝ras follows:

.Q1x/.t /D

Sq.t /.x0 g.x//; t2Œ0; t1

Sq.t ti/Œx.t_i /CIi.x.t_i //CDv.t_i /; t2.ti; tiC1;

.Q2x/.t /D ( Rt

0Tq.t s//.f .s; x.s/; .H x/.s//CBu.s//ds; t2Œ0; t1;

Rt

t_iTq.t s//.f .s; x.s/; .H x/.s//CBu.s//ds; t2.ti; tiC1;

iD1; : : : ; m. We take the controls

uDBT_q.t_k t /R.; _t^tk

k 1;1/P₁^k.x.//;

vDDS_q.t_k t_{k 1}/R.; _t^tk

k 1;2/P₂^k.x.//; (3.1) where

P₁^k.x.//D 8 ˆˆ ˆˆ ˆ<

ˆˆ ˆˆ ˆ:

x1 Sq.t1/.x0 g.x//

Rt₁

0 Tq.t1 s/f .s; x.s/; .H x/.s//ds; kD1;

x_k S_q.t_k t_{k 1}/Œx.t_{k 1}/CI_{k 1}.x.t_{k 1}//

Rtk

tk 1Tq.t_k s/f .s; x.s/; .H x/.s//ds; kD2; : : : ; mC1;

P₂^k.x.//D

(x_k Sq.t_k t_{k 1}/Œx.t_{k 1}/CI_{k 1}.x.t_{k 1}//

Rtk

tk 1T_q.t_k s/f .s; x.s/; .H x/.s//ds; kD2; : : : ; mC1:

For any > 0, we shall show that Q1CQ2 has a fixed point on ˝r, which is a solution of system (1.1). According to (3.1), together with (2.2) and (2.3), we have

ku.t /k 1

MkBkkP1.x.//kandkv.t /k 1

MkDkkP2.x.//k: (3.2)

Using assumptions.H1/–.H5/, we get

kP₁¹.x.//k kx1k C kSq.t1/kk.x0 g.x//k C

Z t1

0 kTq.t1 s/kkf .s; x.s/; .H x/.s//kds kx1k CM.kx0k C kg.x/k/

CM t1 k1kL¹.I;R^C/Crk2kL¹.I;R^C/CK₁rk3kL¹.I;R^C/

kx1k CMkx0k CMˇkxk CMkg.0/k

CM t1 k1kL¹.I;R^C/Crk2kL¹.I;R^C/CK₁rk3kL¹.I;R^C/

and, forkD2; : : : ; mC1,

kP₁^k.x.//k kx_kk C kSq.t_k t_{k 1}/kŒkx.t_{k 1}/k C kI_{k 1}.x.t_{k 1}//k C

Z tk

tk 1

kTq.tk s/kkf .s; x.s/; .H x/.s//kds kx_kk CM.kx.t_{k 1}/k Ceikxk/CM.t_k t_{k 1}/

k1kL¹.I;R^C/Crk2kL¹.I;R^C/CK_krk3kL¹.I;R^C/

kxkk CM.kx.t_{k 1}/k Crei/CM.tk tk 1/

k₁kL¹.I;R^C/Crk₂kL¹.I;R^C/CK_krk₃kL¹.I;R^C/

: Similarly, we get

kP₂^k.x.//k kx_kk CM.kx.t_{k 1}/k Cre_{k 1}/

CM.t_k t_{k 1}/ k1kL¹.I;R^C/Crk2kL¹.I;R^C/CK_krk3kL¹.I;R^C/

; kD2; : : : ; mC1. For anyx2˝r, we obtain

k.Q1x/.t /C.Q2x/.t /k M .kx0k C kg.x/k/

CM t1 k1kL¹.I;R^C/Crk2kL¹.I;R^C/CK₁rk3kL¹.I;R^C/C kBkkuk M.kx0k CˇrC kg.0/k/

CM t1.k1kL¹.I;R^C/Crk2kL¹.I;R^C/CK₁rk3kL¹.I;R^C/C kBkkuk/ fort2Œ0; t1, and

k.Q₁x/.t /C.Q₂x/.t /k

M kx.t_{k 1}/k Ce_{k 1}kxk C kDkkv.t_{k 1}/k

CM.t_k t_{k 1}/

k1kL¹.I;R^C/Crk2kL¹.I;R^C/CK_krk3kL¹.I;R^C/C kBkkuk M.kx.t_{k 1}/k Ce_{k 1}rC kDkkv.t_{k 1}/k CM.t_k t_{k 1}/

k1kL¹.I;R^C/Crk2kL¹.I;R^C/CK_krk3kL¹.I;R^C/C kBkkuk

fort2.tk 1; tk. By the inequalities (3.2), we can find1; 2> 0such that k.Q1x/.t /C.Q2x/.t /k

(1; t 2Œ0; t1;

2; t 2.t_{k 1}; t_k; kD2; : : : ; mC1:

Hence,Q1xCQ2xis bounded. Now, letx; y2˝r. We have

k.Q1x/.t / .Q1y/.t /k kSq.t /kkg.x/ g.y/k Mˇkx yk fort2Œ0; t1and

k.Q1x/.t / .Q1y/.t /k

kSq.t tk 1/kŒkx.t_{k 1}/ y.t_{k 1}/k C kIk 1.x.t_{k 1}// Ik 1.y.t_{k 1}//k M

kx.t_{k 1}/ y.t_{k 1}/k Cdk 1kx yk

for t 2.t_{k 1}; t_k, kD2; : : : ; mC1. Since Mˇ < 1 and M.1Cd_{k 1}/ < 1, kD 2; : : : ; mC1, it follows that Q₁ is a contraction mapping. Letfx_ngbe a sequence in ˝r such that xn!x 2˝r. Since f andg are continuous, i.e., for all > 0, there exists a positive integer n0, such that for n > n0 kf .s; xn.s/; .H xn/.s//

f .s; x.s/; .H x/.s//k andkg.xn/ g.x/k , the continuity ofIi.x/on.ti; tiC1 giveskIi.xn.t_i // Ii.x.t_i //k ,iD1; : : : ; m. Now, for allt2Œ0; t1,

k.Q2xn/.t / .Q2x/.t /k

Z t1

0 kTq.t /kkBBT_q.t1 /R.; _t^t1

0;1/kh

kSq.t1/.g.xn/ g.x//k C

Z t1

0 kTq.t1 s/kf .s; xn.s/; .H xn/.s// f .s; x.s/; .H x/.s//kdsi d C

Z t1

0 kTq.t s/kkf .s; xn.s/; .H xn/.s// f .s; x.s/; .H x/.s//kds

M³kBk²t1.2t1C1/:

Moreover, for allt2.ti; tiC1,iD1; : : : ; m, one has k.Q2xn/.t / .Q2x/.t /k

Z t

ti

kTq.t /kkBBT_q.tiC1 /R.; _t^tiC1

i;1 /k

kSq.tiC1 ti/Œxn.t_i / x.t_i /CIi.xn.t_i // Ii.x.t_i //k C

Z tiC1

t_i kTq.tiC1 s/kf .s; xn.s/; .H xn/.s// f .s; x.s/; .H x/.s//kds/

d C

Z t

t_i kTq.t s/kk.f .s; xn.s/; .H xn/.s// f .s; x.s/; .H x/.s//kds

2

M³kBk².tiC1 ti/.tiC1 tiC1/:

Therefore,Q2 is continuous. Next, we prove the compactness ofQ2. For that, we first show that the set f.Q2x/.t /Wx2˝rgis relatively compact inP C.I; X /. By the assumptions of our theorem, we have

k.Q2x/.t /k

M t1.k1kL¹.I;R^C/Crk2kL¹.I;R^C/CK₁rk3kL¹.I;R^C/C kBkkuk/;

fort2Œ0; t1, and

k.Q2x/.t /k M.t_k t_{k 1}/

k1kL¹.I;R^C/Crk2kL¹.I;R^C/CK_krk3kL¹.I;R^C/C kBkkuk

; for t 2.t_{k 1}; t_k, which gives the uniformly boundedness of f.Q2x/.t /Wx2˝rg. We now show thatQ2.˝r/ is equicontinuous. Functionsf.Q2x/.t /Wx2˝rgare equicontinuous attD0. For anyx2˝r, if0 < r1< r2t1, then

k.Q2x/.r2/ .Q2x/.r1/k

Z r₁

0 kTq.r2 s/ Tq.r1 s/kŒkBu.s/k C kf .s; x.s/; .H x/.s//kds C

Z r2

r₁ kTq.r2 s/kŒkBu.s/k C kf .s; x.s/; .H x/.s//kds Œr1kTq.r2 s/ Tq.r1 s/k CM.r2 r1/

kBkkuk C k1kL¹.I;R^C/Crk2kL¹.I;R^C/CK₁rk3kL¹.I;R^C/

: Similarly, ifti < r1< r2tiC1, then

k.Q2x/.r2/ .Q2x/.r1/k

Z r1

t_i kTq.r2 s/ Tq.r1 s/kŒkBu.s/k C kf .s; x.s/; .H x/.s//kds C

Z r₂ r1

kTq.r2 s/kŒkBu.s/k C kf .s; x.s/; .H x/.s//kds Œ.r1 ti/kTq.r2 s/ Tq.r1 s/k CM.r2 r1/

kBkkuk C k1kL¹.I;R^C/Crk2kL¹.I;R^C/CK_iC1rk3kL¹.I;R^C/

: From .H₁/, it follows the continuity of operator T_q./ in the uniform operator to- pology. Thus, the right hand side of the above inequality tends to zero asr2!r1. Therefore,f.Q2x/.t /Wx2˝rgis a family of equicontinuous functions. According to the infinite dimensional version of the Ascoli–Arzela theorem, it remains to prove that, for anyt2Œ0; bn ft1; : : : ; tmg, the setV .t /WD f.Q2x/.t /Wx2˝rgis relatively compact inP C.I; X /. The caset D0is trivial: V .0/D f.Q2x/.0/Wx./2˝rgis

compact in P C.I; X /. Lett 2.0; t1be a fixed real number andhbe a given real number satisfying0 < h < t1. DefineVh.t /Dn

.Q^h₂x/.t /Wx2˝r

o , .Q^h₂x/.t /D

Z t h 0

Tq.t s/Bu.s/dsC Z t h

0

Tq.t s/f .s; x.s/; .H x/.s//ds DTq.h/

Z t h 0

Tq.t s h/Bu.s/ds CTq.h/

Z t h 0

Tq.t s h/f .s; x.s/; .H x/.s//ds DTq.h/y1.t; h/:

We use same arguments, we fixt2.ti; tiC1, and lethbe a given real number satis- fyingti < h < tiC1, we defineVh.t /Dn

.Q^h₂x/.t /Wx2˝r

o , .Q^h₂x/.t /D

Z t h t_i

Tq.t s/Bu.s/dsC Z t h

t_i

Tq.t s/f .s; x.s/; .H x/.s//ds DTq.h/

Z t h t_i

Tq.t s h/Bu.s/ds CTq.h/

Z t h t_i

Tq.t s h/f .s; x.s/; .H x/.s//ds DTq.h/y2.t; h/:

The compactness of Tq.h/ in P C.I; X /, together with the boundedness of both y1.t; h/andy2.t; h/on˝r, give the relativity compactness of the setVh.t /in P C.I; X /. Moreover, for allt2Œ0; t1,

k.Q2x/.t / .Q^h₂x/.t /k

Z t t h

Tq.t s/Bu.s/dsC Z t

t h

Tq.t s/f .s; x.s/; .H x/.s//ds

hM kBkkuk C k1kL¹.I;R^C/Crk2kL¹.I;R^C/CK₁rk3kL¹.I;R^C/

: Also, for allt2.ti; tiC1,

k.Q2x/.t / .Q^h₂x/.t /k

Z t t h

Tq.t s/Bu.s/dsC Z t

t h

Tq.t s/f .s; x.s/; .H x/.s//ds

hM kBkkuk C k1kL¹.I;R^C/Crk2kL¹.I;R^C/CK_iC1rk3kL¹.I;R^C/

: Choose h small enough. It implies that there are relatively compact sets arbitrar- ily close to the set V .t / for each t 2Œ0; bn ft1; : : : ; tmg. Then, V .t /, t 2 Œ0; bn ft1; : : : ; tmg, is relatively compact in P C.I; X /. Since it is compact at t D0, we

have the relatively compactness ofV .t / inP C.I; X / for allt 2Œ0; bn ft1; : : : ; tmg. Hence, by the Arzela–Ascoli theorem, we conclude that Q2 is compact. From Lemma 2, we ensure that the control system (1.1) has at least one mild solution

ont2Œ0; bn ft1; : : : ; tmg.

4. APPROXIMATE CONTROLLABILITY

In this section, with help of the obtained existence theorem of mild solutions, we show an approximate controllability result for system (1.1).

Theorem 2. If (H1)–(H5) are satisfied andR.; _t^tk

k 1;i/!0in the strong op- erator topology as!0^C,iD1; 2, then the impulsive non-local fractional control system(1.1)is approximately controllable ont2Œ0; bn ft1; : : : ; tmg.

Proof. According to Theorem1,Q₁CQ₂has a fixed point in˝r for any > 0.

This implies that there existsx2.Q₁CQ₂/.x/such that

x.t /D 8 ˆˆ ˆˆ ˆˆ

<

ˆˆ ˆˆ ˆˆ :

Sq.t /.x0 g.x//

CRt

0Tq.t s/Œf.s; x.s/; .H x/.s//CBu.s/ds; t2Œ0; t1;

Sq.t t_{k 1}/Œx.t_{k 1}/CI_{k 1}.x.t_{k 1}//CDv.t_{k 1}/

CRt

tk 1T_q.t s/Œf.s; x.s/; .H x/.s//CBu.s/ds; t 2.t_{k 1}; t_k;

where fort2Œ0; t1we have uDBT_q.t1 t /R.; _0;1^t1 /

x1 Sq.t1/.x0 g.x//

Z t1

0

Tq.t1 s/f.s; x.s/; .H x/.s//ds while forkD2; : : : ; mC1

u

DBT_q.t_k t /R.; _t^tk

k 1;1/

x_k Sq.t_k t_{k 1}/Œx.t_{k 1}/CI_{k 1}.x.t_{k 1}//

Z tk

t_{k 1}

Tq.t_k s/f.s; x.s/; .H x/.s//ds and

vDDS_q.t_k t_{k 1}/R.; _t^tk

k 1;2/

x_k Sq.t_k t_{k 1}/Œx.t_{k 1}/CI_{k 1}.x.t_{k 1}//

Z tk

tk 1

Tq.tk s/f.s; x.s/; .H x/.s//ds

: Furthermore,

x.t1/DSq.t1/.x0 g.x//

C Z t1

0

Tq.t1 s/Œf.s; x.s/; .H x/.s//CBu.s/ds;

x.t_k/DSq.t_k t_{k 1}/Œx.t_{k 1}/CI_{k 1}.x.t_{k 1}//CDv.t_{k 1}/

C Z tk

t_{k 1}

Tq.t_k s/Œf.s; x.s/; .H x/.s//CBu.s/ds;

kD2; : : : ; mC1, with xt1 x.t1/Dx1 t₁

0;1R.; _0;1^t1 /

x1 Sq.t1/.x0 g.x//

Z t1

0

Tq.t1 s/f.s; x.s/; .H x/.s//ds

Sq.t1/.x0 g.x//

Z t1

0

Tq.t1 s/f.s; x.s/; .H x/.s//ds;

xt_k x.t_k/Dx_{k k 1;2}^tk R.; _{k 1;2}^tk /

x_k Sq.t_k t_{k 1}/h

x.t_{k 1}/CI_{k 1}.x.t_{k 1}//i Z tk

t_{k 1}

Tq.t_k s/f

s; x.s/; .H x/.s/

ds

Sq.tk tk 1/ h

x.t_{k 1}/CIk 1

.x.t_{k 1}//

i Z t_k

tk 1

Tq.t_k s/f

s; x.s/; .H x/.s/

ds

tk

k 1;1R.; _{k 1;1}^tk /

x_k Sq.t_k t_{k 1}/h

x.t_{k 1}/CI_{k 1}.x.t_{k 1}//i Z tk

tk 1

Tq.tk s/f

s; x.s/; .H x/.s/

ds

; kD2; : : : ; mC1:

From (2.3) we haveI _t^tk

k 1;iR

; _t^tk

k 1;i

DR

; _t^tk

k 1;i

,iD1; 2, and xt1 x.t1/DR

; _0;1^t1

x1 Sq.t1/.x0 g.x//

Z t1

0

Tq.t1 s/f.s; x.s/; .H x/.s//ds

; (4.1)

xtk x.t_k/Dh R

; _{k 1;1}^tk CR

; _{k 1;2}^tk i

x_k Sq.t_k t_{k 1}/h

x.t_{k 1}/CI_{k 1}.x.t_{k 1}//i Z tk

t_{k 1}

Tq.t_k s/f.s; x.s/; .H x/.s//ds

; kD2; : : : ; mC1: (4.2) Since compactness of both Sq.t /t >0 and Tq.t /t >0 hold, and also boundedness of f,gandI_{k 1}, we can use on (4.1)–(4.2) the fact thatR.; _t^tk

k 1;i/!0in the strong operator topology as !0^C,i D1; 2. This giveskxtk x.t_k/k^˛!0as !0^C,iD1; 2. Hence, the impulsive non-local fractional control system (1.1) is approximately controllable ont2Œ0; bn ft1; : : : ; tmg.

5. OPTIMALITY

LetY be a separable reflexive Banach space andw_f.Y /represent a class of non- empty, closed and convex subsets ofY. The multifunctionwWI !wf.Y /is meas- urable andw./E, whereE is a bounded set ofY. We give the admissible control set as follows:

Uad D˚

.u; v/2L¹.E/L¹.E/ju.t /; v.t /2w.t / a:e: ¤¿: Consider the following impulsive nonlocal fractional control system:

8

<

:

CD^q_tx.t /DAx.t /Cf .t; x.t /; .H x/.t //CBu.t /; t 2.0; bn ft1; t2; : : : ; tmg; x.0/Cg.x/Dx02X;

4x.ti/DIi.x.t_i //CDv.t_i /; iD1; 2; : : : ; m; .u; v/2U_ad;

(5.1) whereB;D 2L¹.I; L.Y; X //. It is clear thatBu;Dv2L¹.I; X /for all.u; v/2 Uad. Letx^u;v be a mild solution of system (5.1) corresponding to controls.u; v/2 U_ad. We consider the Bolza problem .BP /: find an optimal triplet .x⁰; u⁰; v⁰/2 P C.I; X /U_ad such thatJ.x⁰; u⁰; v⁰/J.x^u;v; u; v/, for all.u; v/2U_ad, where

J.x^u;v; u; v/D

mC1

X

iD1

˚.x^u;v.ti//C Z ti

t_{i 1}

L.t; x^u;v.t /; u.t /; v.t //dt

; iD1; : : : ; mC1. The following extra assumptions are needed:

(H6) The functionalLWIXY²!R[ f1gis Borel measurable.

(H7) L.t;;;/is sequentially lower semi-continuous onXY², a.e. onI. (H8) L.t;;/is convex onY²for eachx2X and almost allt2I.

(H9) There is a non-negative function'2L¹.I;R/andc1; c2; c30such that L.t; x; u; v/'.t /Cc1kxk Cc2kuk^pYCc3kvk^pY.

(H10) The functional˚WX !Ris continuous and non-negative.

Theorem 3. If (H6)–(H10) hold together with the assumptions of Theorem1, then the Bolza problem.BP /admits at least one optimal triplet onP CU_ad.

Proof. Assume inffJ.x^u;v; u; v/j.u; v/2U_adg Dı <C1. From (H6)–(H10), J.x^u;v; u; v/

mC1

X

iD1

˚.x^u;v.ti//C Z ti

t_{i 1}

˚'.t /Cc1kx.t /k Cc2ku.t /k^pY Cc3kv.t /k^pY dt

> 1; iD1; : : : ; mC1:

Here, is a positive constant, i.e., ı > 1. By the definition of infimum, there exists a minimizing sequence of feasible tripletsf.xⁿ; uⁿ; vⁿ/g A_ad, where Aad f.x; u; v/jxis a mild solution of system (5.1) corresponding to.u; v/2Uadg, such that J.xⁿ; uⁿ; vⁿ/!ı as m! C1. As f.uⁿ; vⁿ/g U_ad and fuⁿ; vⁿg is bounded inL¹.I; Y /, then there exists a subsequence, still denoted byf.uⁿ; vⁿ/g, and u⁰; v⁰2L¹.I; Y /, such that.uⁿ; vⁿ/^weakly! .u⁰; v⁰/inL¹.I; Y /L¹.I; Y /. Since the admissible control set U_ad is convex and closed, by Marzur lemma, we have .u⁰; v⁰/2Uad. Suppose thatxⁿis a mild solution of system (5.1), corresponding to uⁿandvⁿ, that satisfies

xⁿ.t /DSq.t /.x0 g.xⁿ//C Z t

0

Tq.t s/.f .s; xⁿ.s/; .H xⁿ/.s//CBuⁿ.s//ds;

fort2Œ0; t1, and

xⁿ.t /DSq.t ti/Œxⁿ.t_i /CIi.xⁿ.t_i //CDvⁿ.t_i /

C Z t

t_i

Tq.t s/Œf .s; xⁿ.s/; .H xⁿ/.s//CBuⁿ.s/ds fort 2.ti; tiC1,i D1; : : : ; m. From (H2), the non-linear functionf is bounded and continuous. Then, there exists a subsequence (with the same notation)

ff .s; xⁿ; .H xⁿ/.s//gandf .s; x⁰; .H x⁰/.s//2L¹.I; X /such thatf .s; xⁿ; .H xⁿ/.s//

converges weakly tof .s; x⁰; .H x⁰/.s//. Also, the same arguments on (H₃) and (H5) yield other weak convergences ofg.xⁿ/andIi.xⁿ/tog.x⁰/andIi.x⁰/, respectively.

Let us denote

.P1x/.t /DSq.t /g.x/C Z t

0

Tq.t s/.f .s; x.s/; .H x/.s//CBu.s//ds; t2Œ0; t1;

.P2x/.t /DSq.t ti/Œx.t_i /CIi.x.t_i //CDv.t_i /

C Z t

ti

Tq.t s/Œf .s; x.s/; .H x/.s//CBu.s/ds; t2.ti,tiC1; iD1; : : : ; m:

Obviously,.P1x/.t /and.P2x/.t /are strongly continuous operators. Thus,.P1xⁿ/.t / and.P2xⁿ/.t /strongly converge to.P1x/.t /and.P2x/.t /, respectively. Next, we consider the system

x⁰.t /DSq.t /.x0 g.x⁰//C Z t

0

Tq.t s/.f .s; x⁰.s/; .H x⁰/.s//CBu⁰.s//ds;

t2Œ0; t1, and

x⁰.t /DSq.t ti/Œx⁰.t_i /CIi.x⁰.t_i //CDv⁰.t_i /

C Z t

ti

Tq.t s/Œf .s; x⁰.s/; .H x⁰/.s//CBu⁰.s/ds;

t 2.ti; tiC1, i D1; : : : ; m. It is not difficult to check thatkxⁿ.t / x⁰.t /k !0as n! 1. Therefore, we can infer that xⁿstrongly converges tox⁰ inP C.I; X /as n! 1. From assumptions (H6)–(H10) and Balder’s theorem, we get

D lim

n!1 mC1

X

iD1

˚.xⁿ.ti//C Z t_i

ti 1

L.t; xⁿ.t /; uⁿ.t /; vⁿ.t //dt

mC1

X

iD1

˚.x⁰.ti//C Z ti

ti 1

L.t; x⁰.t /; u⁰.t /; v⁰.t //dt

DJ.x⁰; u⁰; v⁰/; i D1; : : : ; mC1;

which implies thatJattains its minimum at.x⁰; u⁰; v⁰/2P C.I; X /U_ad.

ACKNOWLEDGEMENT

This research is part of first author’s PhD project. Debbouche and Torres are sup- ported by FCT and CIDMA through project UID/MAT/04106/2013.

REFERENCES

[1] Q. Chen, A. Debbouche, Z. Luo, and J. Wang, “Impulsive fractional differential equations with Riemann-Liouville derivative and iterative learning control,”Chaos Solitons Fractals, vol. 102, pp. 111–118, 2017, doi:10.1016/j.chaos.2017.03.024.

[2] N. D. Cong and H. T. Tuan, “Generation of nonlocal fractional dynamical systems by frac- tional differential equations,”J. Integral Equations Appl., vol. 29, no. 4, pp. 585–608, 2017, doi:

10.1216/JIE-2017-29-4-585.

[3] A. Debbouche and V. Antonov, “Approximate controllability of semilinear Hilfer fractional differ- ential inclusions with impulsive control inclusion conditions in Banach spaces,”Chaos Solitons Fractals, vol. 102, pp. 140–148, 2017, doi:10.1016/j.chaos.2017.03.023.

[4] A. Debbouche and J. J. Nieto, “Sobolev type fractional abstract evolution equations with nonlocal conditions and optimal multi-controls,”Appl. Math. Comput., vol. 245, pp. 74–85, 2014, doi:

10.1016/j.amc.2014.07.073.

[5] A. Debbouche, J. J. Nieto, and D. F. M. Torres, “Optimal solutions to relaxation in multiple control problems of Sobolev type with nonlocal nonlinear fractional differential equations,” J. Optim.

Theory Appl., vol. 174, no. 1, pp. 7–31, 2017, doi:10.1007/s10957-015-0743-7.

[6] A. Debbouche and D. F. M. Torres, “Approximate controllability of fractional delay dynamic inclusions with nonlocal control conditions,”Appl. Math. Comput., vol. 243, pp. 161–175, 2014, doi:10.1016/j.amc.2014.05.087.

[7] A. Debbouche and D. F. M. Torres, “Sobolev type fractional dynamic equations and optimal multi- integral controls with fractional nonlocal conditions,”Fract. Calc. Appl. Anal., vol. 18, no. 1, pp.

95–121, 2015, doi:10.1515/fca-2015-0007.

[8] H. Gou and B. Li, “Local and global existence of mild solution to impulsive fractional semilin- ear integro-differential equation with noncompact semigroup,”Commun. Nonlinear Sci. Numer.

Simul., vol. 42, pp. 204–214, 2017, doi:10.1016/j.cnsns.2016.05.021.

[9] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,Theory and applications of fractional differential equations. Elsevier Science B.V., Amsterdam, 2006.

[10] S. Liu, A. Debbouche, and J. Wang, “On the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths,”J. Comput. Appl. Math., vol. 312, pp.

47–57, 2017, doi:10.1016/j.cam.2015.10.028.

[11] S. Liu, J. Wang, and Y. Zhou, “Optimal control of noninstantaneous impulsive dif- ferential equations,” J. Franklin Inst., vol. 354, no. 17, pp. 7668–7698, 2017, doi:

10.1016/j.jfranklin.2017.09.010.

[12] A. Pazy, Semigroups of linear operators and applications to partial differential equations.

Springer-Verlag, New York, 1983. doi:10.1007/978-1-4612-5561-1.

[13] H. Qin, X. Zuo, J. Liu, and L. Liu, “Approximate controllability and optimal controls of fractional dynamical systems of order1 < q < 2in Banach spaces,”Adv. Difference Equ., pp. 2015:73, 17, 2015, doi:10.1186/s13662-015-0399-5.

[14] B. Radhakrishnan and K. Balachandran, “Controllability of neutral evolution integrodifferential systems with state dependent delay,”J. Optim. Theory Appl., vol. 153, no. 1, pp. 85–97, 2012, doi:

10.1007/s10957-011-9934-z.

[15] R. Sakthivel, R. Ganesh, Y. Ren, and S. M. Anthoni, “Approximate controllability of nonlinear fractional dynamical systems,”Commun. Nonlinear Sci. Numer. Simul., vol. 18, no. 12, pp. 3498–

3508, 2013, doi:10.1016/j.cnsns.2013.05.015.

[16] X.-B. Shu and Q. Wang, “The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order1 < ˛ < 2,”Comput. Math. Appl., vol. 64, no. 6, pp.

2100–2110, 2012, doi:10.1016/j.camwa.2012.04.006.

[17] R. Triggiani, “A note on the lack of exact controllability for mild solutions in Banach spaces,”

SIAM J. Control Optimization, vol. 15, no. 3, pp. 407–411, 1977, doi:10.1137/0315028.

Authors’ addresses

Sarra Guechi

Guelma University, Department of Mathematics, 24000 Guelma, Algeria E-mail address:guechi.sara@yahoo.fr

Amar Debbouche

Guelma University, Department of Mathematics, 24000 Guelma, Algeria E-mail address:amar debbouche@yahoo.fr

Delfim F. M. Torres

University of Aveiro, DMat, CIDMA, 3810-193 Aveiro, Portugal E-mail address:delfim@ua.pt