Approximate controllability and complete. controllability of semilinear fractional functional differential systems with control

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1 Wen and Zhou Advances in Difference Equations 28) 28:375 R E S E A R C H Open Access Approximate controllability and complete controllability of semilinear fractional functional differential systems with control Yanhua Wen * and Xian-Feng Zhou * Correspondence: wenyanhua45@63.com School of Mathematical Sciences, Anhui University, Hefei, China Abstract This paper is concerned with the approximate controllability and complete controllability of semilinear fractional functional differential systems with control involving Caputo fractional derivative. By using the operator semigroup theory and the fixed point theorem, we establish sufficient conditions for each of these types of controllability. The results are obtained under the assumption that the corresponding linear system is approximately controllable and completely controllable, respectively. In the end, an example is presented to illustrate the obtained theory. Keywords: Fractional functional differential system; Approximate controllability; Complete controllability; Mild solution Introduction In this paper, we assume that X is a Hilbert space with the norm.eti =[,T]. Denote C to be the Banach space of continuous functions from [ h,]intox with the norm x = sup t [ h,] xt), x C. The aim of this paper is to study the controllability approximate and complete controllability) of semilinear fractional functional differential system with control of the form: C D q t xt)=axt)but)f t, x t, ut)), t I =[,T], ) x θ)=φθ), h θ, where C D q t is the Caputo fractional derivative of order < q <,thestatevariablex ) takes values in X, thecontrolfunctionu ) 2 I, U) takes values in a Hilbert space U. A is the infinitesimal generator of an analytic semigroup {Tt)} t of bounded operators on X, B : U X is a bounded linear operator, and f : I C U X is a given function satisfying certain assumptions. If x :[ h, T] X is a continuous function, then x t is an element in C defined by x t θ)=xt θ), θ [ h,],andφ C. The controllability theory plays an important role in abstract control systems. Many researchers investigated the approximate or complete controllability of control systems see [ 2] and the references therein). In particular, many authors [2 9] studied the approximate or complete controllability for various semilinear fractional evolution systems. In particular, Sakthivel et al. [7] formulated a new set of sufficient conditions for The Authors) 28. This article is distributed under the terms of the Creative Commons Attribution 4. International icense which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Wen and Zhou Advances in Difference Equations 28) 28:375 Page 2 of 8 the approximate controllability of the semilinear fractional differential system C D q t xt)=axt)but)f t, xt)), x) = x, t J =[,T], 2) where C D q t is the Caputo fractional derivative of order q, ), A is the infinitesimal generator of a C semigroup Tt) of bounded operators on the Hilbert space X, B : U X is a bounded linear operator, U is a Hilbert space, f : J X X is a given function. In [2, 2], the authors paid attention to the approximate controllability of semilinear delay control systems in which the nonlinear terms depend on both state function and control function under the assumption that the corresponding linear systems are approximately controllable. On the other hand, Sukavanam et al. [22] formulated some sufficient conditions for the approximate controllability of a semilinear fractional delay control system. Moreover, the approximate controllability of fractional order semilinear systems with bounded delay when the nonlinear term is independent of a control function was addressed in [23]. The approximate controllability for a class of Riemann iouville fractional differential inclusions via fractional calculus, multi-valued analysis, semigroup theory, and the fixed-point technique was investigated by Yang and Wang in [24]. Also, Sakthivel [25] dealtwith theexactcontrollabilityforfractionalneutralcontrolsystems by using a fixed point analysis approach. On the other hand, in order to study the various fractional systems, introducing a suitableconceptofmildsolutionswillbeofgreatimportance.zhouandjiao [26] studied the existence of mild solution of fractional neutral evolution equations by the fractional power of operatorsand some fixedpoint theorems.also, Wangand Zhou [27] investigated a new mild solution for a class of fractional evolution equations, and then the existence of optimal pairs of the considered system was also obtained. Furthermore, Wang and Zhou [8] discussed the complete controllability of fractional evolution systems based on the fractional calculus, properties of characteristic solution operators, and fixed point theorems. Motivated by the above work, in this paper we first use the above suitable concept of mildsolutionin [26, 27] andthenadoptamethodsimilartopaper [2] which investigates the approximate controllability of integer semilinear functional differential equations. In the end, we establish different sufficient conditions for the approximate and complete controllability of the semilinear fractional functional differential system ). This paper is organized as follows. Section 2 is devoted to some preliminaries. In Sect. 3, we give the approximate controllability result of system ). In Sect. 4, weformulatesufficient conditions for the complete controllability of system ) by using the Banach fixed point theorem when the semigroup {Tt)} t is not compact. An example is presented to demonstrate the approximate controllability result in Sect Preliminaries In this section, we introduce some notations, definitions and lemmas which will be used throughout this paper.

3 Wen and Zhou Advances in Difference Equations 28) 28:375 Page 3 of 8 Definition 2. [28]) The fractional integral of order α with the lower limit for a function g is defined as I α gt)= gs) ds, t >, α >, 3) Ɣα) t s) α provided the right-hand side is pointwise defined on [, ), where Ɣ is the gamma function. Definition 2.2 [28]) The Caputo derivative of order α with the lower limit for a function g can be written as C D α gt)= t g n) s) ds, t >, n <α < n. 4) Ɣn α) t s) α n If g is an abstract function with value in the Hilbert space X,thenintegralswhichappear in Definitions 2.and2.2aretaken in Bochner ssense. By comparing with the fractional differential equations given in [26], we give the following definition of a mild solution of system ). Definition 2.3 [26]) A function x C[ h, T]; X) is called a mild solution of system ) if on [ h, T]itsatisfies xt)=s q t)φ) t s)q T q t s)bus)f s, x s, us))) ds, t [, T], 5) x θ)=φθ), h θ, where S q t)= T q t)=q φ q θ)t t q θ ) dθ, θφ q θ)t t q θ ) dθ, φ q θ)= q θ q ψ q θ q ), ψ q θ)= π ) n qn Ɣnq ) θ n! n= sinnπq), θ, ) is a probability density function. In addition, φ q θ) is the probability density function defined as φ q θ), θ, ), and φ q θ) dθ =. Remark 2. For ξ [, ], θ ξ φ q θ) dθ = Ɣξ) Ɣqξ). emma 2. [6, 26]) The operators S q t) and T q t) have the following properties:

4 Wen and Zhou Advances in Difference Equations 28) 28:375 Page 4 of 8 i) For any t, the operators S q t) and T q t) are linear and bounded operators, that is, for any x X, Sq t)x M x and Tq t)x Ɣ q) x. ii) {S q t)} t and {T q t)} t are strongly continuous. iii) For every t >, S q t) and T q t) are also compact operators if Tt), t >is compact. Definition 2.4 System ) is said to be approximately controllable completely controllable) on the interval I if RT, φ)=x RT, φ)=x), where RT, φ)={x T φ, u)) : u ) 2 I, U)}. In order to deal with our problems, we introduce the following two operators: Ɣ T = T s) q T q T s)bb Tq T s) ds, R μ, Ɣ T ) ) = μi Ɣ T, μ >, where B, Tq t) denote the adjoint of B and T qt), respectively. By [7, 29], we know that the linear fractional control system C D q t xt)=axt)but), t I, x) = φ) 6) is approximately controllable on I if and only if H) μrμ, Ɣ T ) as μ in the strong operator topology. emma 2.2 Hölder s inequality) Assume p, q and p q =.If f p I, R), g q I, R), then fg I, R) and fg I) f p I) g q I). emma 2.3 Schauder s fixed point theorem) If Q is a closed bounded and convex subset of a Banach space E and : Q Q is completely continuous, then has a fixed point in Q. 3 Approximate controllability In this section, we first prove that the operator F μ,<μ whichisdefinedbelow)has a fixed point based on the Schauder s fixed point theorem. Then we consider the approximate controllability of system ) under the condition that the linear fractional system 6) is approximately controllable. In the Banach space CI, C) CI, U), let Y r = { x, u) CI, C) CI, U) x, u) = xt ut) r, t I }, where r is a positive constant. To prove the main results of this section, we impose the following hypotheses:

5 Wen and Zhou Advances in Difference Equations 28) 28:375 Page 5 of 8 H) Tt) is a compact operator for every t >. H2) The function f : I C U X is continuous, and there exist a constant q, q), λ i t) q I, R ) and g i C U, R ), i =,2,...,m,suchthat f t, xt, u) λ i t) sup { g i x t, u): x t } ut) r, t, xt, u) I C U. H3) For each μ >, lim sup r r c i μ sup{ g i x t, u): x t ut) r } ) =. H4) The function f : I C U X is continuous and uniformly bounded, and there is a constant N >such that f t, ϕ, u) N, t, ϕ, u) I C U. For the sake of convenience, we also introduce the following notations. k = max {, MM BT q }, M B = B, a = q, ), Ɣ q) q a i = 3kM B Ɣ q) a) q) T Ɣ q) a) λ q i, q [,T] b i = 3T a) q ) Ɣ q) a) λ q, i q [,T] c i = max{a i, b i }, d = 3kM B xt M φ ), Ɣ q) d 2 =3M ) φ, d = max{d, d 2 }. For μ >,wedefinetheoperatorf μ on CI, C) CI, U)as F μ x, u)=z, v), 7) where vt)=b Tq T t)r μ, Ɣ T ) px, u), 8) zt)=s q t)φ) z θ)=φθ), h θ, t s) q T q t s) Bvs)f s, x s, us) )) ds, t >, 9) px, u)=x T S q T)φ) T s) q T q T s)f s, x s, us) ) ds. We next prove that the operator F μ has a fixed point in CI, C) CI, U).

6 Wen and Zhou Advances in Difference Equations 28) 28:375 Page 6 of 8 Theorem 3. If the hypotheses H) H2) are satisfied, then for each <μ, the operator F μ has a fixed point in CI, C) CI, U). Proof The proof of Theorem 3. is divided into several steps. Step. We first prove that for an arbitrary < μ, there is a constant r = r μ)> such that F μ : Y r Y r.et ψ i r)=sup { g i x t, u): x t ut) r,xt, u) C U }. ) It follows from H3) and ) that there exists r such that r c i μ ψ ir ) d μ, ) that is, d m μ c i μ ψ ir ) r. 2) Direct calculation gives that t s) q q [, t], t [, T], q, q). By emma 2.i), H2) and ), using Hölder s inequality, we obtain that t s) q T q t s)f s, x s, us) ) ds Ɣ q) m t s) q λ i s)ψ i r) ds t s) q ) ) q m q ds λ i ψ Ɣ q) q i r) [,T] T a) q ) Ɣ q) a) q q i r). 3) [,T] If x, u) Y r, then it follows from 8), 2)and3)that vt) = B Tq T t)r μ, Ɣ T ) x T S q T)φ) T s) q T q T s)f s, x s, us) ) ds) ) M B T a) q ) x T M φ λ μ Ɣ q) Ɣ q) a) q i ψ q i r ) [,T] [ ] [ ] d μ 3k c i ψ i r ) d c i ψ i r ) 3k 3kμ r 3k 4)

7 Wen and Zhou Advances in Difference Equations 28) 28:375 Page 7 of 8 and Thus, zt θ) S q t θ)φ) φ θ t θ s) q T q t θ s) Bvs)f s, x s, us) )) ds M ) φ MM BT q Ɣ q) v T a) q) Ɣ q) a) q d 3 k v c i ψ i r ) 3 [ ] d c i ψ i r ) k v 3 q i r ) [,T] μr 3 r 3 2r 3. 5) F μ x, u) ) t) = z, v) = zt vt) r. 6) Therefore, F μ : Y r Y r. Step 2. We will prove that F μ : Y r Y r is continuous. et {y n, u n )} Y r and y n, u n ) y, u), y, u) Y r.byh2), we have f s, y n s, un s)) f s, y s, us)) as n, s I and f s, y n s, u n s) ) f s, y s, us) ) m 2 λ i s)ψ i r ). 7) It follows from the ebesgue dominated convergence theorem, 8) and9) thatforeach t [, T], there exists a constant l such that F μ y n, u n)) t) F μ y, u) ) t) = z n t z t v n t) vt) θ t θ s) q T q t θ s) [ B v n s) vs) ) f s, y n s, un s) ) f s, y s, us) )] ds B Tq T t)r μ, Ɣ T )[ p y n, u n) py, u) ] l θ which implies t θ s) q f s, y n s, un s) ) f s, y s, us) ) ds, n, 8) F μ y n, u n) F μ y, u), n. 9) This means that F μ is continuous. Step 3. Next, we will show that V = { F μ x, u) ) ):x, u) Y r }

8 Wen and Zhou Advances in Difference Equations 28) 28:375 Page 8 of 8 is equicontinuous on [, T]. Indeed, for t θ < t 2 θ T and x, u) Y r,8), 9)and 3)imply vt 2 ) vt ) = B Tq T t 2) Tq T t ) ) R μ, Ɣ T ) x T S q T)φ) T s) q T q T s)f s, x s, us) ) ds) M B T μ q T t 2 ) Tq T t ) ) x T M φ ) T a) q ) λ Ɣ q) a) q i ψ q i r ) [,T] = I 2) and z t2 θ) z t θ) = S q t 2 θ)φ) S q t θ)φ) 2 θ t 2 θ s) q T q t 2 θ s) Bvs)f s, x s, us) )) ds θ t θ s) q T q t θ s) Bvs)f s, x s, us) )) ds Sq t 2 θ)φ) S q t θ)φ) 2 θ t 2 θ s) q T q t 2 θ s)f s, x s, us) ) ds t θ θ[ t2 θ s) q t θ s) q ] T q t 2 θ s)f s, x s, us) ) ds θ t θ s) q [ T q t 2 θ s) T q t θ s) ] f s, x s, us) ) ds 2 θ t 2 θ s) q T q t 2 θ s)bvs) ds t θ θ[ t2 θ s) q t θ s) q ] T q t 2 θ s)bvs) ds θ t θ s) q [ T q t 2 θ s) T q t θ s) ] Bvs) ds = I I 2 I 3 I 4 I 5 I 6 I 7. 2) We now need to check I i ast 2 t, i =,,2,...,7. For I and I, by emma 2.ii), I, I ast 2 t. For I 2 and I 3, similar to 3), by emma 2.i), H2) and emma 2.2,wehave I 2 = 2 θ t θ t 2 θ s) q T q t 2 θ s)f s, x s, us) ) ds

9 Wen and Zhou Advances in Difference Equations 28) 28:375 Page 9 of 8 Ɣ q) 2 θ t θ = t 2 t ) a) q ) Ɣ q) a) q t2 θ s) q ) ) q q m ds λ i q i r ) [,T] q [,T] ψ i r ) ast 2 t 22) and I 3 = θ Ɣ q) [ t2 θ s) q t θ s) q ] T q t 2 θ s)f s, x s, us) ) ds θ θ Ɣ q) q i r ) [,T] = Ɣ q) θ Ɣ q) a) q λ i ψ q i r ) [,T] t 2 t ) a) q ) Ɣ q) a) q [ t θ s) q t 2 θ s) q ] m λ i s)ψ i r ) ds t θ s) q t 2 θ s) ) ) q q ) q ds ) q m t θ s) a t 2 θ s) a ds λ i t θ) a t 2 θ) a t 2 t ) a) q q i r ) [,T] q [,T] ψ i r ) ast 2 t. 23) For t θ =,<t 2 θ T,itcanbeeasilyseenthatI 4 =.Fort θ >andε >small enough, by emma 2.i), H2) and emma 2.2,weobtain I 4 = θ θ ε θ t θ ε sup s [,t θ ε] t θ s) q [ T q t 2 θ s) T q t θ s) ] f s, x s, us) ) ds t θ s) q T q t 2 θ s) T q t θ s) f s, x s, us) ) ds t θ s) q T q t 2 θ s) T q t θ s) ) f s, x s, us) ) ds T q t 2 θ s) T q t θ s) θ ε 2 θ m t θ s) q λ i s)ψ i r ) ds Ɣ q) t θ ε sup s [,t θ ε] Tq t 2 θ s) T q t θ s) θ ε m t θ s) q λ i s)ψ i r ) ds t θ s) q ) ) q q ds

10 Wen and Zhou Advances in Difference Equations 28) 28:375 Page of 8 q i r ) [,T] q i r ) [,T] 2 θ t θ s) q ) ) q q ds Ɣ q) t θ ε = sup Tq t 2 θ s) T q t θ s) t θ) b ε a ) q s [,t θ ε] a) q q i r ) [,T] 2 Ɣ q)a ) q εa) q ) q i r ). 24) [,T] Assumption H) and emma 2.ii), iii) imply the continuity of T q t), t >intheuniform operator topology, then it is easy to obtain that I 4 tendstozeroast 2 t andε. On the other hand, by 4), we know that v = B Tq T t)r μ, Ɣ T ) x T S q T)φ) T s) q T q T s)f s, x s, us) ) ds) M B T a) q ) x T M φ μ Ɣ q) Ɣ q) a) q ) q i r ) [,T] 25) is bounded. Then, by using similar methods as we did to I 2, I 3 and I 4, it follows I 5 = 2 θ t θ M B v Ɣ q) t 2 θ s) q T q t 2 θ s)bvs) ds 2 θ t θ MM B v Ɣ q) t 2 t ) q t 2 θ s) q ds ast 2 t, 26) θ[ I 6 = t2 θ s) q t θ s) q ] T q t 2 θ s)bvs) ds M B v Ɣ q) = MM B v Ɣ q) θ MM B v Ɣ q) t 2 t ) q [ t θ s) q t 2 θ s) ) q ] ds [ t θ) q t 2 θ) q t 2 t ) q] ast 2 t 27)

11 Wen and Zhou Advances in Difference Equations 28) 28:375 Page of 8 and I 7 = θ θ ε θ t θ ε t θ s) q [ T q t 2 θ s) T q t θ s) ] Bvs) ds t θ s) q T q t 2 θ s) T q t θ s) M B v ds t θ s) q T q t 2 θ s) T q t θ s) ) Bvs) ds sup T q t 2 θ s) T q t θ s) M B v t θ) q ε q s [,t θ ε] q 2MM B v Ɣ q) εq ast 2 t, ε. 28) In consequence, F μ x, u)t 2 ) F μ x, u)t ) = z t2 z t vt 2 ) vt ) tends to zero independently of x, u) Y r as t 2 t. This means that V = {F μ x, u)) ):x, u) Y r } is equicontinuous. Since the other cases t θ < t 2 θ <,t θ <<t 2 θ are very simple, we only consider the case t θ < t 2 θ.inconclusion,f μ [Y r ] is equicontinuous and also bounded. Step 4. It remains to prove that for any t [, T], Vt) ={F μ x, u))t) :x, u) Y r } is relatively compact. Obviously, for t =,V) is compact. et < t T be fixed and η be a given real number satisfying < η < t.forη, t)and δ >, define an operator F μ η,δ as F μ η,δ x, u)) t) = φ q θ)t t q θ ) φ) dθ q δ Bvs)f s, x s, us) )) ) ds, vt) η = T η q δ )[ φ q θ)t t q θ η q δ ) φ) dθ q δ δ θt s) q φ q θ)t t s) q θ ) η T t s) q θ η q δ ) Bvs)f s, x s, us) )) ] ) ds, vt) δ θt s) q φ q θ) = T η q δ ) zt, η), vt) ), 29) where x, u) Y r.sincetη q δ), η q δ >iscompact,zt, η) andvt) areboundedony r, the set V η,δ t)= { F μ η,δ x, u)) } t):x, u) Y r is relatively compact in C U for η, t)and δ >.Furthermore,forx, u) Y r, F μ x, u) ) t) F μ η,δ x, u)) t) δ = φ q θ)t t q θ ) δ dθφ) q θt s) q φ q θ)t t s) q θ ) Bvs) ds

12 Wen and Zhou Advances in Difference Equations 28) 28:375 Page 2 of 8 q δ t η δ t η M φ δ δ qm θt s) q φ q θ)t t s) q θ ) f s, x s, us) ) ds θt s) q φ q θ)t t s) q θ ) Bvs) ds θt s) q φ q θ)t t s) q θ ) f s, x s, us) ) ds φ q θ) dθ qm M B v t s) q ) q ds) q qm M B v t s) q ds t η qm t η [ M φ M M B v T q qm T a) q ) a ) q M M B v η q Ɣ q) t s) q ) q ds) q t s) q ds δ θφ q θ) dθ q i r ) [,T] θφ q θ) dθ q i r ) [,T] ] δ q i r ) θφ q θ) dθ [,T] qm Ɣ q)a ) q ηa) q ) δ θφ q θ) dθ θφ q θ) dθ q i r ) [,T] asδ, η. 3) Here the last inequalityis based on Remark 2.. This means that there are relatively compact sets arbitrarily close to the set Vt), t, T]. Thus, for each t, T], Vt) is relatively compact in C U. By the Arzelá Ascoli theorem, F μ [Y r ] is relatively compact in CI, C) CI, U). We obtain that F μ is a completely continuous operator. Hence, emma 2.3 shows that F μ has a fixed point. The proof is complete. Theorem 3.2 Assume that the hypotheses of Theorem 3. and H4) are satisfiedand that the linear system 6) is approximately controllable on I. Then system ) is approximately controllable on I. Proof et x μ ), ū μ )beafixedpointiny r, i.e., F μ x μ ), ū μ) = x μ ), ū μ), where x μ t)=s q t)φ) x μ θ)= φθ), h θ t s) q T q t s) Bū μ s)f s, x μ s, ūμ s) )) ds, t >, 3)

13 Wen and Zhou Advances in Difference Equations 28) 28:375 Page 3 of 8 and the control function ū μ t)=b Tq T t)r μ, Ɣ T ) p x μ, ū μ), 32) p x μ, ū μ )=x T S q T)φ) T s)q T q T s)f s, x μ s, ū μ s)) ds. By Definition 2.3, any fixed point of F μ is a mild solution of ) under the control ū μ t). Then, by 3), 32)and the fact I Ɣ TRμ, ƔT )=μrμ, ƔT ), we have x μ T)=x T p x μ, ū μ) T s) q T q T s)bū μ s) ds T s) q T q T s)bb Tq T s)r μ, Ɣ T ) p x μ, ū μ) ds = x T p x μ, ū μ) = x T p x μ, ū μ) Ɣ T R μ, Ɣ T ) p x μ, ū μ) = x T Ɣ T R μ, Ɣ T ) ) I p x μ, ū μ) = x T μr μ, Ɣ T ) p x μ, ū μ). 33) Furthermore, by condition H4), itfollows f s, x μ s, ū μ s) ) ) 2 2 ds NT 2. 34) Consequently, {f s, x μ s, ū μ s))} is bounded in 2 I, X). Then there exists a subsequence, still denoted by {f s, x μ s, ū μ s))},whichweaklyconvergestofs)in 2 I, X). Define w = x T S q T)φ) It follows from 35)that T s) q T q T s)f s) ds. 35) p x μ, ū μ) w = T s) q T q T s) f s, x μ s, ūμ s) ) f s) ) ds = sup t s) q T q t s) f s, x μ s, ūμ s) ) f s) ) ds. 36) tt As Step 3, Step 4 in the proof of Theorem 3., using the Arzelá Ascoli theorem one can obtain that an operator g ) s)q T q s)gs) ds is compact. Thus, the right-hand side of 36)tendstozeroasμ.ThenH), 33)and36)imply x μ ) T) x T = μr μ, Ɣ T p x μ, ū μ) = μr μ, Ɣ T ) ) w)μr μ, Ɣ T p x μ, ū μ) w ) ) ) μr μ, Ɣ T w) μr μ, Ɣ T p x μ, ū μ) w μr μ, Ɣ T ) w) p x μ, ū μ) w 37)

14 Wen and Zhou Advances in Difference Equations 28) 28:375 Page 4 of 8 as μ. This proves that system ) is approximately controllable on I. Theproofis complete. 4 Complete controllability In this section, we formulate sufficient conditions for complete controllability of the semilinear fractional system ) under the assumption that the linear system 6)iscompletely controllable. In this case, the condition of compactness of Tt) is not made. To prove our results, let us assume that H5) The function f : I C U X is continuous, and there exists a constant > such that f t, ϕ, u) ϕ C u ), t, ϕ, u) I C U. H6) There exists a constant such that f t, ϕ, u ) f t, ϕ 2, u 2 ) ϕ ϕ 2 C u u 2 ) for ϕ, u ), ϕ 2, u 2 ) C U. H7) The linear system 6) is completely controllable. emma 4. [25, 3]) The linear fractional control system 6) is completely controllable on I if and only if there exists γ >such that Ɣ T x, x γ x 2 inthehilbertspacex, i.e., Ɣ T ) γ. Define the operator F on CI, C) CI, U)as F x, u)=z, v), 38) where vt)=b Tq T t) Ɣ T ) px, u), 39) zt)=s q t)φ) z θ)=φθ), h θ, t s) q T q t s) Bvs)f s, x s, us) )) ds, 4) px, u)=x T S q T)φ) T s) q T q T s)f s, x s, us) ) ds. Theorem 4. Assume that the hypotheses H5), H6) and H7) are satisfied. If MB M 2 qt q γɣ 2 q) M2 B M3 qt 2q γɣ 3 q) MT q ) <, 4) Ɣ q) then the operator F has a fixed point in CI, C) CI, U).

15 Wen and Zhou Advances in Difference Equations 28) 28:375 Page 5 of 8 Proof First we show that F maps CI, C) CI, U)intoitself.ByH5), emma 2.i) and emma 4.,thereexisttwoconstantsl, l 2 >suchthat vt) M B T x T M φ T s) q f s, xs, us) ) ) ds γɣ q) Ɣ q) M B x T M φ MT q γɣ q) Ɣ q) x C u ) ) l x C u ) 42) and zt) M φ Ɣ q) Ɣ q) t s) q Bvs) ds t s) q f s, xs, us) ) ds M φ MM BT q Ɣ q) C x C u ) MT q Ɣ q) x C u ) l 2 x C u ), 43) respectively. It follows from 42)and43) that there exists a constant C 3 such that F x, u) = v z l 3 x C u ). 44) This means F maps CI, C) CI, U)intoitself. We next prove that F is a contraction mapping. For any x, u), y, w) CI, C) CI, U), it holds F x, u) F y, w) v v 2 z z 2 v v 2 t s) q T q t s)b v s) v 2 s) ) ds t s) q T q t s) f s, x s, us) ) f s, y s, ws) )) ds = I I 2 I 3. 45) It follows from emma 2.i), emma 4. and H6) that I = v v 2 = B Tq T t) Ɣ T ) T T s) q T q T s) f s, x s, us) ) f s, y s, ws) )) ds M B γ M 2 q 2 Ɣ 2 q) T s) q f s, x s, us) ) f s, y s, ws) ) ds M BM 2 qt q γɣ 2 q) x y u w ), 46)

16 Wen and Zhou Advances in Difference Equations 28) 28:375 Page 6 of 8 I 2 = t s) q T q t s)b v s) v 2 s) ) ds MT q M B Ɣ q) v v 2 M2 B M3 qt 2q γɣ 3 q) x y u w ), 47) and I 3 = t s) q T q t s) f s, x s, us) ) f s, y s, ws) )) ds Ɣ q) t s) q f s, xs, us) ) f s, y s, ws) ) ds MT q Ɣ q) x y u w ). 48) Then 45) 48)imply F x, u) F y, w) MB M 2 qt q γɣ 2 q) M2 B M3 qt 2q γɣ 3 q) MT q ) Ɣ q) x y u w ). 49) Combining 4)with49)yieldsthatF is a contraction mapping. Consequently, F has a fixed point in CI, C) CI, U) by the Banach fixed point theorem, which is a mild solution of system ). This completes the proof. Remark 4. It can be easily seen that inequality 4) is satisfied if the constant is small enough. Theorem 4.2 Assumethat theassumptions H5), H6) and H7) aresatisfied. Then system ) is completely controllable on I. Proof If x, ū ) is a fixed point of the operator F,then33)holdswithμ =, i.e., x T)= x T for any x T X. Hence,system) is completely controllable on I. We omit the details here. 5 Anexample As an application of our results, we consider the fractional partial differential equation of the form q t xt, z)= z 2 xt, z)bz)ut)f t, xt h, z), ut)), t [, ], z [, π], xt,)=xt, π)=, t >, 5) xt, z)=φt, z), h t, here q t is the Caputo fractional derivative of order < q <withrespecttot, φ is continuous, u U = 2 [, ] is continuous, X = 2 [, π], b X and f is a given continuous and uniformly bounded function.

17 Wen and Zhou Advances in Difference Equations 28) 28:375 Page 7 of 8 Put xt) =xt, ), i.e., xt)z) =xt, z). Define the function f : [, ] X U X by f t, x t, u)z) =f t, xt h, z), ut)). et B : U X be a bounded linear operator defined by Bu)z)=bz)u, z [, π], u U, bz) X and define an operator A : X X by Av = v with the domain DA)= { v X v ) 2 [, π], v, v are absolutly continuous, v X and v) = vπ)= }. It follows Av = n 2 v, e n )e n, n= v DA), 2 where e n z)= sin nz, z π, n =,2,...Itiswellknownthattheoperator A generates a compact semigroup Tt), t π >,whichisgivenby Tt)v = e n2t v, e n )e n, v X. n= Formoredetails, pleasereferto [3]. Therefore, system 5) canbewrittentotheabstract form ) and assumption H) is satisfied. Then under the condition imposed on f and the linear system corresponding to 5) is approximately controllable on [, ], see [7], system 5) is approximately controllable on [, ] by Theorem 3.2. Acknowledgements The authors would like to give their thanks to the reviewers for their valuable comments and suggestions. Funding This work is supported by the National Natural Science Foundation of China 475, 3727, and 63), the Natural Science Foundation of Anhui Province 5885MA, 6885MA2, and 7885MA5), the Program of Natural Science Research for Universities of Anhui Province KJ26A23) and the Natural Science Fund of Colleges and Universities in Anhui Province KJ28A47). Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally in this article. They read and approved the final manuscript. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 4 August 28 Accepted: 9 October 28 References. Curtain, R., Zwart, H.J.: An Introduction to Infinite Dimensional inear Systems Theory. Springer, New York 995) 2. Mahmudov, N.I.: Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. Comput. Math. Appl. 62, ) 3. Klamka, J.: Controllability of non-densely defined functional differential systems with delays. Nonlinear Dyn. 56, ) 4. Ahmed, N.: Dynamic Systems and Control with Applications. World Scientific, Hackensack 26)

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