# Approximate Controllability of Neutral Measure Evolution Equations with Nonlocal Conditions

Authors: Lina Ma; Haibo Gu; Yiru Chen
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we consider a kind of neutral measure evolution equations with nonlocal conditions. By using semigroup theory and fixed point theorem, we can obtain sufficient conditions for the controllability results of such equations. Finally, an example is given to verify the reliability of the results.


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## Body

## 1. Introduction

In the past decades, the theory of impulsive differential equations has been fully developed. An impulsive differential equation is used to simulate the evolution process of a system under perturbation during continuous evolution [1–3]. However, this type of system allows only a limited number of discontinuities within a limited range. As a result, it cannot simulate some complex phenomena, such as Zeno’s behavior. However, the dynamic system with discontinuous trajectory is modeled by a measure differential equation or measure-driven equation [4–10]. Measure differential equations (MDEs) were studied in the early days [11–18]. In 1971, Das first studied measure differential equations. For a complete introduction of measure differential systems, we can refer to [12, 13].On the other hand, the complete controllability of several nonlinear dynamic systems, such as stochastic systems of fractional order and dynamic systems of impulsive differential equations, has been extensively studied. Recently, some authors had discussed existence, stability, and nonlocal controllability of the measure evolution equation [9, 15, 19–22]. In the past few years, the existence and controllability of fractional abstract functional differential development systems with nonlocal conditions have been fully studied [2, 23–31]. However, the controllability problem of neutral measure evolution equations with nonlocal conditions is seldom studied. Therefore, whether the appropriate solution of the system exists for any given control u and whether the system is approximately controllable are studied.In the paper, we will study the following neutral measure evolution equations with nonlocal conditions:(1)dxt−qt,xt=Axtdt+But+ft,xtdgt,t∈J=0,a,xs=ϕs+px,s∈−r,0,where f,q:0,a×G−r,0;X⟶X,xtθ=xt+θ,θ∈−r,0,r>0, p:G−r,a;X⟶X is a specified function. The state variable x· takes values in Banach space X with the norm ⋅. A:DA⊂X⟶X generates a uniformly bounded analytic semigroup Ttt≥0 in Banach space X. g:0,a⟶R is a nondecreasing left-continuous function. The control function ut takes values in another Banach space U, where U is a control set. The set G−r,0;X and G−r,a;X represent the regulated functions space on −r,0 and −r,a, respectively. The rest of this paper is organized as follows. In Section 2, some notations and preparation are given about Kurzweil–Henstock–Stieltjes integrals and regulated functions. In Section 3, we obtain the existence results for measure evolution system (1) by using Schauder’s fixed point theorem. In Section 4, based on Krasnoselskii’s fixed point theorem, we establish a controllability result for mild solutions of system (1). An example is given to prove validity of the results we obtained in Section 5.


## 2. Preliminaries

In this section, we will review some concepts and main results regarding Kurzweil–Henstock–Stieltjes integrals and regulated functions.Consider a functionδ:a,b⟶R+. A tagged partition of the interval a,b with division points a=t0≤t1≤⋯≤tm=b and tags τi∈ti−1,ti,i=1,…,m is called δ-fine if τi−δτi≤ti≤τi+δτi,i∈1,…,m.Definition 1.
(See [22]). Let g:a,b⟶R. A function f:a,b⟶Rn is said to be Kurzweil-Henstock-Stieltjes integrable with respect to (w.r.t.) g on a,b (shortly, K–H–Stieltjes integrable) if there exists a vector I∈X such that for every ε>0, there is a gauge δε on a,b with(2)∑i=1mfτigti−gti−1−I<ε,for every δε-fine partition ti−1,ti,ξi:i=1,2,…,m of a,b. The K–H–Stieltjes integrability is preserved on all subintervals of a,b. The function ∫abftdgt is called the K–H–Stieltjes primitive of f w.r.t. g on a,b. Let KgpJ;Xp>1 be a space of all p-ordered K–H–Stieltjes integral regulated functions from J to X with respect to g, with the norm ⋅Kgp defined by(3)fKgp=KHS∫0afspdgs1/p.Definition 2.
(See [7]). Let X be a Banach space with a norm ⋅ and a,b be a closed interval of the real line. A function f:a,b⟶X is called regulated on a,b if the limits(4)lims⟶t−fs=ft−,t∈a,b,lims⟶t+fs=ft+,t∈a,b,exist and are finite. The space of all regulated functions f:a,b⟶X is denoted by Ga,b;X. It is well known that the set of discontinuities of a regulated function is at most countable and that the space Ga,b;X is a Banach space endowed with the norm f∞=supt∈a,bft. Let r>0, for any element z∈G−r,0;X, and we define the norm z∗=sups∈−r,0zs.
LetY be another separable reflexive Banach space where control function u takes values. Let E⊂Y be bounded, and admissible control set U=KgpJ;Ep>1.Definition 3.
(See [7]). A set A⊂Ga,b;X is called equiregulated if for every ε>0 and t0∈a,b, there is a θ>0 such that(i)
Ifx∈A,t∈a,b and t0−θ<t<t0, then xt0−−xt<ε(ii)
Ifx∈A,t∈a,b and t0<t<t0+θ, then xt−xt0+<εLemma 1.
(See [7]). Consider the functions f:J⟶X and g:J⟶R such that g is regulated and ∫abfdg exists. Then, the function ht=∫abfdg,t∈a,b is regulated and satisfies(5)ht+=ht+ftΔ+gt,t∈a,b,ht−=ht−ftΔ−gt,t∈a,b,where Δ+gt=gt+−gt and Δ−gt=gt−gt−, where gt− and gt+ denote the left limit and the right limit of the function g at t, respectively.Lemma 2.
(See [7]). Let X be a Banach space. Assume that A⊂Ga,b;X is equiregulated, and for every t∈a,b, the set xt:t∈A is relatively compact in X. Then, the set A is relatively compact in Ga,b;X.Lemma 3.
(See [7]). Let X be a Banach space and E⊆X be a bounded, closed, and convex set. If the operator N:E⟶E is completely continuous, then N has a fixed point on E.Lemma 4.
(See [7]). Let xnn=1∞ be a sequence of functions from a,b to X. If xn converges pointwisely to x0 as n⟶∞ and the sequence xnn=1∞ is equiregulated, then xn converges uniformly to x0.
For0<α<1, −Aα can be defined as a closed linear invertible operator with its domain D−Aα being dense in X. We denote by Xα the Banach space D−Aα endowed with norm xα=−Ααx which is equivalent to the graph norm of −Aα. We have Xβ↪Xα for 0<α<β, and the embedding is continuous [32].Lemma 5.
(See [32]). The following properties hold:(1)
If0<α<β, then Xα⊂Xβ and the embedding is compact whenever the resolvent operator of A is compact(2)
For every0<α<1, there exists Cα>0 such that −AαTt≤Cα/tαLemma 6.
(See [31]). Let M be a closed convex nonempty subset of a Banach space S,·. We suppose that P and Q map M into S such that(i)
Px+Qy∈M∀x,y∈M(ii)
P is continuous and PM is contained in a compact set(iii)
Q is a contraction with constant k<1
Then, there is ay∈M with Py+Qy=y.


## 3. Existence Results

In this section, we will study the existence of the mild solution of system (1). We first give the definition of solutions for measure system (1). Using the methods same as in [33], we can obtain the following definition.Definition 4.
The functionx∈GJ;X is called a mild solution of system (1) on J if it satisfies the following measure integral equation:(6)xt=Ttϕ0+px−q0,x0+qt,xt+∫0tATt−sqs,xsds+∫0tTt−sBus+fs,xsdgs.
We introduce the following assumptions:(H1): letM=supt∈0,aTt, and Tt is a compact operator for every t>0(H2): for eacht∈0,a, the function q:0,a×G−r,0;X⟶X is continuous and there exists a constant α∈0,1, and H,H1>0 such that q∈D−Aα, and for any ϕ1,ϕ2,ϕ∈G−r,0;X,t∈0,a, the function −Aαq·,ϕ is strongly measurable and −Aαqt,· satisfies the Lipschitz condition,(7)−Aαqt,ϕ1−−Aαqt,ϕ2≤Hϕ1−ϕ2∗,and the inequality(8)−Aαqt,ϕ≤H1ϕ∗+1(H3): the functionf:0,a×G−r,0;X⟶X satisfies the following:(i)
f·,ϕ is measurable for all ϕ∈G−r,0;X, and ft,·:G−r,0;X⟶KgpJ;X is continuous for a.e.t∈J(ii)
There is a functionm∈KgpJ;X and a nondecreasing continuous function Φ:R+⟶R+ such that(9)ft,xt≤mtΦx,for allxt∈G−r,0;X, almost all t∈J, and liml⟶+∞infΦl/l=φ<+∞(H4):p:G−r,a;X⟶X is continuous and compact, and there exist positive constants e and f such that(10)px≤ex∞+f,(H5):B:E⟶X is a linear and bounded operator, so there is a positive constant MB such that B≤MBTheorem 1.
If assumptions (H1)–(H5) hold, then problem (1) has a mild solution provided that(11)Me+M+1−A−αH1+C1−αH1aαα+Mφga−g01/qmKgp<1.Proof.
We define the operatorQ:GJ;X⟶GJ;X by(12)Qxt=Ttϕ0+px−q0,x0+qt,xt+∫0tATt−sqs,xsds+∫0tTt−sBus+fs,xsdgs.
Letl>0 be a constant and Bl=GJ;X:x∞≤l, where Bl is a bounded, closed, and convex set, QBl=Qx:x∈Bl.Step I: we prove that there exists a constantl>0 such that QBl⊂Bl. Assuming that this conclusion is not true, for each l>0, there will exist xl∈Bl,tl∈J such that Qxltl>l.According to (H1)–(H5),(13)Qxltl=Ttlϕ0+pxl−q0,x0+qtl,xtll+∫0tlATtl−sqs,xslds+∫0tlTtl−sBus+fs,xsldgs≤Mϕ0+pxl+Mq0,x0+qtl,xtll+∫0tl−A1−αTtl−s−Aαqs,xslds+∫0tlTtl−sBusdgs+∫0tlTtl−sfs,xsldgs≤Mϕ0+exl+f+M−A−αH1x0+1+−A−αH1xl+1+C1−αH1×xl+1∫0tltl−sα−1ds+MMB∫0tlusdgs+M∫0tlfs,xsldgs≤Mϕ0+el+f+M+1−A−αH1l+1+C1−αH1l+1aαα+MMBga−g01/quKgp+Mga−g01/qΦxl∫0tlmspdgs1/p≤Mϕ0+el+f+M+1−A−αH1l+1+C1−αH1l+1aαα+MMBga−g01/q‖u‖Kgp+Mga−g01/qΦlmKgp.In the formula (13), q>0,1/q+1/p=1, combining (13) with the fact l<Qxltl, we can obtain(14)l<Qxltl≤Mϕ0+el+f+M+1−A−αH1l+1+C1−αH1l+1aαα+MMBga−g01/q‖u‖Kgp+Mga−g01/qΦlmKgp.Dividing both sides of (14) by l and taking limit as l⟶∞, we have(15)Me+M+1−A−αH1+C1−αH1aαα+Mφga−g01/qmKgp>1,which contradicts.Step II:QBl is equiregulated.(16)Qxt−Qxt0+≤Tt−Tt0+ϕ0+px−q0,x0+qt,xt−qt0+,xt0++∫0t0+−ATt−s−Tt0+−sqs,xsds+∫t0+t−ATt−sqs,xsds+∫0t0+Tt−s−Tt0+−sBusdgs+∫t0+tTt−sBusdgs+∫0t0+Tt−s−Tt0+−sfs,xsdgs+∫t0+tTt−sfs,xsdgs=I1+I2+I3+I4+I5+I6+I7+I8,where(17)I1=Tt−Tt0+ϕ0+px−q0,x0,I2=qt,xt−qt0+,xt0+,I3=∫0t0+−ATt−s−Tt0+−sqs,xsds,I4=∫t0+t−ATt−sqs,xsds,I5=∫0t0+Tt−s−Tt0+−sBusdgs,I6=∫t0+tTt−sBusdgs,I7=∫0t0+Tt−s−Tt0+−sfs,xsdgs,I8=∫t0+tTt−sfs,xsdgs.The combination of the compactness of semigroupTt,t>0 and its strong continuity shows the continuity of Tt in the sense of uniform operator topology and px,qt,xt,ft,xt are bounded, and applying dominated convergence theorem, I1,I3,I5,I7⟶0, as t⟶t0+, independently on particular choices of x·. Combining 1=−A−α−Aα and −A=−Aα−A1−α with (H2), I2,I4⟶0.(18)I6=∫t0+tTt−sBusdgs≤MMBuKgpgt−gt0+1/q,ast⟶t0+,I6⟶0. Moreover, let Ht=∫0tmsdgs, according to Lemma 1, and Ht is a regulated function; then,(19)I8≤∫t0+tTt−sfs,xsdgs≤MΦl∫t0+tmsdgs≤MΦlHt−Ht0+⟶0,ast⟶t0+. Also, we can follow the similar procedure to show Qxt0−−Qxt⟶0 as t⟶t0− for each t0∈0,a. Therefore, QBl is equiregulated on J.Step III:Q:Bl⟶Bl is a continuous operator. Let xn be a convergent sequence in Bl with xn⟶x∈Bl as n⟶∞. According to hypothesis (H2)–(H5) and the boundedness of Tt, we have, for each s∈J,(20)fs,xsn⟶fs,xs,qs,xsn⟶qs,xs,pxn⟶px,asn⟶∞, and then, by the dominated convergence theorem, we get(21)Qxn−Qx≤Mpxn−px+qt,xtn−qt,xt+∫0t−A1−αTt−s−Aαqs,xsn−qs,xsds+∫0tTt−sfs,xsn−fs,xsdgs≤Mpxn−px+qt,xtn−qt,xt+C1−α∫0tt−sα−1−Aαqs,xsn−qs,xsds+M∫0tfs,xsn−fs,xsdgs⟶0.In addition, the analysis same as in Step II demonstrates thatQxnn=1∞ is equiregulated on J. This property and the abovementioned verification together with Lemma 4 imply that Qxn converges uniformly to Qxt as n⟶∞, namely,(22)Qxn−Qx∞=supt∈JQxnt−Qxt⟶0,asn⟶∞.Therefore,QBl is a continuous operator on J.Step IV: for everyt∈J, the set Vt=Qxt:x·∈Bl is relatively compact in X.
The case wheret=0 is trivial, since V0=ϕ0+px,x∈Bl. Let t0≤t≤a be fixed, and let η be a given real number satisfying 0<η<t. For every x·∈Bl, we define(23)Qηxt=Ttϕ0+px−q0,x0+qt,xt+Tη∫0t−ηATt−s−ηqs,xsds+Tη∫0t−ηTt−s−ηBus+fs,xsdgs.
SinceTη is compact, the set Vηt=Qηxt:x·∈Bl is relatively compact in X for every η, 0<η<t. On the other hand, for every x·∈Bl in view of assumption (H2)-(H3), we have(24)Qx−Qηx=∫t−ηt−A1−αTt−s−Aαqs,xsds+∫t−ηtTt−sBusdgs+∫t−ηtTt−sfs,xsdgs≤C1−αH1l+1∫t−ηtt−sα−1ds+MMB∫t−ηtusdgs+M∫t−ηtfs,xsdgs≤C1−αH1l+1ηαα+MMBga−ga−η1/quKgp+MΦl∫0tmsdgs−∫0t−ηmsdgs⟶0.
By the left continuity ofg and Lemma 1, as η⟶0+, Qx−Qηx⟶0. Therefore there are relatively compact sets arbitrarily close to the set Vt. Hence, for each t∈J, Vt is a relatively compact set in X. Step II and Step IV together with Lemma 2 imply that the set QBl is relatively compact in GJ;X. Hence, Q is a completely continuous operator on Bl. By Schauder’s fixed point theorem (Lemma 3), Q has a fixed point in Bl, which is a mild solution of measure control system (1). The proof is completed.


## 4. Controllability Result

In this section, to investigate the controllability of system (1), we first give the definition of controllability.Definition 5.
The system (1) is said to be controllable on the interval J if for every initial function x0∈G−r,0;X and x1∈X, there exists a control u∈KgpJ;Ep>1 such that the mild solution xt of (1) satisfies xa=x1.
We assume the following conditions:(H6): for allϕ1,ϕ2∈G−r,0;X, we assume that function f satisfies(25)fs,ϕ1−fs,ϕ2≤K1ϕ1−ϕ2∗,and for allφ1,φ2∈G−r,a;X, function p satisfies(26)pφ1−pφ2≤Lpφ1−φ2.(H7): the linear operatorW:KgpJ;U⟶X, defined by Wu=∫0aTa−sBusdgs, has an invertible operator W−1 taking values in KgpJ;U/KerW, and there exists a positive constant M′ such that BW−1≤M′(H8):N=MΦl′ga−g01/qmKgp+MM′ga−g0x1+Mϕ0+el′+f+H1−A−α×x0+1+−A−αH1l′+1+C1−αaαH1l′+1/α+MΦl′ga−g01/qmKgp+Mel′+f+H1−A−αx0+1+H1−A−αl′+1+C1−αaαH1l′+1/α<l,wherel′=l+Mϕ0(H9):k:=MLp+H−A−α+aαC1−α/α<1Theorem 2.
If hypotheses (H1)–(H9) are satisfied, system (1) is controllable on J.Proof.
Using assumption (H7) for an arbitrary functionx·, we define the control(27)uxt=W−1x1−Taϕ0+px−q0,x0−qa,xa−∫0aATa−sqs,xsds−∫0aTa−sfs,xsdgst.
In what follows, it suffices to show that when using this control, the operatorF defined by(28)Fxt=ϕt+px,t∈−r,0,Fxt=Ttϕ0+px−q0,x0+qt,xt+∫0tATt−sqs,xsds+∫0tTt−sBuxs+fs,xsdgs,t∈J,has a fixed point x· from which it follows that this fixed point is a mild solution of system (1). Clearly, xa=Fxa=x1, from which we conclude that the system is controllable. Let xt=yt+ϕ^t,t∈−r,a, where ϕ^t is taken as ϕt+px for t∈−r,0 while for t∈J, it is defined as Ttϕ0.
We define the operatorsF1 and F2 by(29)F1yt=0,t∈−r,0,∫0tTt−sfs,ys+ϕ^sdgs+∫0tTt−sBW−1x1−Taϕ0+py+ϕ^−q0,x0−qa,ya+ϕ^a−∫0aATa−τqτ,yτ+ϕ^τdτ−∫0aTa−sfτ,yτ+ϕ^τdgτsdgs,t∈J,F2yt=0,t∈−r,0,Ttpy+ϕ^−q0,x0+qt,yt+ϕ^t+∫0tATt−sqs,ys+ϕ^sds,t∈J.
Obviously, the operatorF has a fixed point if only if operator F1+F2 has a fixed point. Thus, we shall employ Krasnoselskii’s fixed point theorem.Step I: we have to showF1+F2Bl⊂Bl, i.e., any ϕ1,ϕ2∈Bl implies that F1ϕ1+F2ϕ2∈Bl. It is easy from hypotheses (H1–H9) and Lemma 1 to see that(30)F1ϕ1t+F2ϕ2t=∫0tTt−sfs,ϕ1s+ϕ^sdgs+∫0tTt−sBW−1x1+Taϕ0+pϕ1+ϕ^+q0,x0+qa,ϕ1a+ϕ^a+∫0aATa−τqτ,ϕ1τ+ϕ^τdτ+∫0aTa−sfτ,ϕ1τ+ϕ^τdgτsdgs+Ttpϕ2+ϕ^+q0,x0+qt,ϕ2t+ϕ^t+∫0tATt−sqs,ϕ2s+ϕ^sds≤MΦl′ga−g01/qmKgp+MM′ga−g0x1+Mϕ0+el′+f+H1−A−αx0+1+−A−αH1l′+1+C1−αaαH1l′+1α+MΦl′ga−g01/qmKgp+Mel′+f+H1−A−αx0+1+H1−A−αl′+1+C1−αaαH1l′+1α=N<l,and thus, condition (i) in Lemma6 is verified.Step II: next, we need to show that operatorF1 is continuous.(31)F1ϕ1t−F1ϕ2t=∫0tTt−sfs,ϕ1s+ϕ^s−fs,ϕ2s+ϕ^sdgs+∫0tTt−sBW−1Ttpϕ1s+ϕ^s−pϕ2s+ϕ^s+qa,ϕ1a+ϕ^a−qa,ϕ2a+ϕ^a+∫0aATa−τqτ,ϕ1τ+ϕ^τ−qτ,ϕ2τ+ϕ^τdτ+∫0aTa−sfτ,ϕ1τ+ϕ^τ−fτ,ϕ2τ+ϕ^τdgτsdgs≤Mga−g0K11+MM′ga−g0+M′MLp+H−A−α+HC1−αaααϕ1−ϕ2.Therefore, givenε>0, there exists δε>0 such that ϕ1−ϕ2<δ implies F1ϕ1t−F1ϕ2t<ε, which proves that operator F1 is continuous.Step III: we shall show thatF1 maps Bl into an equicontinuous family. For y∈Bl,θ1,θ2∈J,0<θ2<θ1≤a, we have(32)F1yθ1−F1yθ2=∫0θ1Tθ1−sfs,ys+ϕ^sdgs−∫0θ2Tθ2−sfs,ys+ϕ^sdgs+∫0θ1Tθ1−sBuxsdgs−∫0θ2Tθ2−sBuxsdgs≤∫0θ2Tθ1−s−Tθ2−sfs,ys+ϕ^sdgs+∫θ2θ1Tθ1−sfs,ys+ϕ^sdgs+∫0θ2Tθ1−s−Tθ2−sBuxsdgs+∫θ2θ1Tθ1−sBuxsdgs≤sups∈0,aTθ1−s−Tθ2−s∫0θ2fs,ys+ϕ^sdgs+MΦl′Hθ1−Hθ2+Lsups∈0,aTθ1−s−Tθ2−sgθ2−g0+LMgθ1−gθ2,whereHt=∫0tmsdgs, L=M′x1+Mϕ0+el′+f+H1−A−αx0+1+H1−A−αl′+1+H1C1−αaαl′+1/α+MΦl′ga−g01/qmKgp. F1yθ1−F1yθ2⟶0, as θ2⟶θ1. Hence, F1Bl is equicontinuous. Since the case θ2<θ1<0 or θ2<0<θ1 is very simple, the proof of the equicontinuities for the two cases is omitted.Subsequentially, we shall show thatF1Bl is precompact. Let 0<t≤a be fixed and ε be a real number satisfying 0<ε<t. For y∈Bl, we define(33)F1εyt=Tε∫0t−εTt−ε−sfs,ys+ϕ^sdgs+Tε∫0t−εTt−ε−sBW−1x1−Ta×ϕ0+py+ϕ^−q0,x0−qa,ya+ϕ^a−∫0aATa−τqτ,yτ+ϕ^τdτ−∫0aTa−τfτ,yτ+ϕ^τdgτdgs.Tt is a compact operator, and Yεt=F1εyt:y∈Bl is relatively compact in X. For every ε,0<ε<t. Moreover, for every y∈Bl, we have(34)F1yt−F1εyt=∫t−εtTt−sfs,ys+ϕ^sdgs+∫t−εtTt−sBW−1x1−Ta−q0,x0−qa,ya+ϕ^a−∫0aATa−τqτ,yτ+ϕ^τdτ−∫0aTa−τfτ,yτ+ϕ^τdgτdgs.ε⟶0+,F1yt:Y∈Bl is precompact in X. F1Bl is uniformly bounded. By the Arzela–Ascoli theorem, it is concluded from the uniform boundedness, equicontinuity, and precompactness of the set F1Bl that F1Bl¯ is compact.Step IV:FBl is equiregulated on J.(35)Fxt−Fxt0+≤Tt−Tt0+ϕ0+px−q0,x0+qt,xt−qt0+,xt0++∫0t0+−ATt−s−Tt0+−sqs,xsds+∫t0+t−ATt−sqs,xsds+∫0t0+Tt−s−Tt0+−sBusdgs+∫t0+tTt−sBusdgs+∫0t0+Tt−s−Tt0+−sfs,xsdgs+∫t0+tTt−sfs,xsdgs=I1+I2+I3+I4+I5+I6+I7+I8,where(36)I1=Tt−Tt0+ϕ0+px−q0,x0,I2=qt,xt−qt0+,xt0+,I3=∫0t0+−ATt−s−Tt0+−sqs,xsds,I4=∫t0+t−ATt−sqs,xsds,I5=∫0t0+Tt−s−Tt0+−sBusdgs,I6=∫t0+tTt−sBusdgs,I7=∫0t0+Tt−s−Tt0+−sfs,xsdgs,I8=∫t0+tTt−sfs,xsdgs.According to Step II of Theroem1, we can know that FBl is equiregulated on J.Step V: we show that operatorF2 is a contraction with constant k, and we have(37)F2ϕ1t−F2ϕ2t≤Mpϕ1+ϕ^−pϕ2+ϕ^+qt,ϕ1t+ϕt^−qt,ϕ2t+ϕt^+∫0tATt−sqs,ϕ1s+ϕs^−qs,ϕ2s+ϕs^ds≤MLpϕ1−ϕ2+H−A−αϕ1−ϕ2+aαHC1−ααϕ1−ϕ2≤MLp+H−A−α+aαC1−ααϕ1−ϕ2≔kϕ1−ϕ2.
Therefore, all the conditions of Krasnoselskii’s fixed point theorem are satisfied, and thus, operatorF1+F2 has a fixed point in Bl. From this, it follows that operator F has a fixed point and, hence, system (1) is controllable on J. This completes the proof.


## 5. Example

Consider the following equation:(38)dtxt,θ−∫0πUθ,yxts,ydy=∂2∂θ2xt,θdt+ut+ft,xts,θdgt,t∈J=0,1,θ∈0,π,xt,0=xt,π=0,t∈0,1,xs,θ=ϕs,θ+∫−1/20hsln1+xs,θds,θ∈0,π,s∈−12,0.LetX=L20,π;R, we define A:X⟶X by Ax=∂2/∂θ2xt,θ and DA=x·∈X:x,∂x/∂θ are absolutely continuous, and ∂2x/∂θ2∈X,xt,0=xt,π=0. Then, A generates a strongly continuous semigroup Tt,t≥0. Furthermore, A has a discrete spectrum, and the eigenvalues are −n2,n∈N, with corresponding normalized eigenvectors xnz=2/π1/2sinnz. We also use the following properties:(i)
For eachx∈X,Ttx:=∑n=1∞e−n2tx,xnxn. In particular, T· is a uniformly stable semigroup and TtL20,π≤e−t≤1.(ii)
For eachx∈X,−A−1/2x=∑n=1∞1/nx,xnxn. In particular, −A1/2L20,π=1.(iii)
The operator−A1/2 is given by −A1/2x=∑n=1∞nx,xnxn, on the space D−A−1/2=x·∈X,∑n=1∞nx,xnxn∈X.Obviously, Lemma5 and (H1) are satisfied. Let Uhxθ=∫0πUθ,yxydy, for x∈X,θ∈0,π. Then, function f:0,1×G−1/2,0;X⟶X is continuous. Let Ft,xtθ=ft,xtθ=asinxtθ,t∈0,1,θ∈0,π, where a is a constant. We define(39)gt=1−12,0≤t≤1−12,…,1−1n,1−1n−1≤t≤1−1n,n>2,n∈N,…,1,t=1.Letmt=a,Φl=l, and we can see that φ=1.(40)ft,xtθ=∫0πa2sin2xtθdθ1/2≤∫0πa2xtθ2dθ1/2=axt,ft,xtn−ft,xt=∫0πa2sinxtnθ−sinxtθ2dθ1/2≤a∫0πxtnθ−xtθ2dθ1/2≤axtn−xt.Therefore, (H3) is satisfied.p:G−1/2,1;X⟶X, and pϕθ=∫−1/20hsln1+ϕsθds,ϕ∈G−1/2,1;X,ϕsθ=xs,θ. Supposing that h∈L−1/2,0;R and letting ϕn,n∈N be a convergence sequence on G−1/2,1;X,(41)pϕn−pϕX2=∫0πpϕnθ−pϕθ2dθ,=∫0π∫−1/20hsln1+ϕnsθ−ln1+ϕsθds2dθ≤∫−1/20hs2ds∫−1/20∫0πϕnsθ−ϕsθ2dθds⟶0,asn⟶∞. Thus, p is a continuous operator. Using the same method, we can obtain that, for any ϕ∈G−1/2,1;X,pϕθ+η−pϕθX⟶0, as η⟶0. Moreover, pϕX≤∫−1/20hs2ds1/2ϕ∞=eϕ∞. Therefore, (H4) is satisfied.(iv)
The functionUθ,y,θ,y∈0,π is measurable and ∫0π∫0πU2θ,ydydθ<∞.(v)
The function∂θUθ,y is measurable, and U0,y=Uπ,y=0; let(42)H¯=∫0π∫0π∂θUθ,y2dydθ1/2<∞.From (iv) it is clear thatUh is a bounded linear operator on X. Furthermore, Uhx∈D−A1/2, and −A−1/2UhL20,π<∞.In fact, from the definition ofUh and (v), it follows that(43)Uhx,xn=∫0πxnθ∫0πUθ,yxydydθ=1n2π1/2U¯x,cosnθwhere U¯ is defined by U¯xθ=∫0π∂θUθ,yxydy.From (v) we know thatU¯:X⟶X is a bounded linear operator with U¯L20,π≤H¯. Hence, we can write −A1/2UhxL20,π=U¯xL20,π, which implies that (H2) holds. Hence, according to Theorem 1, system (38) has a mild solution provided that the condition of Theorem 1 holds. Moreover, we can impose suitable hypotheses to verify the assumptions stated in Theorem 2. Thus, system (38) is controllable on 0,1.


## 6. Conclusions

In this paper, the issue on approximate controllability of neutral measure evolution equations has been addressed, which can model a large class of hybrid systems without any restriction on their Zeno behavior. Firstly, by adopting Schauder’s fixed point theorem, the existence results of mild solutions for this type of measure control system corresponding to some control function are obtained. Then, the approximate controllability results are provided. Finally, we also use an example to illustrate the main result. Furthermore, we will investigate measure functional evolution equations of Sobolev type in the next work.



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