R E S E A R C H
Open Access
Nonlinear impulsive evolution equations
with nonlocal conditions and optimal
controls
Lanping Zhu
*and Qianglian Huang
*Correspondence:
School of Mathematics, Yangzhou University, Yangzhou, 225002, China
Abstract
This paper is concerned with controlled nonlinear impulsive evolution equations with nonlocal conditions. The existence of PC-mild solutions is proved, but the uniqueness cannot be obtained. By constructing approximating minimizing sequences of functions, the existence of optimal controls of systems governed by nonlinear impulsive evolution equations is also presented.
MSC: 34G20; 34K35; 47D06; 93C25
Keywords: nonlinear impulsive equations; nonlocal conditions; feasible pairs; optimal controls
1 Introduction
Since the end of last century, impulsive evolution equations and impulsive optimal con-trols have attracted much attention because many evolutional processes are subject to abrupt changes in the form of impulses (see [–]). Indeed, we can find numerous appli-cations in demography, control, mechanics, electrical engineering fields, and so on. Re-cently, there has been significant development in impulsive differential equations and op-timal controls. For more details, we refer to the monographs of Ahmed [, , ], Xiang [, , ], Ashyralyev [, ], Sharifov [–, ], Mardanov [], and the references therein. There are many papers discussing the impulsive differential equations and impulsive op-timal controls with the classic initial condition:x() =x(see [, , , , ]). How does one look for suitable optimal controls when the uniqueness of solutions of controlled im-pulsive equations cannot be obtained? In spite of some attempts and efforts, the previous method is not entirely satisfactory. New techniques must be given. In particular, nonlocal differential equations have been studied extensively in recent years [, , , ]. But the existence of optimal controls of systems governed by nonlocal impulsive evolution equa-tions with the initial conditionx() +g(x) =xseems to be rarely involved. In this paper, we consider the following controlled nonlocal impulsive equation:
⎧ ⎪ ⎨ ⎪ ⎩
x(t) =Ax(t) +f(t,x(t)) +B(t)u(t), t∈[,T],t=ti,
x() +g(x) =x, u∈Uad,
x(ti) =Ji(x(ti)), i= , , . . . ,q, <t<t<· · ·<tq<T,
(.)
whereA:D(A)⊆X→Xis the infinitesimal generator of strongly continuous semigroup S(t) fort> in a real Banach spaceX,f is a nonlinear perturbation,Ji(i= , , . . . ,q) is
a nonlinear map and x(ti) =x(ti+) –x(t–i), g is a givenX-valued function, the control
u∈Uad,Uadis a control set which we will introduce in Section .
By introducing a reasonable mild solution for (.) that can be represented by integral equation, we show the existence of feasible pairs. Moreover, a limited Lagrange problem of system governed by (.) is investigated. Due to the lack of uniqueness of feasible pairs, we mainly apply the idea of constructing approximating minimizing sequences of functions twice and derive the existence of optimal controls. It is different from the usual approach that the Banach contraction mapping theory is applied directly to prove the existence of optimal controls if the uniqueness of feasible pairs cannot be obtained. Therefore our re-sults essentially generalize and develop many previous ones in this field.
This paper is organized as follows. In Section , we give some associated notations and recall some concepts and facts about the measure of noncompactness, fixed point theo-rem and impulsive semilinear differential equations. In Section , the existence results of controlled nonlocal evolution equations are obtained. In Section , the existence of op-timal controls for a Lagrange problem is established. Finally, an example is presented to illustrate our results in the last section.
2 Preliminaries
Let Xbe a real Banach space.L(X) is the class of (not necessary bounded) linear op-erators in X. We denote by C([,T];X) the Banach space of all continuous functions from [,T] to X with the norm u =sup{u(t),t∈ [,T]} and by L([,T];X) the Banach space of all X-valued Bochner integrable functions defined on [,T] with the norm u=
T
u(t)dt. LetPC([,T];X) ={u: [,T]→X:u(t) is continuous att= tiand left continuous att=tiand the right limitu(t+i) exists fori= , , . . . ,q}. It is easy to
check thatPC([,T];X) is a Banach space with the normuPC=sup{u(t),t∈[,T]}
andC([,T];X)⊆PC([,T];X)⊆L([,T];X).
We introduce the Hausdorff measure of noncompactnessαdefined on each bounded subsetofEby
α() =inf
R> ; there are finite pointsx,x, . . . ,xn∈Ewith⊂ n
i=
B(xi,R) .
The mapQ:D⊂E→Eis said to beα-contraction if there exists a positive constant k< such thatα(QD)≤kα(D) for any bounded closed subsetD⊂E, whereEis a Banach space.
The following fixed point theorem plays a key role in the proof of our main results.
Lemma .([]: Darbo-Sadovskii) If D⊂E is bounded closed and convex,the continuous map Q:D→D is anα-contraction,then the map Q has at least one fixed point in D.
LetY be another separable reflexive Banach space where controlsutake values. De-notedPf(Y) by a class of nonempty closed and convex subsets ofY. We suppose that the
multivalued mapw: [,T]→Pf(Y) is measurable,w(·)⊂E, whereEis a bounded set ofY,
and the admissible control setUad=Spw={u∈Lp(E)|u(t)∈w(t), a.e.},p> . ThenUad=∅,
Throughout this work, we suppose that
(A) The linear operatorA:D(A)⊆X→Xgenerates a compactC-semigroup
{S(t) :t≥}. Hence, there exists a positive numberMsuch that
M=sup≤t≤TS(t).
For use in the sequel, we introduce the following definition.
Definition . A functionx∈PC([,T];X) is said to be a PC-mild solution of the nonlo-cal problem (.), if it satisfies
x(t) =S(t)x–g(x)
+
t
S(t–s)fs,x(s)+B(s)u(s)ds+ <ti<t
S(t–ti)Ji
x(ti)
,
≤t≤T.
In addition, letrbe a finite positive constant, and setBr:={x∈X:x ≤r}andWr:= {y∈PC([,T];X) :y(t)∈Br,∀t∈[,T]}.
3 Controlled impulsive differential equations
In this section, we consider the following controlled nonlocal impulsive problem:
⎧ ⎪ ⎨ ⎪ ⎩
x(t) =Ax(t) +f(t,x(t)) +B(t)u(t), t∈[,T],t=ti,
x() +g(x) =x, u∈Uad,
x(ti) =Ji(x(ti)), i= , , . . . ,q, <t<t<· · ·<tq<T.
(.)
First, we give the following hypotheses:
(F) () f : [,T]×X→Xis a Carathéodory function,i.e., for allx∈X,
f(·,x) : [,T]→Xis measurable and for a.e.t∈[,T],f(t,·) :X→Xis continuous.
() For finite positive constantr> , there exists a functionϕr∈L(,T;R)such that
f(t,x)≤ϕr(t)
for a.e.t∈[,T]andx∈Br.
(B) B: [,T]→L(Y,X)is essentially bounded,i.e.,B∈L∞([,T],L(Y,X)). (G) g:PC([,T];X)→Xis a continuous and compact operator.
(J) Ji:X→Xsatisfies the following Lipschitz condition:
Ji(x) –Ji(x)≤hix–x, i= , , . . . ,q,
for some constantshi> andx,x∈PC([,T];X).
(R) M{x+supx∈Wrg(x)+BuL+
≤i≤q[Ji()+hir] +ϕrL} ≤r.
Remark . From the assumption (B) and the definition ofUad, it is also easy to verify that
Bu∈Lp([,T];X) withp> for allu∈U
ad. Therefore,Bu∈L([,T];X) andBuL< +∞.
Theorem . Assume that there exists a constant r> such that the conditions(A), (F), (B), (G), (J),and(R)are satisfied.Then the nonlocal problem(.)has at least one PC-mild solution on[,T].
In order to obtain the existence of solutions for controlled impulsive evolution equations and optimal controls, we need the following important lemma, which can easily be proved (refer to Lemma . and Corollary . in Chapter of []).
Lemma . Suppose S(t)is a compact C-semigroup on Banach space X and the operator B satisfies the condition(B).If p> and define
(u)(·) =
Proof of Theorem. The proof is divided into the following three steps. Step. Letx∈Xbe fixed. We define a mappingHonPC([,T];X) by
≤t≤T. It follows from the properties of theC-semigroup and all assumptions in The-orem . thatHis well defined and mapsPC([,T];X) into itself. Furthermore, one can also easily see thatx∈PC([,T]) is a PC-mild solution of controlled nonlocal impulsive differential equation (.) if and only ifxis a fixed point ofH. Therefore, we shall prove thatHhas a fixed point inPC([,T]).
Step. We show thatHmapsWrinto itself. By conditions (F), (B), (G), (J), one obtains,
Step. In the sequel, we show thatHis continuous inWr. Lettingx,x∈Wr, we have
This, together with (F)(), (G), (J), shows thatHis continuous. Step. Now we prove thatHhas at least one fixed point. Set
x(t) =S(t)
α(Wr) = . Similarly, it follows from (A), Arzela-Ascoli’s theorem, and Lemma . that
the mappingis also compact inWr. Therefore,α(Wr) = .
This shows thatHis anα-contraction. Now by Lemma . we immediately deduce that the mappingHhas a fixed point inWr,i.e., (.) has at least one mild solution. This completes
the proof.
However, they need the locally Lipschitz continuity off. Here, without the Lipschitz as-sumption off, we make full use of the technique of the compactness as regards the solu-tion operator to obtain the solvability of the controlled nonlocal impulsive equasolu-tion (.). Therefore, our results improve those in [, ] and the references therein, and they have broader applications.
Remark . The uniqueness of solution of the controlled impulsive differential equation (.) cannot be obtained. Therefore, we can denote bySol(u) all solutions of system (.) inWr, for anyu∈Uad.
Assume that
(G) g(x) =g(s, . . . ,sm,x(s), . . . ,x(sm)) =
m
j=cjx(sj), wherecj,j= , , . . . ,m, are given con-stants, and <s<s<· · ·<sm<T.
Corollary . Let conditions(A), (F), (B), (G),and(J)be satisfied.Then the nonlocal im-pulsive equation(.)has at least one PC-mild solution on[,T]provided that
M
x+
m
j=
|cj|r+BuL+
≤i≤q
Ji()+hir
+ϕrL ≤r.
Proof It is easy to see that ifg(x) =g(s, . . . ,sm,x(s), . . . ,x(sm)) =
m
j=cjx(sj), then the
con-dition (G) holds. Thus all the concon-ditions in Theorem . are satisfied. Then the nonlocal impulsive equation (.) has at least one PC-mild solution on [,T]. This completes the
proof.
4 Existence of optimal controls
In the section, we give the existence of optimal controls for system (.).
Letxu∈Wr denote the PC-mild solution of system (.) corresponding to the control
u∈Uad, we consider the following limited Lagrange problem (P):
Findx∈W
r⊆PC([,T];X) andu∈Uadsuch that
Jx,u≤Jxu,u, for allu∈Uad,
where
Jxu,u=
T
lt,xu(t),u(t)dt
andx∈W
rdenotes the PC-mild solution of system (.) corresponding to the control
u∈U
ad.
We make the following assumption:
(L) () The functionl: [,T]×X×Y→R∪ ∞is Borel measurable;
() l(t,·,·)is sequentially lower semicontinuous onX×Yfor a.e.t∈[,T]; () l(t,x,·)is convex onYfor eachx∈Xand a.e.t∈[,T];
() there are two constantsc≥,d> andφ∈L([,T];R)such that
Remark . A pair (x(·),u(·)) is said to feasible if it satisfies system (.) forx(·)∈Wr.
Remark . If (xu,u) is a feasible pair, thenxu∈Sol(u)⊂W r.
Theorem . Assume that condition(L)is satisfied.Under the conditions of Theorem., the problem(P)has at least one optimal feasible pair.
Proof The proof is divided into the following four steps. Step. For anyu∈Uad, set
J(u) = inf
xu∈Sol(u)J
xu,u.
If there are finite elements inSol(u), there exists somexu∈Sol(u), such thatJ(xu,u) = infxu∈Sol(u)J(xu,u) =J(u).
If there are infinite elements in Sol(u), there is nothing to prove in the case ofJ(u) = infxu∈Sol(u)J(xu,u) = +∞.
We assume thatJ(u) =infxu∈Sol(u)J(xu,u) < +∞. By assumption (L), one hasJ(u) > –∞. By the definition of the infimum there exists a sequence {xu
n}∞n= ⊆Sol(u), such that J(xu
n,u)→J(u) asn→ ∞.
Since{(xu
n,u)}∞n=is a sequence of feasible pairs, we have
xun(t) =S(t)x–g
xun+
t
S(t–s)fs,xun(s)+B(s)u(s)ds
+
<ti<t
S(t–ti)Ji
xun(ti)
, ≤t≤T. (.)
Step . Now we will prove that there exists some xu ∈ Sol(u) such that J(xu,u) = infxu∈Sol(u)J(xu,u) =J(u). To show it, we first prove that{xu
n}∞n=is precompact inPC([,T]; X) for eachu∈Uad. For this purpose, note that
xun=xun+xun+xun.
From Step in the proof of Theorem ., we know that{xun}∞n=and{xun}∞n=are both precompact subsets ofPC([,T];X). Moreover, by condition (J), we see that is Lips-chitz continuous inPC([,T];X) with Lipschitz constantM(qi=hi). Thus, according to
the properties of the Hausdorff measure of noncompactness, we conclude that
βxun∞n=≤β
xun∞n=+β
xun∞n=+β
xun∞n=
≤M
q
i= hi
βxun∞n=.
Since the condition (R) holds, M(qi=hi) < . Thus the above inequality implies that
β({xu
n}∞n=) = . Consequently, the set{xun}∞n=is precompact inPC([,T];X) foru∈Uad.
Without loss of generality, we may suppose thatxu
n→xu, asn→ ∞inPC([,T];X) for
g,Ji, andf with respect to the second argument, and using the dominated convergence and using the assumption [L] and Balder’s theorem, we have
J(u) = lim flexive Banach space. Thus there exists a subsequence, and without loss of generality we may suppose that{un}∞n=converges weakly to someu∈Lp([,T];Y) asn→ ∞.
Note thatUadis closed and convex, so it follows from the Mazur lemma thatu∈Uad.
Forn≥,xun is the mild solution for (.) corresponding tou
n, whereJ(un) attains its
minimum. Then (xun,u
n) is a feasible pair and satisfies the following integral equation:
Similar to Step in the proof of Theorem ., we deduce that β({xun}∞
n=) = , i.e.,
{xun}∞
n= ⊆PC([,T];X) is precompact. Thus there exists a subsequence, relabeled as
{xun}∞
Remark . Constructing approximating minimizing sequences of functions twice plays a key role in the proof of looking for optimal controls, which enable us to deal with the multiple solution problem of feasible pairs. More importantly, this will allow us to study more extensive and complex evolution equations and optimal controls problems. More-over, we have the following consequences.
Corollary . Assume that conditions(A), (F), (B), (G), (J),and(L)are satisfied.Then the problem(P)has at least one optimal feasible pair on[,T]provided that
M
Proof If condition (G) holds, then we conclude to the existence of feasible pairs on [,T] from Corollary .. Similar to the proof of Theorem ., we can obtain the optimal feasible
pair on [,T]. This completes the proof.
Corollary . Assume that conditions(A), (F), (B), (J),and(L)are satisfied and g(x) =x. Then the problem(P)has at least one optimal control on[,T]provided that
M
We use the following assumptions instead of (J) and (R):
(J) Ji:X→X,i= , , . . . ,q, are continuous and compact mappings. (R) M{x+supx∈Wrg(x)+BuL+supx∈Wr
q
i=Ji(x(ti))+ϕrL} ≤r.
We may apply Schauder’s second fixed point theorem to obtain the existence of PC-mild solutions. Note thatHis a continuous mapping fromWrtoWr. We need to prove
thatHis a compact mapping. In fact, we already proved thatandare both compact operators in Theorem .. The same idea can be used to prove the compactness ofdue to the assumption (J) and the Ascoli-Arzela theorem. The rest of the proof is similar to that of Theorem .. So we can obtain the following result.
Corollary . Let(A), (F), (B), (G), (J), (R),and(L)be satisfied.Then the problem(P)has at least one optimal control on[,T].
5 An example
In this section, we shall give one example to illustrate our theory.
Example . Consider the following semilinear partial differential system:
⎧
Further, we assume thatωsatisfies:
Now we assume that:
() f: [,T]×X→Xis a continuous function defined by
f(t,z)(y) =Ft,z(y), ≤t≤T, ≤y≤.
Moreover, for givenr> , there exists an integrable functionφr: [,T]→Rsuch thatf(t,z) ≤φr(t)fort∈[,T],z∈Br.
() g:PC([,T];X)→Xis defined by
g(x)(y) =
T
ωt,y,ξ,x(t,ξ)dt dξ, y∈[, ].
From Theorem . in [], we directly see thatgis well defined and it is a continuous and compact map by the above conditions (i) and (ii) about the functionω.
() Ji:X→Xis a continuous function for eachi= , , . . . ,q, defined by
Ji(x)(y) =Ji
x(y).
Here we takeJi(x(y)) = (αi|x(y)|+ti)–,αi> ,i= , , . . . ,q, <t<t<· · ·<tq<T,
y∈[, ]. ThenJiis Lipschitz continuous with constanthi=αi/ti,i= , , . . . ,q,i.e., the assumption (J) is satisfied.
Let us observe that the problem (.) may be reformulated as
⎧ ⎪ ⎨ ⎪ ⎩
x(t) =Ax(t) +f(t,x(t)) +B(t)u(t), t∈[,T],t=ti,
x() =g(x),
x(ti) =Ji(x(ti)), i= , , . . . ,q,
(.)
with the cost function
Jxu,u=
T
xu(t)X+u(t)Ydt,
where (xu,u) is a feasible pair. If the inequality
M
sup
x∈Wr
g(x)+uL+
≤i≤q
Ji()+r·αi/ti
+φrL
≤r
holds, (.) satisfies all the assumptions given in our former theorems. Therefore our re-sults can be used to deal with (.).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
The authors are grateful to the editor and anonymous reviewers for their valuable comments and suggestions. Moreover, this research was supported by the Natural Science Foundation of China (11201410) and the Natural Science Foundation of Jiangsu Province (BK2012260 and BK20141271).
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