R E S E A R C H
Open Access
Stability analysis for optimal control problems
governed by semilinear evolution equation
Hongyong Deng
1and Wei Wei
1,2**Correspondence:
wwei@gzu.edu.cn
1Department of Mathematics,
Guizhou University, Guiyang, 550025, P.R. China
2Department of Mathematics,
Guizhou Minzu University, Guiyang, 550025, P.R. China
Abstract
In this paper, the stability of solutions of optimal control for the distributed parameter system governed by a semilinear evolution equation with compact control set in the spaceL1(0,T;E) is discussed. The stability results for optimal control problems with respect to the right-hand side functions are obtained by the theory of set-valued mapping and the definition of essential solutions for optimal control problems. Keywords: optimal control; stability; semilinear evolution equation;C0-semigroup; set-valued mapping
1 Introduction
The stability analysis of systems governed by differential equations in modern mathemat-ics is very important for practical applications (see [, ]), especially in numerical compu-tation (see [, ]).
In recent years, there has been growing interest in stability analysis of optimal control problems for the ODE system or PDE system (see [, ]). But most of the results are based on the existence and uniqueness of an optimal control for problems, there are only a small number of articles discussing a stability analysis for optimal control problems without the uniqueness in view of set-valued analysis. In recent years, Yuet al.discussed the stabil-ity of optimal controls with respect to the right-hand side function based on set-valued mapping (see []). But all results are established under the control admissible setU[,T] assumed as the compact set ofC([,T];Rm) in the optimal control problem for an ODE
system. Moreover, we have done some work concerning the control admissible setU[,T] assumed as the compact set ofL(,T;Rm) in order to include many cases of practical sit-uations (see []). Therefore, it is natural for us to ask whether these results are still valid for infinite dimensional controlled systems.
This paper studies the existence and stability properties of solutions of optimal control problems governed by a semilinear evolution equation, and it is stated as follows.
Problem(P): GivenT> ,yˆ(·)∈L(,T;X),y
∈X, andyT∈X, find an optimal control ¯
u∈U[,T] such that
J(u¯)≤J(u) for allu∈U[,T], ()
where
J(u) =y(T) –yTX+
T
y(t) –yˆ(t)Xdt ()
subject to
˙
y(t) =Ay(t) +f(t,y(t),u(t)), t∈[,T], y() =y,
()
whereAis infinitesimal generator ofC-semigroup on a Banach spaceX, the set of
ad-missible controls in the spaceL(,T;E) is defined by
U[,T] =u|u(t) is measurable w.r.t.t,u(t)∈U⊂E, ()
whereEis another separable reflexive Banach space from which the controlsutake values. The paper is organized as follows. In Section , we give some properties ofC-semigroup
and some results as regards compact sets. In Section , the existence of an optimal control is obtained. In Section , for the reader’s convenience, the set-valued mapping theory is recalled, then we show stability results for the optimal control problem in the sense of the Baire category. In the last section, some examples demonstrate the applicability of our results.
2 Preliminaries
Throughout the paper, constants ≤p< +∞andT> are given. LetXbe a Banach space andeAttheC
-semigroup generated by a linear operatorA. In this section, we recall some
related results about semigroups and compact sets from [, ] and [] which will be used in this paper.
Proposition .([]) There exist constantsω≥and M≥such that
eAt≤Meωt for≤t< +∞.
DenoteLp(,T;X) the space of measurable functions from [,T] intoXequipped with
the norm
yLp(,T;X)=
T
y(t)pXdt
p ,
andC([,T];X) the space of continuous functions from [,T] intoXequipped with the norm
yC([,T];X)= sup
t∈[,T]
y(t)X.
Lemma . ([]) A set V is called totally bounded if for every > ,there exist some
δ> ,a metric space W,and a mapping:V →W such that(V)is totally bounded, and d(x,y) <if d((x),(y)) <δwhenever x,y∈V.
Denote (τhu)(t) =u(t+h) forh> , the compactness ofLp(,T;X) is recalled as follows.
() The set{ t
t u(t)dt|u∈S}is relatively compact inX,for any <t<t<T.
() The limitτhu–uLp(,T–h;X)→ash→,uniformly foru∈S.
Now we consider the compactness of the admissible set ().
Proposition . Suppose that U⊆E is compact andτhu–uL(,T–h;E)→as h→
uniformly for u∈U[,T].ThenU[,T]is compact in L(,T;E).
Proof It follows from the compactness ofU⊆Ethat there exists a constantM> such that
Banach theorem, there exists a bounded linear functionalP∈E∗such that
P= , P(u) =uE.
By the continuity and linearity ofP, we have
If{un} ⊆U[,T] andun→uinL(,T;E), namely,
T
un(t) –u(t)Edt→,
thenu(t)E is measurable in [,T] and there exists a subsequence of{un}, denoted by
{unk}, such that
unk(t) –u(t)E→ a.e.t∈[,T] ask→+∞,
i.e.,unk(t) –u(t)E→ for allt∈A, whereAis some subset of [,T] andμ([,T]\A) = . SinceUis compact, for any fixedt∈Aandunk(t)∈U, there existsu(t)∈Usuch that
unk(t)→u(t) inEask→+∞.
Without loss of generality, we define
ˆ
u(t) = ⎧ ⎨ ⎩
u(t), t∈A,
u(t), t∈[,T]\A,
()
wheretcan be chosen by any point ofA, thenu(t) =uˆ(t) a.e.t∈[,T]. It is clear that
ˆ
u(t)∈Ufor allt∈[,T].uanduˆ are equivalent inL(,T;E). Namely,
unk→ ˆu inL
(,T;E) ask→+∞,
thenU[,T] is closed. From the above, we complete this proposition. We give an example of the compact set inL(,T;E) in the following.
Example . Letu∈U[,T]⊂L(,T;W,p()). For any fixedx∈,u(t,x) is a
piece-wise continuous with respect totcontaining only a finite number of discontinuous points. S(x) ={u(t,x)|t∈[,T]},S(x) is a bounded closed set inW,p(), whereis a bounded
open subset ofRnand∂isC. ThenU[,T] is compact inL(,T;Lp()).
Proof From the compact embedding theorem of [], that is,W,p() is compactly
em-bedded inLp(), writtenW,p()⊂⊂Lp(), sinceS(x) is a bounded closed set inW,p(),
thenS(x) is compact inLp().
Suppose thatD={t, . . . ,tN} ⊂[,T] andt= <t<t<· · ·<tN<T=tN+. We define
˜
u(t,x) = ⎧ ⎨ ⎩
u(t,x), t∈[,T]\{t, . . . ,tN},x∈,
u(ti– ,x), t=ti,i= , , , . . . ,N,x∈.
Without loss of generality, we may assumeh> , then
τhu–uL(,T–h;Lp())
= T–h
=
3 Existence of optimal control
In order to study the optimal control problem (P), we assume that:
Definition . For givenu∈U[,T], a functiony: [,T]→Xis called a mild solution of () ify∈C([,T];X) satisfies
y(t) =eAty
+
t
eA(t–s)fs,y(s),u(s)ds, t∈[,T].
This solution is denoted byy(·,u(·)).
From Proposition . of [], we have the following theorem.
Theorem . Suppose assumptions(Hf), (HA),and(Hu)hold.Then,for any u∈U[,T],
the Cauchy problem()has a unique mild solution y∈C([,T];X).
Moreover, we also have the following result.
Theorem . Suppose assumptions(Hf), (HA),and(Hu)hold.y∈C([,T];X)is the mild
solution of system(), then the map u(·)→y(·,u(·))is continuous from L(,T;E)into
C([,T];X).
Proof Supposeyk(t) andy(t) are mild solutions of the system () with respect toukandu,
respectively. Then
yk(t) =eAty+
t
eA(t–s)fs,yk(s),uk(s)
ds
and
y(t) =eAty+
t
eA(t–s)fs,y(s),u(s)ds.
Since
uk→u inL(,T;E) ask→+∞,
by the Chebyshev inequality of Theorem .. in [], we have
uk(t) –u(t)E→ for almost allt∈[,T] ask→+∞,
and the functionf is continuous w.r.t.u, it follows from Corollary .. of [] that
f(t,y,uk) –f(t,y,u)X→ for almost allt∈[,T] ask→+∞,
and combining the Riesz Theorem ..(i) and Theorem .. of [], then for any subse-quence{uk}we can extract a further subsequence{uk}such that
uk(t) –u(t)E→ a.e.t∈[,T],
then
ft,y(t),uk(t)
Also from Proposition ., there exist constantsω≥ andM≥ such that
eAt≤Meωt for ≤t≤T. ()
Due to assumption (Hf) and the Lebesgue dominated convergence theorem, it follows that
lim
k→∞
T
MeωTfs,y(s),u k(s)
–fs,y(s),u(s)Xds= . ()
Since the subsequence of{uk}is arbitrary, we can conclude that
lim
k→∞
T
MeωTfs,y(s),uk(s)
–fs,y(s),u(s)Xds= , ()
namely, for any> , there existsN> such that
T
MeωTfs,y(s),uk(s)
–fs,y(s),u(s)Xds≤ ()
holds wheneverk≥N. So
yk(t) –y(t)X =
teA(t–s)fs,yk(s),uk(s)
–fs,y(s),u(s)ds
X
≤
t
MeωTfs,yk(s),uk(s)
–fs,y(s),uk(s)Xds
+ T
MeωTfs,y(s),uk(s)
–fs,y(s),u(s)Xds
≤MeωT t
L(t)yk(t) –y(t)Xds+, ()
and using the Gronwall inequality, we obtain
yk(t) –y(t)X≤eMe ωT t
L(s)ds≤eMTeωT TL(s)ds, ∀t∈[,T]. ()
That is,y(·,uk)→y(·,u)∈C([,T];X).
From Proposition ., we have the following lemma.
Lemma . If assumption(Hu)holds,U[,T]is compact in L(,T;E).
Now, we discuss the existence of an optimal control for problem (P).
Theorem . Suppose assumptions(Hu), (HA),and(Hf)hold,then problem(P)admits at
least one optimal control.
Proof Assume that there exists a minimizing sequence of{uk} ⊆U[,T] such that
lim
SinceU[,T] is compact, there exists a subsequence{uk}of{uk}andu¯∈U[,T] such
that
uk→ ¯u inL(,T;E) ask→+∞. ()
From Theorem . we have
yuk(·)→yu¯(·) inC
[,T];Xask→+∞, ()
whereyuk(·,·) andyu¯(·,·) are solutions of () with respect toukandu¯, respectively. Hence
yuk(T)→yu¯(T) inXask→+∞
and
yuk(T) –yTX→yu¯(T) –yTX inRask→+∞,
then
yuk(T) –yTX→yu¯(T) –yTX inRask→+∞.
Thanks to (), there exists a subsequence of{uk}, denoted by{uk}, such that
uk(t) –u¯(t)E→ for almost allt∈[,T]. ()
It follows from () and () that
yuk(t) –yˆ(t)X→y¯u(t) –yˆ(t)X a.e.t∈[,T].
Sinceuk(t)∈U(compact set inE) andyk(·)∈C([,T];X), the image of the compact set
underyis compact (see Theorem .- of []). For allk, there exists a positive constant M, such that
yuk(t) –yˆ(t)X≤M. ()
From the Lebesgue dominated convergence theorem (see Theorem .. of []), we have
lim
k→+∞ T
yuk(t) –yˆ(t)
Xdt=
T
y¯u(t) –ˆy(t)
Xdt. ()
From all the above, we have
J(u¯) =yu¯(T) –yT
X+
T
yu¯(t) –yˆ(t)
Xdt
= lim
k→+∞yuk(T) –yT
X+klim→+∞
T
yuk(t) –ˆy(t)
Xdt
= lim
k→+∞J(uk
) = inf
u∈U[,T]J(u).
4 Stability analysis of optimal control problems
In this section, we will use the set-valued mapping theory to study the stability of the optimal control. At first, we recall some definitions and a lemma on set-valued mappings for convenience of the reader (see [, , ]). LetWandZbe metric spaces.
Definition . A set-valued mapping F:W →Z is called upper (respectively, lower)
semicontinuous atx∈Wif and only if for each open setGinZwithG⊃F(x) (respectively, G∩F(x)=∅), there existsδ> such thatG⊃F(x) (respectively,G∩F(x)=∅) for any x∈Wwithρ(x,x) <δ. It is said to be upper (respectively, lower) semicontinuous inW if and only if it is upper (respectively, lower) semicontinuous at any point ofW.
A set-valued mappingF:W→Zis continuous atxif it is both upper semicontinuous
and lower semicontinuous atx, and that it is continuous if and only if it is continuous at every point ofW.
Definition . A set-valued mappingF:W→Z is called compact upper
semicontinu-ous (called USCO) ifF(x) is nonempty compact, for eachx∈W, andFis upper semicon-tinuous.
Definition . A set-valued mappingF:W→Zis called closed ifGraph(F) is closed,
whereGraph(F) ={(x,z)∈W×Z|z∈F(x)}is the graph ofF.
Lemma . If the set-valued mapping F:W→Zis closed and Z is compact,then F is an
USCO mapping.
Definition . A subsetQ⊂Wis called a residual set if it contains a countable intersec-tion of open dense subsets ofW.
IfWis a complete metric space, any residual subset ofWmust be dense inW.
Lemma . Let W be a complete metric space,and F:W →Z be an USCO mapping,
then there exists a dense residual subset Q of W such that F is lower semicontinuous at each x∈Q.
Now, we consider the stability of optimal controls. Denote
Y=f |f satisfies conditions of (Hf)
.
For everyf,f∈Y, we define
ρ(f,f) = sup (t,y,u)∈[,T]×X×U
f(t,y,u) –f(t,y,u)X.
Then the space (Y,ρ) is a complete metric space []. Let
S(f) =u¯| ¯uis the optimal control of problem (P) associated withf ∈Y.
Then the correspondencef →S(f) is a set-valued mappingS:Y→U[,T].
Theorem . If assumptions(Hu), (HA),and(Hf)hold,S(f)=∅for each f ∈Y.
The following proposition is important in studying the stability of optimal controls.
Proposition . Let{fk}be any sequence of Y such that fk→f in Y and{uk}any sequence
ofU[,T]such that uk→u in L(,T;E),then yfk(·,uk(·))→yf(·,u(·))in C([,T];X)as k→+∞.
Proof By Definition . of a mild solution of Cauchy problem () we have
y(t)yf
wheneverk≥N. The inequality () implies that
holds for anyk≥N, and it follows from the Gronwall inequality that
yk(t) –y(t)X≤eMe ωT t
L(s)ds≤eMTeωT TL(s)ds, ∀t∈[,T]. ()
This impliesyk→yinC([,T];X), which completes the proof of the theorem.
We denote
Jfk(uk) =yk(T) –yT
X+
T
yk(t) –ˆy(t)Xdt.
One can easily obtain the following proposition from Proposition ..
Proposition . Let{fk}be any sequence of Y such that fk→f in Y and{uk}any sequence
ofU[,T]such that uk→u in L(,T;E).Then Jfk(uk)→Jf(u)as k→+∞.
Theorem . Suppose that(Hu), (HA),and(Hf)hold.Then S:Y→U[,T] is an USCO
mapping.
Proof By the compactness ofU[,T] and Lemma ., we only need to show that Graph(S) =(f,u)∈Y×U[,T]|u∈S(f)
is closed. Let{fk} ⊂Y withfk→f inY and{¯uk} ⊂S(fk) withu¯k→ ¯uinL(,T;E). Now
we need to show thatu¯∈S(f). Due tou¯k∈S(fk), we have
Jfk(u¯k)≤Jfk(u) for allu∈U[,T]. () Proposition . yields
Jfk(u¯k)→Jf(u¯) ()
and
Jfk(u)→Jf(u) for allu∈U[,T]. () Combining (), (), and (), we have
Jf(u¯)≤Jf(u) for allu∈U[,T], ()
namely,u¯∈S(f). The proof is completed.
We introduce the following definition for considering the stability of solutions of an optimal control problem.
optimal control problem (P) associated with f is called essential iff its solutions are all essential.
From [], we can obtain the following theorem.
Theorem . The optimal control problem(P)associated with f is essential if and only if S:Y→U[,T]is lower semicontinuous at f∈Y.
Now we need to consider thatSis lower semicontinuous inY. From Lemma . and Theorem ., we have the following lemma.
Lemma . There exists a dense residual subset Q⊂Y such that S is lower semicontinuous at each f ∈Q,namely,S is continuous at each f ∈Q.
SinceSis continuous at eachf ∈QandY is a complete metric space,Qis a second category []. Lemma . and Theorem . yield the generic stability in the sense of the Baire category.
Theorem . There exists a dense residual subset Q of Y such that for any f ∈Q,S(f) is stable in the sense of Hausdorff metric and Jf is robust with respect to f ∈Q.So every
optimal control problem associated f ∈Y can be closely approximated arbitrarily by an essential optimal control problem.
We need to note that when the solutionS(f) is a singleton, the result also holds. By Theorem . we can also see that anyf ∈Y can be closely approximated by an essential optimal control problem.
Remark . In this paper, we only discuss the stability results as regards the optimal con-trol problem with quadratic cost functional. In fact, the results also hold for the optimal control problem with general cost functional such as Bolza problems under some assump-tions.
5 Example
Our main result can be applied to the controlled systems of heat equations and wave equa-tions with theC-semigroup. We will state optimal control problems with parabolic
con-trolled systems and hyperbolic concon-trolled systems, respectively, in the following.
Problem(P): GivenT> ,yˆ(t,x), andyT(x), find an optimal controlu¯ ∈U[,T] such
that
J(u¯)≤J(u) for allu∈U[,T], () where
J(u) =
y(T,x) –yT(x)dx+
T
y(t,x) –yˆ(t,x)dx dt ()
subject to ⎧ ⎪ ⎨ ⎪ ⎩
∂y
∂t(t,x) =y(t,x) +f(t,y(t,x),u(t,x)), t∈[,T],x∈,
y(,x) =y(x), x∈,
y|∂= ,
where⊂Rnis a bounded domain with a smooth boundary∂, letX=L() andA=
C-semigroupeAton the infinite dimensional Banach spaceX. Suppose thaty∈Xand
the set of admissible controls in the spaceL(,T;E) is defined by () whereE=L(). With the assumptions (Hu) and (Hf) holding, the stability results are valid.
Problem(P): GivenT> ,yˆ(t,x), andyT(x), find an optimal controlu¯∈U[,T] such
ThenAis the infinitesimal generator of aC-semigroupeAt on the infinite dimensional
Banach spaceX, the set of admissible controls in the spaceL(,T;E) is defined by ()
where E=L(). With the assumptions (H
u) and (Hf) holding, the stability results are
also valid.
The following example is provided to show that not all optimal control problems are essential.
subject to the distributed parameter system
⎧
generate aC-semigroupeAt. The mild solution can be given by
v(t,x) =eAtv+
t
Let
Similarly to Example .,U[, ] is compact. In order to simplify the calculation, letv= .
() Letf(t,v(t,x),u(t,x)) =u(t,x)∈U[, ], we see that theAgenerate aC-semigroup
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 11261011).
Received: 15 October 2014 Accepted: 16 March 2015
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