R E S E A R C H
Open Access
Some optimal control problems of heat
equations with weighted controls
Shufang Liu
*, Dandan Liu and Guoqi Wang
*Correspondence:
[email protected] School of Mathematics and Statistics, Central South University, Changsha, 410075, P.R. China
Abstract
In this paper, the time and norm optimal control problems of controlled heat equations with a weight function are considered. For the time optimal problems, we study the following two cases: one is for equations with multi-domain control under null controllability, and the other is for equations under approximate null
controllability. We prove the solvability, and obtain the bang-bang principle of the time optimal controls for aforementioned both cases. For the norm optimal control problems, we focus on equations with multi-time and multi-domain control, and present the solvability of these problems.
MSC: 35K05; 49J20
Keywords: heat equation; weight function; optimal control problem; multi-domain
1 Introduction
LetT be a positive number andbe an open bounded domain with smooth boundary inRN,N≥. LetK∈Z+,{E
i} ≡ {Ei}Ki=be a sequence of Lebesgue measurable subsets
of (,T) and{ωi} ≡ {ωi}Ki=be a sequence of positive Lebesgue measurable subsets of
withωi∩ωj=∅, for alli,j∈ {, , . . . ,K}andi=j. Denote byχEi,χωi the characteristic
function ofEi,ωi, respectively, for eachi∈ {, , . . . ,K}. Consider the following controlled
heat equation with a weight function:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂ty(x,t) –y(x,t) =ρ(x)
K
i=χEi(t)χωi(x)ui(x,t), in×(,T),
y(x,t) = , on∂×(,T),
y(x, ) =y(x), in,
(.)
where ρ∈L() is a weight function satisfying <ρ(x)≤ for a.e. x∈, and =
y∈L() is a given function. We denote the solution to (.) by y(·,·;{χEiχωiui},y).
For simplicity, when Ei = (,T) for all i ∈ {, , . . . ,K}, we write y(·,·;{χωiui},y) for
y(·,·;{χEiχωiui},y); furthermore, whenK= , writeω,y(·,·;χωu,y) forωi,y(·,·;{χωiui},
y), respectively.
The weight functionρ in equation (.) is meaningful, which stands for the different influence of the control function in different location.
As is well known, optimization is one of the most important problems in control theory and there exist some work on this topic (see,e.g., [–]). Roughly speaking, the goal of
timization is to improve a variable in order to maximize a benefit (or minimize a cost). The time and norm optimal control problems are important and interesting branches of opti-mization. For the deterministic systems, the reader can refer [] to obtain recent results and find open problems. The reader can also refer [–] for controlled heat equations. For the stochastic ones, the norm optimal control problems were considered in [, ] for controlled stochastic ordinary differential equations, and in [] for controlled stochastic heat equations.
In this paper, we shall consider the time and norm optimal control problems of heat equations with a weight function. In Section , we consider two kind time optimal control problems: one is for equations with multi-domain control under null controllability, and the other is for equations under approximate null controllability. We obtain the bang-bang principle of the time optimal controls for these two problems. In Section , we consider the norm optimal problems with multi-time and multi-domain control, and we obtain the solvability of these problems.
2 Time optimal control problems
In this section, we first state two time optimal control problems, and then study the solv-ability of these problems, obtain the bang-bang property of the time optimal controls. Throughout this section, for alli∈ {, , . . . ,K},Ei= (,T), and
ui∈Uadi ≡
u∈L∞, +∞;L() |u(t)L()≤Mia.e.t∈(, +∞)
. (.)
WhenK= , for simplicity, we writeUadforUadi .
In the following, we consider the following two time optimal control problems subject to (.):
Problem (TP)
T∗=infT|y·,T;{χωiui},y = ,ui∈U i
adfor alli∈ {, , . . . ,K}
.
Problem (TP) ForK= ,
Tε∗=infT|y(·,T;χωu,y)∈ ¯B(,ε),u∈Uad
.
Here and in what follows, we denote byB(u,r) the open ball inL() with centeru∈L()
and radiusr> , and byB(u,¯ r) the closed ball inL() with centeru∈L() and radius
r> .
In order to obtain the solvability of Problem (TP), we assume that there exists a constant M> such that
Mi≤M for alli∈ {, , . . . ,K}. (.)
Notice that the hypothesis (.) is reasonable: for a single control system
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂ty(x,t) –y(x,t) =ρ(x)χω(x)u(x,t), in×(, +∞),
its optimal time
T∗≡infT|y(·,T;χωu,y) = ,u∈Uad
→
asM→ ∞.
It is obvious that Problem (TP) is related to null controllable problem of (.), while Problem (TP) is related to approximately controllable problem of (.). It is well known that, whenK= andρ≡, the system (.) is null controllable for the measurable control domainω(see []), even if the characteristic functionχωcan be relaxed by a measurable
functionβ∈L() with ≤β ≤ for a.e.x∈and
β(x)dx=α||(see []). Here
α∈(, ) is a given constant and||is the Lebesgue measure of. It is natural that there exist a positive constantTand a controlu∈L∞(,T;L()) such thaty(x,T;u,y
) = (see
[, ]).
The following result is related to the solvability of Problem (TP).
Theorem . Let{Mi}be a given positive real number sequence satisfying(.).Then there
exists T∗> ,such that T∗is the solution to Problem(TP).Moreover,for each i= , , . . . ,K, there exists a unique u∗i ∈L∞(,T∗;L()),such that
u∗iL()=Mi for a.e. t∈
,T∗ with Mi≤M for all i∈ {, , . . . ,K}, (.)
i.e.,the time optimal controls sequence of Problem(TP)has the bang-bang property.
The following lemma is needed in proving Theorem ., which comes from [, ].
Lemma . Let E⊂[,T]andω⊂ be two positive measurable sets. Then,for each y∈L(),there is a bounded control function u(·)∈L∞(,T;L())with
uL∞(,T;L())≤CyL(),
such that the solution to the equation
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂ty(x,t) –y(x,t) =ρ(x)χE(t)χω(x)u(x,t), in×(,T),
y(x,t) = , on∂×(,T), y(x, ) =y(x), in,
satisfies y(·,T;u,y) = .Here C=C(,T,|E|,|ω|)is a constant.
We are now in the position to prove Theorem ..
Proof of Theorem. Since the proof is long, we separate it into two steps. Step . For fixedi∈ {, , . . . ,K}, consider the following system:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
By Lemma . of [], we know that there exist a control ui ∈U
i
ad and T, such that
y(·,T,χωiui,y) = . we also know that =y∈L
() is a given function. Therefore,
<T∗≡infT|y·,T;{χωiui},y = ,u i∈Ui
adfor alli∈ {, , . . . ,K}
≤infT|y(·,T;χωiui,y) = ,ui∈U
i
ad
<∞.
Hence, there exists a sequence{Tn}, such that{Tn}is a monotone decreasing sequence
withy(·,Tn;{χωiu n
i},y) = and
Tn→T∗≡inf
T|y·,T;{χωiui},y = ,ui∈U i
adfor alli∈ {, , . . . ,K}
.
Without loss of generality, we assume that Tn ≤ T∗ + for all n ∈ N. Then yn ≡
y(·,·;{χωiu n
i},y) is a solution to the following equation:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂tyn(x,t) –yn(x,t) =ρ(x)
K
i=χωi(x)u n
i(x,t), in×(,Tn),
yn(x,t) = , on∂×(,T n),
yn(x, ) =y(x),yn(x,Tn) = , in.
Now, denote
˜ uni(x,t) =
⎧ ⎨ ⎩
uni(x,t), (x,t)∈×(,Tn),
, (x,t)∈×[Tn, +∞).
Then
˜ yn(x,t) =
⎧ ⎨ ⎩
yn(x,t), (x,t)∈×(,T n),
, (x,t)∈×[Tn, +∞),
solves the following system:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂ty˜n(x,t) –˜yn(x,t) =ρ(x)
K
i=χωi(x)u˜ n
i(x,t), in×(, +∞),
˜
yn(x,t) = , on∂×(, +∞),
˜
yn(x, ) =y
(x), in,
˜
yn(x,t) = , in×[Tn, +∞).
Moreover, by the definition ofy˜n, it is easy to see thaty˜nsolves the following system:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂ty˜n(x,t) –˜yn(x,t) =ρ(x)Ki=χωi(x)u˜ n
i(x,t), in×(,T∗+ ),
˜
yn(x,t) = , on∂×(,T∗+ ), ˜
yn(x, ) =y
(x), in,
˜
Note that˜unL()≤Mfor alln∈N. Then there exist a subsequence{˜un} ⊂L∞(,T∗+ ;L()) of{˜un
}andu˜ ∈L∞(,T∗+ ;L()) such that
˜ un
→ ˜u weakly∗inL∞
,T∗+ ;L() asn→ ∞.
Similarly, sinceun
L()≤Mfor alln∈N, there exist a subsequence{˜un}of{˜un}and ˜
u
∈L∞(,T∗+ ;L()) such that
˜ un
→ ˜u weakly∗inL∞
,T∗+ ;L() asn→ ∞.
By inductive argument, for eachi∈ {, , . . . ,K}, there exist a subsequence{˜uni i }of{˜u
ni–
i }
andu˜i ∈L∞(,T∗+ ;L()) such that
˜ uni
i → ˜ui weakly∗inL∞
,T∗+ ;L() asni→ ∞.
By the diagram argument, for alli∈ {, , . . . ,K}, we can abstract a subsequence{˜unn i }of
{˜uni}such that
˜ unn
i → ˜ui weakly∗inL∞
,T∗+ ;L() asn→ ∞.
Sinceωi∩ωj=∅for alli,j∈ {, , . . . ,K}withi=j, one can get
ρ(x)
K
i= χωiu˜
nn i →ρ(x)
K
i= χωiu˜
i weakly∗inL∞
,T∗+ ;L() asn→ ∞.
On the other hand,y˜nnis the solution to the following system:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂ty˜nn(x,t) –y˜nn(x,t) =ρ(x)
K
i=χωi(x)u˜ nn
i (x,t), in×(,T∗+ ),
˜
ynn(x,t) = , on∂×(,T∗+ ),
˜
ynn(x, ) =y
(x), in,
˜
ynn(x,t) = , in×[Tnn,T∗+ ).
Then there exist a subsequence of{˜ynn}, still so denoted, andy˜such that
˜
ynn→ ˜y weakly inL×,T∗+ asn→ ∞
and ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂ty˜(x,t) –˜y(x,t) =ρ(x)
K
i=χωi(x)u˜
i(x,t), in×(,T∗+ ),
˜
y(x,t) = , on∂×(,T∗+ ),
˜
y(x, ) =y
(x), in.
Note thaty˜nn= in×[T
nn,T∗+ ). Since forn∈N, one hasy˜nn= in×[Tnn,T∗+ ),
˜
We gety˜= in×[T
nn,T∗+ ). We gety˜= in [T∗,T∗+ ) sinceTnn→T∗asn→ ∞.
Take
ui =u˜i|×(,T∗), and y=y˜|×(,T∗).
By the fact thaty˜∈C((,T];L()),yis the solution to the following system: ⎧
⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂ty(x,t) –y(x,t) =ρ(x)
K
i=χωi(x)u
i(x,t), in×(,T∗),
y(x,t) = , on∂×(,T∗),
y(x, ) =y
(x), y(x,T∗) = , in,
which implies that{u
i}are the desired controls respect to the optimal timeT∗.
Step . In the following, we shall show that the time optimal control of Problem (TP) has the bang-bang property. Otherwise, we suppose that there existi∈ {, , . . . ,K}and
a subsetE⊂[α,T∗–α] with positive measure for someα> and a positive numberε,
such that
u∗i∈Ui
ad and Mi–u
∗
iL()≥ε, for eachtin the setE,
whereu∗iis the time optimal control respect toT∗. It is obvious that the solution to (.) satisfies
y·,T∗;χωiu ∗ i
,y = ,
and, for eacht∈E,B(u∗ i(t),
ε
)⊂B(,Mi).
We denote byetthe semigroup generated bywith the Dirichlet boundary condition.
Set
hδ=
δ
e(δ–σ)ρ K
i= χωiu
∗
i(σ)dσ+
eδ–I y. (.)
Considering the following system:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ zδ
t(x,t) –zδ(x,t) =ρ(x)χEδ(t)χωi(x)wδ(x,t), in×(,T
∗–δ),
z(x,t) = , on∂×(,T∗–δ), z(x, ) = –hδ(x), in,
(.)
whereEδ is the set{t|t+δ∈E}. By Lemma ., there exist positive constantsδ and
L=L(,T,|E|,|ωi|), such that, for eachδwith <δ≤δ, there is a controlwδ in the
spaceL∞(,T∗–δ;L()) with the estimate
wδL∞(,T∗–δ;L())≤LhδL(),
and the solution to (.) satisfies
On the other hand, by (.), there exists a positive numberδsuch that, for each positive
numberδwithδ≤δ, one has
hδL()≤
ε
L.
Therefore, for eachδ≤δ≡min{δ,δ}, there exists a controlwδsatisfying
wδL∞(,T∗–δ;L())≤
ε
, (.)
and the corresponding solution to (.) satisfies (.). Set
vi(x,t) =
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
u∗i(x,δ+t) +wδ(x,t), fori=iand (x,t)∈×Eδ,
u∗i(x,δ+t), fori=iand (x,t)∈×((,T∗–δ) –Eδ),
u∗i(x,δ+t), fori=iand (x,t)∈×(,T∗–δ),
, otherwise.
It is obvious thatvi∈Uadi for alli∈ {, , . . . ,K}. Consider the following system:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
yt(x,t) –y(x,t) =ρ(x)
K
i=χωi(x)vi(x,t), in×(, +∞),
y(x,t) = , on∂×(, +∞), y(x, ) =y(x), in.
(.)
For any <δ<min{T∗,δ}, it is easy to check that the solution to above equation satisfies
y·,T∗–δ;{vi},y =e(T ∗–δ)
y+ρ(x) K
i=
T∗–δ
e(T∗–δ–σ)χ
ωivi(σ)dσ
=e(T∗–δ)y+ K
i=
T∗–δ
e(T∗–δ–σ)ρ(x)χωiu ∗
i(δ+σ)dσ
+ T∗–δ
e(T∗–δ–σ)ρ(x)χ
Eδχωiwδ(σ)dσ
=e(T∗–δ)y+ K
i=
T∗
δ
e(T∗–σ)ρ(x)χωiu∗i(σ)dσ+e
(T∗–δ)h
δ
=e(T∗–δ)y+ K
i=
T∗
δ
e(T∗–σ)ρ(x)χωiu ∗ i(σ)dσ
+ δ
e(T∗–σ)ρ(x)
K
i= χωiu
∗
i(σ)dσ+e
T∗y –e(T
∗–δ)
y
=eT∗y+
T∗
e(T∗–σ)ρ(x)
K
i= χωiu
∗ i(σ)dσ
which shows that{vi}are the desired controls such thaty(·,T∗–δ;{χωivi},y) = . It
con-tradicts the definition ofT∗and we have proved the theorem.
Remark . There are some relations between the optimal control problem of (.) and shape design problem. For more about shape design problem see [, –].
Before stating the results on Problem (TP), we define the following reachable set:
R(T) =y(·,T;χωu,y)|u∈Uad
(.)
for eachT∈(, +∞).
Theorem . For any given positive constantε,Problem(TP)has a solution Tε∗,and
R(Tε∗)∩ ¯B(,ε)has only one point belonging to the boundary of B(,ε).Moreover,the
cor-responding time optimal control u∗εis unique and has the bang-bang property.
Proof Since the proof is long, we separate it into the following several steps.
Step . We shall show that there exists at least one time optimal control,i.e., there exists at least oneu∗∈Uadsuch thaty(·,Tε∗;u∗,y)∈ ¯B(,ε).
Let{Tn}be a monotone decreasing sequence such thatTn→Tε∗asn→+∞, then there
exists a sequence{un} ⊂Uadsuch thaty(·,Tn;χωun,y)∈ ¯B(,ε). Set
˜ un(t) =
⎧ ⎨ ⎩
un(t), t∈(,Tn),
, t∈[Tn,T).
SinceM∈L∞(,T;L()),{˜un}is a bounded sequence inL∞(,T;L()). Then there
existu˜∗∈L∞(,T;L()) and a subsequence of{un}, still so denoted, such thatun→u∗
weakly∗inL∞(,T;L()). Moreover,
ρχωu˜n→ρχωu˜∗ weakly∗inL∞
,T;L() asn→ ∞.
Therefore the solutionyn(·,·;χωu˜n,y) to the following system:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂tyn(x,t) –yn(x,t) =ρ(x)χω(x)u˜n(x,t), in×(,T),
yn(x,t) = , on∂×(,T),
yn(x, ) =y(x), in,
satisfies yn(·,t;χωu˜n,y)∈ ¯B(,ε) fort≥Tn. Denote byy∗ the solution to the following
system:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂ty∗(x,t) –y∗(x,t) =ρ(x)χω(x)u˜∗(x,t), in×(,T),
y∗(x,t) = , on∂×(,T),
Then
yn→y∗ weakly inL
,T;H() ∩H,T;L() ,
strongly inC[,T];L() asn→ ∞,
for anyδ> . Sinceyn(·,t;χωu˜n,y)∈ ¯B(,ε) fort≥Tn, y∗(·,t;χωu˜∗,y)∈ ¯B(,ε) for all
t≥Tnandn∈N. Hencey∗(·,Tε∗;χωu˜∗,y)∈ ¯B(,ε). Set
u∗(t) =u˜∗(t), t∈,Tε∗.
Theny∗satisfies the following system:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂ty∗(x,t) –y∗(x,t) =ρ(x)χω(x)u∗(x,t), in×(,T),
y∗(x,t) = , on∂×(,T),
y∗(x, ) =y(x), in,
andy∗(·,Tε∗;χωu∗,y)∈ ¯B(,ε).
Claim:u∗(t)L()≤M(t) for a.e.t∈[,Tε∗]. Indeed, let{ζk}k∈Nbe the countable density subset ofL(). Denote byLthe Lebesgue point ofun(t),ζk,t∈[,Tε∗], where·,·is the
inner product ofun(t) andζkinL(). Sinceun(t),ζk,u∗(t),ζk ∈L∞(,Tε∗), for each
t∈E≡ ∞
n,k=
Lun,ζk ∩ ∞
k=
Lu∗,ζk ,
we have
lim
δ→
δ
t+δ
t–δ
un(t),ζk
dt=un(t),ζk
,
lim
δ→
δ
t+δ
t–δ
u∗(t),ζk
dt=u∗(t),ζk
.
By virtue of
δ
t+δ
t–δ
un(t),ζk
dt→
δ
t+δ
t–δ
u∗(t),ζk
dt asn→ ∞,
un→u∗weakly∗inL∞(,Tε∗;L()), and the arbitrary ofδ> , we get
un(t),ζk
=lim
δ→
δ
t+δ
t–δ
un(t),ζk
dt →
lim
δ→
δ
t+δ
t–δ
u∗(t),ζk
dt=u∗(t),ζk
asn→ ∞.
Since{ζk}is dense inL(), we have
un(t),ζ
→u∗(t),ζ
for eachζ∈L(). That impliesu
n(t)→u∗(t) weakly inL(), and henceu∗(t)L()≤
lim infn→∞un(t)L()≤M. Applying |E|=Tε∗,
we obtainu∗(t)L()≤Mfor a.e.t∈[,Tε∗]. That proves the claim.
Step . We show thatR(Tε∗)∩ ¯B(,ε) has only one point. If so, it is obvious that this point
belongs to the boundary ofB(,ε).
In Step we getR(Tε∗)∩ ¯B(,ε)= ∅. By contradiction, we assume thatR(Tε∗)∩ ¯B(,ε) has at least two points,i.e., there existy≡y(·,Tε∗;χωu∗,y),y≡y(·,Tε∗;χωu∗,y)∈R(Tε∗)∩
¯
B(,ε) withy=y. It is obvious thatu∗=u∗inUad. Define
ˆ u≡u
∗ +u∗
.
Sinceuˆ∈Uad, andB(,ε) is strongly convex inL(), we getyˆ≡y(·,Tε∗;χωu,ˆ y) =y+y
is an inner point ofB(,ε),i.e., there existsγ > such thatB(y,ˆ γ)⊂B(,ε). For anyξ> , define
hξ≡ ˆy–y
·,Tε∗–ξ;χωu,ˆ y .
It is easy to check that
hξ =e(T
∗
ε–ξ)eξ–Iy
+
Tε∗–ξ
eξ–Ie(Tε∗–ξ–σ)ρχωu(ˆ σ)dσ
+ Tε∗
Tε∗–ξ
e(Tε∗–σ)ρχ
ωu(ˆ σ)dσ.
Hence,ξ can be chosen small enough such thathξL()≤γ. Therefore, we can get y(·,Tε∗–ξ;χωu,ˆ y)∈ ¯B(,ε). This contradicts the optimal timeTε∗. Subsequently, the set
R(Tε∗)∩ ¯B(,ε) has only one point, and this point belongs to the boundary ofB(,ε).
Step . The time optimal controlu∗has the bang-bang property.
SinceR(Tε∗)∩ ¯B(,ε) has only one point (denote this point byy∗=y(·,Tε∗;χωu∗,y)),
andR(Tε∗) andB(,¯ ε) are two convex sets, by hyperplane separation theorem, there exists
η∗∈L() such that
sup y∈R(Tε∗)
y,η∗≤ inf z∈ ¯B(,ε)
z,η∗≤y∗,η∗. (.)
Notice thaty∈R(Tε∗) can be written by
y·,Tε∗;χωu,y =eT ∗
εy
+
Tε∗
e(Tε∗–σ)ρχωu(σ)dσ.
Then (.) can be written as
sup ¯ u∈U
Tε∗
e(T∗ε–σ)ρχωMu(¯ σ),η∗dσ≤ Tε∗
Here
¯
u∗∈U≡
¯
u∈L∞,Tε∗;L() |u(t)¯ L()≤ for a.e.t∈
,Tε∗
,
and
u∗(t) =Mu¯∗(t) for allt∈,Tε∗
. (.)
Hence, we have
sup ¯ u∈U
Tε∗
¯
u(σ),e(Tε∗–σ)ρχ
ωMη∗
dσ≤
Tε∗
¯
u∗(σ),e(Tε∗–σ)ρχ
ωMη∗
dσ.
For givent∈E, choosing
¯ u(t) =
⎧ ⎨ ⎩
¯
u∗(t), fort∈(,Tε∗)\(t–λ,t+λ), ζ, fort∈(t–λ,t+λ)⊂(,Tε∗),
whereζ∈L(), we get
sup
ζ∈L()
ζ,e(Tε∗–t)ρχ
ωMη∗
≤u¯∗(t),e(T ∗
ε–t)ρχ
ωMη∗
,
i.e.,
e(Tε∗–t)ρχ
ωη∗L() = sup
ζ∈L()
ζ,e(Tε∗–t)ρχ
ωη∗
≤u¯∗(t),e(Tε∗–t)ρχ
ωη∗
≤u¯∗(t)L()e
(Tε∗–t)ρχ
ωη∗L(). This implies that
u¯∗(t)L()= (.)
byu¯∗∈U. Equation (.), together with (.) and|E|=Tε∗, yields
u∗(t)L()=M for a.e.t∈
,Tε∗
.
From the above, we get the time optimal control’s bang-bang property. That completes the
proof.
3 Norm optimal control problems with multi-time and multi-domain controls In this section, letT∈R+,K∈Z+be given finite constants, andKbe time partition of
[,T] defined by
For anyi∈ {, , . . . ,K}, setIi= (ti–,ti]. TakingEi=Ii, we can rewrite (.) as
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂ty(x,t) –y(x,t) =ρ(x)
K
i=χIi(t)χωi(x)ui(x,t), in×(,T),
y(x,t) = , on∂×(,T),
y(x, ) =y(x), in.
(.)
It is obvious that the system is null controllable (see [, ]). For any given partitionK, by standard minimizing sequence method, there exists a solution to the following norm optimal control problem:
N(K)≡inf
K
i=
ρχIiχωiuiL∞(,T;L())
y·,T;{χIiχωiui},y = ,
{ui}Ki=⊂L∞
,T;L()
. (.)
We are interested in the partition’s existence of the following norm optimal control prob-lem:
Problem (NP)
NK∗ ≡infN(K)|Kis defined by partition (.)
.
We have the following solvability result on Problem (NP).
Theorem . For any K> ,there exists at least one solution to Problem(NP).
Proof It is obviously thatNK∗ <∞. Let{nK}be the partition sequence such that
NnK →NK∗ asn→ ∞.
Then there exists a control sequence{un,un, . . . ,unK}∞n=, such that
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂tyn(x,t) –yn(x,t) =ρ(x)
K
i=χIin(t)χωi(x)u n
i(x,t), in×(,T),
yn(x,t) = , on∂×(,T),
yn(x, ) =y
(x),yn(x,T) = , in,
with ρ
K
i= χIn
iχωiu n i
L∞(,T;L())
=NnK for alln∈N.
Since{tn
} ⊂[,T] is a bounded sequence, there exist a subsequence of{tn}, still so
de-noted, andt
such thattn →tasn→ ∞. Also, there exist a subsequence of{tn}, still
so denoted, andt
satisfyingtn→t asn→ ∞. Moreover, sincetn≤tnfor alln∈N, we
havet
≤t. By the same line, we can get{ti}Ki=satisfying
Denote the above partition byK. We also can defineIifor eachi∈ {, , . . . ,K}. Naturally, one has
IinIi→ asn→ ∞. (.)
Here and in what follows, we defineIinIi≡(In
i –Ii)∪(Ii –Iin) for alli,n.
On the other hand, since{un,un, . . . ,unK}∞n= is bounded,{uni}∞n= is also a bounded se-quence inL∞(,T;L()) for alli= , . . . ,K. Then, for eachi= , . . . ,K, there exist a
sub-sequence of{un
i}∞n=, still so denoted, andui ∈L∞(,T;L()) such that
uni →ui weakly∗inL∞,T;L() asn→ ∞. (.)
For eachv∈L(,T;L()), we have
T
ρχIinχωiu n
i –χIiχωiu i v dx dt
= T
ρχIn iχωiu
n
i –χIiχωiu n i v dx dt
+ T
ρχI
iχωiu n
i –χIiχωiu i v dx dt
= T
ρ(χIn
i –χIi)χωiu n iv dx dt
+ T
ρuni –ui χI
iχωiv dx dt. (.)
By (.), we get
T
uni –ui ρχI
iχωiv dx dt→ asn→ ∞. (.)
By (.) and the absolutely continuity ofv∈L(,T;L()), we have
Tρ(χIn
i –χIi)χωiu n iv dx dt
≤uniL∞(,T;L())ρ(χIin–χIi)χωivL(,T;L())
=uniL∞(,T;L())
IinIi
|ρχωiv| dx
dt
≤un
iL∞(,T;L())
IinIi
|v|dx
dt
→ asn→ ∞. (.)
Equation (.), together with (.) and (.), yields
T
ρχIn iχωiu
n
i –χIiχωiu
That implies, for alli∈ {, . . . ,K},
ρχIinχωiu n
i →ρχIiχωiu
i weakly∗inL∞
,T;L() asn→ ∞.
Therefore, one gets
ρ K
i=
χIinχωiu n i →ρ
K
i= χI
iχωiu
i weakly∗inL∞
,T;L() asn→ ∞ (.)
and ρ
K
i= χI
iχωiu i
L∞(,T;L()) ≤lim inf
n→∞
ρ
K
i= χIn
iχωiu n i
L∞(,T;L())
=NK∗. (.)
Letybe the solution to the following system:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂ty(x,t) –y(x,t) =ρ(x)
K
i=χIi(t)χωi(x)u
i(x,t), in×(,T),
y(x,t) = , on∂×(,T),
y(x, ) =y
(x), in.
(.)
By (.) we get
yn→y weakly inL,T;H() ∩H,T;L() ,
strongly inC[δ,T];L() ,
for everyδ> . Sinceyn(T) = , immediately we have
y(T) = . (.)
Combining (.), (.) and (.), we complete the proof.
Now, let us consider an alteration of system (.):
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂ty(x,t) –y(x,t) =ρ(x)
K
i=χIi(t)χωσ(i)(x)ui(x,t), in×(,T),
y(x,t) = , on∂×(,T),
y(x, ) =y(x), in,
where
By the aforementioned discussion, this system is null controllable, and for given parti-tionKand mapσ, there exists at least a solution to the following norm optimal control
problem:
N(σ)≡inf
K
i=
ρχIiχωσ(i)uiL∞(,T;L())
y·,T;{χIiχωσ(i)ui},y = ,
{ui}Ki=⊂L∞
,T;L()
.
Let us consider the solvability of the following norm optimal control problem:
N≡infN(σ)|σis defined by (.). (.)
Since the number ofσ is finite, it is obvious that there existsσ∗satisfyingN∗=N(σ∗). Hence, we obtain the following result.
Proposition . Let{Ii}Ki=,{ωk}Kk=be defined as before.Then there exists at least a solution
to the problem(.).
Acknowledgements
The authors are grateful to the anonymous referees for helpful comments and suggestions, which greatly improved the presentation of this paper.
Competing interests
The authors declare that there are no competing interests regarding the publication of this article.
Authors’ contributions
All authors 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: 28 April 2017 Accepted: 22 September 2017
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