Equivalence of three different kinds of optimal control problems for heat equations and its applications

Abstract

This paper presents an equivalence theorem for three different kinds of
optimal control problems, which are optimal target control problems, optimal
norm control problems and optimal time control problems. Controlled systems in
this study are internally controlled heat equations. With the aid of this
theorem, we establish an optimal norm feedback law and build up two algorithms
for optimal norms (together with optimal norm controls) and optimal time (along
with optimal time controls), respectively.

Full text

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arXiv:1110.3885v2 [math.OC] 20 Oct 2011
Equivalence of three different kinds of optimal control
problems for heat equations and its applications
Gengsheng Wang∗
Yashan Xu†
Abstract
This paper presents an equivalence theorem for three different kinds of optimal
control problems, which are optimal target control problems, optimal norm control
problems and optimal time control problems. Controlled systems in this study are
internally controlled heat equations. With the aid of this theorem, we establish an
optimal norm feedback law and build up two algorithms for optimal norms (together
with optimal norm controls) and optimal time (along with optimal time controls),
respectively.
AMS Subject Classifications. 35K05, 49N90
Keywords.
equations
optimal controls, optimal norm, optimal time, feedback law, heat
1Introduction
We begin with introducing the controlled system. Let T be a positive number and
Ω ⊆ Rdbe a bounded domain with a smooth boundary ∂Ω. Let ω be an open and non-
empty subset of Ω. Write χωfor the characteristic function of ω. Consider the following
controlled heat equation:


of Mathematicsand Statistics,
(wanggs62@yeah.net) The author was partially supported by National Basis Research Program of
China (973 Program) under grant 2011CB808002 and the National Natural Science Foundation of China
under grant 11161130003 and 11171264.
†Schoolof MathematicalSciences,Fudan
(yashanxu@fudan.edu.cn) This work was partially supported by NNSF Grant 10801041, 10831007.

∂ty − △y = χωχ(τ,T)u in
y = 0
y(0) = y0
Ω × (0,T),
on ∂Ω × (0,T).
inΩ,
(1.1)
∗SchoolWuhan University,Wuhan, 430072,China.
University,KLMNS, Shanghai200433, China.
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Here y0∈ L2(Ω), u ∈ L∞(0,T;L2(Ω)), τ ∈ [0,T) and χ(τ,T)stands for the characteristic
function of (τ,T). In this equation, controls are restricted over ω×(τ,T). It is well known
that for each u ∈ L∞(0,T;L2(Ω)) and each y0 ∈ L2(Ω), Equation (1.1) has a unique
solution in C([0,T];L2(Ω)). We denote, by y(·;χ(τ,T)u,y0), the solution of Equation (1.1)
corresponding to the control u and the initial state y0. Throughout this paper, ? · ? and
< ·,· > stand for the usual norm and inner product of the space L2(Ω), respectively.
Next, we will set up, for each y0 ∈ L2(Ω), three kinds of optimal control problems
associated with Equation (1.1). For this purpose, we take a target zd∈ L2(Ω) such that
?
The set on the right hand side of (1.2) is called the attainable set of Equation (1.1). Then
we introduce the following target sets:
zd/ ∈
y(T;χ(0,T)u,0) : u ∈ L∞(0,T;L2(Ω))
?
. (1.2)
B(zd,r) = {ˆ y ∈ L2(Ω) : ?ˆ y − zd? ≤ r}, r > 0.
For each M ≥ 0, each r > 0 and each τ ∈ [0,T), we define three sets of controls as follows:
• Uτ,M= {v ∈ L∞(0,T;L2(Ω)) : ?v(t)? ≤ M for a.e. t ∈ (τ,T)};
• UM,r= {v : ∃ τ ∈ [0,T) s.t. v ∈ Uτ,M and y(T;χ(τ,T)v,y0) ∈ B(zd,r)};
• Ur,τ= {v ∈ L∞(0,T;L2(Ω)) : y(T;χ(τ,T)v,y0) ∈ B(zd,r)}.
For each u ∈ UM,r, we set
? τM,r(u) = sup{τ ∈ [0,T) : u ∈ Uτ,Mand y(T;χ(τ,T)u,y0) ∈ B(zd,r)}.
• (OP)τ,M: inf{?y(T;χ(τ,T)u,y0) − zd?2: u ∈ Uτ,M};
(1.3)
Three kinds of optimal control problems studied in this paper are as follows:
• (TP)M,r: sup{? τM,r(u) : u ∈ UM,r};
We call (OP)τ,Mas an optimal target control problem, which is a kind of optimal control
problem with the observation of the final state (see [9], page 177). The problem (NP)r,τ
is an optimal norm control problem, which is related to the approximate controllability
(see [4]). The problem (TP)M,ris an optimal time control problem. The aim of con-
trols in (TP)M,ris to delay initiation of active control as late as possible, such that the
corresponding solution reaches the target B(zd,r) at the ending time T (see [11]).
The above three problems provide the following three values, respectively:
• (NP)r,τ: inf{?u?L∞(τ,T;L2(Ω)): u ∈ Ur,τ}.
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• r(τ,M) ≡ inf{?y(T;χ(τ,T)u,y0) − zd? : u ∈ Uτ,M};
• τ(M,r) ≡ sup{? τM,r(u) : u ∈ UM,r};
The value r(τ,M) is called the optimal distance to the target for (OP)τ,M; while values
τ(M,r) and M(r,τ) are called the optimal time for (TP)M,rand the optimal norm for
(NP)r,τ, respectively. The optimal controls to these problems are defined as follows:
• M(r,τ) ≡ inf{?u?L∞(τ,T;L2(Ω)): u ∈ Ur,τ}.
• u∗is called an optimal control to (OP)τ,Mif u∗= χ(τ,T)v∗for some v∗∈ Uτ,Msuch
that ?y(T;χ(τ,T)v∗,y0) − zd? = r(τ,M);
• u∗is called an optimal control to (TP)M,rif u∗= χ(τ(M,r),T)v∗for some v∗∈ Uτ(M,r),M
such that y(T;χ(τ(M,r),T)v∗,y0) ∈ B(zd,r).
• u∗is called an optimal control to (NP)r,τif u∗= χ(τ,T)v∗for some v∗∈ Ur,τ and
?u∗?L∞(τ,T;L2(Ω))= M(r,τ).
Throughout the paper, the following notation will be used frequently:
rT(y0) ≡ ?y(T;0,y0) − zd?. (1.4)
The main purpose of this study is to present an equivalence theorem for the above-
mentioned three kinds of optimal control problems and its applications. This theorem
can be stated, in plain language, as follows:
• (OP)τ,M⇔ (TP)M,r(τ,M)⇔ (NP)r(τ,M),τwhen M > 0 and τ ∈ [0,T);
• (NP)r,τ⇔ (OP)τ,M(r,τ)⇔ (TP)M(r,τ),rwhen r ∈ (0,rT(y0)) and τ ∈ [0,T);
• (TP)M,r⇔ (NP)r,τ(M,r)⇔ (OP)τ(M,r),Mwhen M > 0 and r ∈ [r(0,M),rT(y0)).
Here, by (P1) ⇔ (P2), we mean that problems (P1) and (P2) have the same optimal
controls. Based on the equivalence theorem, the study of one kind of optimal control
problem can be carried out by investigating one of the other two kinds of optimal control
problems. In particular, one can use some existing fine properties for optimal target
controls to derive properties of optimal norm controls and optimal time controls.
An important application of the equivalence theorem is to build up a feedback law
for norm optimal control problems. We will roughly present this result in what follows.
Notice that Problem (NP)r,τdepends on τ ∈ [0,T) and y0∈ L2(Ω), when r and zdare
fixed. To stress this dependence, we denote, by (NP)r,τ
initial state y0. Throughout this paper, we let A be the operator on L2(Ω) with domain
y0, the problem (NP)r,τwith the
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D(A) = H1
for the semigroup generated by A. By the equivalence theorem and some characteristics
of the target optimal control problems, we construct a map F : [0,T) × L2(Ω) → L2(Ω)
holding properties:
(i) For each y0∈ L2(Ω) and each τ ∈ [0,T), the evolution equation
?
y(0) = y0,
0(Ω)?H2(Ω) and defined by Ay = △y for each y ∈ D(A). Write {et△: t ≥ 0}
˙ y(t) − Ay(t) = χωχ(τ,T)(t)F(t,y(t)),t ∈ (0,T)
has a unique mild solution, which will be denoted by yF,τ,y0(·). Here χωis treated as an
operator on L2(Ω) in the usual way.
(ii) For each y0∈ L2(Ω) and each τ ∈ [0,T), χ(τ,T)(·)F(·,yF,τ,y0(·)) is the optimal control
to Problem (NP)r,τ
Consequently, the map F is an optimal feedback law for the family of optimal norm
control problems as follows:
With the aid of the equivalence theorem, we also build up two algorithms for the opti-
mal norm, along with the optimal control, to (NP)r,τand the optimal time, together with
the optimal control, to (TP)M,r, respectively. These algorithms show that the optimal
norm and the optimal control to (NP)r,τcan be approximated through solving a series
of two-point boundary value problems, and the same can be said about the optimal time
and the optimal control to (TP)M,r.
It deserves to mention that all results obtained in this paper still stand when Equation
(1.1) is replaced by


where a ∈ L∞(Ω × (0,T)) and Ω is convex (see Remark 2.13).
The equivalence between optimal time and norm control problems have been studied
in [13], [7] and [5] and the references therein. The optimal time control problem studied in
these papers is to initiate control from the beginning such that the corresponding solution
(to a controlled system) reaches a target set in the shortest time. Though problems studied
in the current paper differ from those in [13], our study is partially inspired by [13]. To
the best of our knowledge, the equivalence theorem of the above-mentioned three kinds
of optimal control problems has not been touched upon. Moreover, the feedback law and
the algorithms established in this paper seem to be new.
The rest of the paper is organized as follows: Section 2 presents the equivalence
theorem and its proof. Section 3 provides the above-mentioned two algorithms. In section
4, we build up an optimal norm feedback law.
y0.
?(NP)r,τ
y0: τ ∈ [0,T), y0∈ L2(Ω)?.

∂ty − △y + ay = χωχ(τ,T)u in
y = 0
y(0) = y0
Ω × (0,T),
on ∂Ω × (0,T),
inΩ,
(1.5)
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2 Equivalence of three optimal control problems
Throughout this section, the initial state y0is fixed in L2(Ω). For simplicity, we write
y(·;χ(τ,T)u) and rTfor y(·;χ(τ,T)u,y0) and rT(y0) (which is defined by (1.4)), respectively.
The purpose of this section is to prove the following equivalence theorem:
Theorem 2.1. When M > 0 and τ ∈ [0,T), the problems (OP)τ,M, (TP)M,r(τ,M)and
(NP)r(τ,M),τhave the same optimal control; When r ∈ (0,rT) and τ ∈ [0,T), the problems
(NP)r,τ, (OP)τ,M(r,τ)and (TP)M(r,τ),rhave the same optimal control; When M > 0 and
r ∈ [r(0,M),rT), the problems (TP)M,r, (NP)r,τ(M,r)and (OP)τ(M,r),Mhave the same
optimal control.
2.1 Some properties on optimal target control problems
Lemma 2.2. Let M ≥ 0 and τ ∈ [0,T). Then, (i) (OP)τ,Mhas optimal controls; (ii)
r(τ,M) > 0; (iii) u∗is an optimal control to (OP)τ,Mif and only if u∗∈ L∞(0,T;L2(Ω)),
with u∗= 0 over (τ,T), satisfies
?T
where p∗is the solution to the equation:


with y∗(·) solving the equation:


Proof. (i) and (iii) have been proved in [9] (see the proof of Theorem 7.2, Chapter III in
[9]). The remainder is to show (ii). For this purpose, we let u∗be an optimal control to
(OP)τ,Mand write y∗(·) for y(·;χ(τ,T)u∗,y0). Then it holds that r(τ,M) = ?y∗(T) − zd?
and
y∗(T) ∈?y(T;χ(0,T)u,y0) : u ∈ L∞(0,T;L2(Ω))?.
On the other hand, by the null controllability for the heat equation (see, for instance, [3]
or [6]), one can easily check that
?y(T;χ(0,T)u,y0) : u ∈ L∞(0,T;L2(Ω))?=?y(T;χ(0,T)u,0) : u ∈ L∞(0,T;L2(Ω))?.
5
0
< χ(τ,T)(t)χωp∗(t),u∗(t) > dt =max
v(·)∈Uτ,M
?T
0
< χ(τ,T)(t)χωp∗(t),v(t) > dt,(2.1)

∂tp∗+ △p∗= 0
p∗= 0
p∗(T) = −(y∗(T) − zd) in
in
on ∂Ω × (0,T),
Ω
Ω × (0,T),
(2.2)

∂ty∗− △y∗= χωχ(τ,T)u∗
y∗= 0
y∗(0) = y0
in
on ∂Ω × (0,T),
inΩ.
Ω × (0,T),
(2.3)
(2.4)