CANADIAN APPLIED
MATHEMATICS QUARTERLY
Volume 14, Number 3, Fall 2006
NECESSARY CONDITIONS FOR OPTIMAL
CONTROL SYSTEMS GOVERNED BY
SEMI-LINEAR ELLIPTIC EQUATIONS
HANG GAO AND MICHAEL Y. LI
ABSTRACT. We study an optimal control problem for systems governed by a class of elliptic equations that may possess
multiple solutions. Necessary conditions for optimal control
of the elliptic problem are obtained by constructing a suitable
sequence of approximating parabolic control problems.
1 Introduction and main results We consider the following control problem for systems governed by a class of semi-linear elliptic equations
(
−A y = f (x, y(x), u(x)), x ∈ Ω
(1.1)
y ∂Ω = 0,
where the operator A has the form
Ay =
n
X
aij (x) yxj
i,j=1
xi
.
The region Ω ⊂ Rn is bounded with smooth boundary ∂Ω. The control function u(x) satisfies u(x) ∈ U for all x ∈ Ω, where U ⊂ Rn
is a bounded closed convex set. A y(x) is a solution to (1.1) if y ∈
W 2,p (Ω), p > 1, −Ay = f (x, y(x), u(x)), a.e. x ∈ Ω, and y|∂Ω = 0. We
make the following assumptions.
(P1) aij ∈ C 1 (Ω), aij (x) = aji (x), and
n
X
i,j=1
aij (x) ηi ηj ≥ λ |η|2 ,
∀η ∈ Rn .
c
Copyright Applied
Mathematics Institute, University of Alberta.
239
240
HANG GAO AND MICHAEL Y. LI
(P2) f : Ω × R × U → R satisfies that
f (x, y, k1 u1 + k2 u2 ) = k1 f (x, y, u1 ) + k2 f (x, y, u2 ),
fy (x, ·, u) ∈ C(R), f (·, y, u) is measurable on Ω, and there exists
a monotonically increasing function c : R+ → R+ such that
|f (x, y, u)| ≤ c(|y|).
In [3], the following problem
(
−div |∇y|p−2 ∇y = a(x) |y|p y + b(x),
(1.2)
y ∂Ω = 0,
x∈Ω
was considered in with p > 1, where y(x) represents distribution of
temperature. When p = 2, (1.2) is the Dirichlet problem of a semilinear elliptic equation. The right hand side of the equation is linear
with respect to both a(x) and b(x). When the control function is chosen
to be either the heat coefficient a(x) or the heat source b(x), system (1.2)
satisfies our assumptions (P1) and (P2).
A function y ∗ ∈ W 2,p (Ω) is said to be a upper solution to (1.1) if
(1.3)
(
−A y ≥ f (x, y(x), u(x)),
y ∂Ω ≥ 0.
a.e. x ∈ Ω
Similarly, a lower solution y∗ ∈ W 2,p (Ω) satisfies (1.3) with both inequalities reversed. Define
(1.4)
U = { u(x) : u(x) ∈ U, u(·) is measurable on Ω }
as the set of admissible controls. We make the following additional
assumption for (1.1).
(P3) The problem (1.1) has an upper solution y ∗ (x) and a lower solution y∗ (x) for all u ∈ U. Both y ∗ and y∗ are independent of
u.
The following lemma is similar to Theorem 1 in [1].
Lemma 1.1. Assume that the assumptions (P1)–(P3) are satisfied. Then,
for each u ∈ U, problem (1.1) has at least two solutions y1 (x) and y2 (x)
such that y∗ (x) ≤ y1 (x) ≤ y2 (x) ≤ y ∗ (x).
NECESSARY CONDITIONS FOR OPTIMAL CONTROL
241
Set
P = {(y, u) : u ∈ U, (y, u) satisfies (1.1)}.
Any (y, u) ∈ P will be called an admissible pair. We define the following
functional on P
Z
2
[y(x, u) − B(x)] dx.
(1.5)
J(y, u) =
Ω
If there exists (y, u) ∈ P such that
J(y, u) = inf{J(y, u) : (y, u) ∈ P},
then u is called an optimal control with y as an optimal state. We also
call (y, u) an optimal pair.
Problem E. Assume that (y, u) is an optimal pair. What conditions
do u and y satisfy?
This problem has been studied for systems governed by semi-linear
elliptic equations (see [4, 5, 8, 12]) under the following assumption
on f
(1.6)
fy (x, y, u) ≤ 0 for all (x, y, u).
Condition (1.6) implies that the state variable is continuously dependent
on the control variable. It plays an essential role in previous studies of
Problem E; if (1.6) is not satisfied, then the problem (1.1) may have
multiple solutions for each u. In the present paper, the restriction (1.6)
is relaxed. To deal with the non-uniqueness, we approximate the elliptic
control problem (1.1) with a suitable sequence of parabolic control systems. Necessary conditions for optimal control are first obtained for the
approximating parabolic systems, which, in the limit, lead to necessary
conditions for optimal control of the elliptic system. The main results
of the present paper are stated in the following.
Theorem 1.1. Assume that (P1)–(P3) are satisfied. Suppose that
(¯
y (x), u
¯(x)) is an optimal pair and that fy (x, y(x), u(x)) < λ, where
λ is given in (P1). Then there exist ξ(x) and ψ(x) such that kξk2,Ω ≤ C
and ψ(x) satisfies the equation
(1.7)
(
−A ψ(x) = fy (x, y(x), u(x)) ψ(x) − 2 [y(x) − A(x)] ,
ψ ∂Ω = 0.
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HANG GAO AND MICHAEL Y. LI
Furthermore, the following maximum principle holds
(1.8)
H(x, u
¯(x)) = max H(x, u),
u∈U
a.e. x ∈ Ω,
in which,
H(x, u) := ψ(x) f (x, y(x), u) + ξ(x)u.
The existence of optimal control is proved under a more relaxed assumption:
(P2’) f (·, y, u) is measurable on Ω, f (x, · , ·) and fy (x, · , ·) are continuous on R × U, and there exists a continuous function c :
[0, +∞) → [0, +∞), such that
| f (x, y, u) | ≤ c(|y|).
Theorem 1.2. Suppose that (P1), (P2’) and (P3) are satisfied. Then
the problems (1.1) and (1.5) have at least one optimal pair.
2 Approximation by parabolic control systems
(0, 1), Γ = ∂Ω × (0, 1), and
Let Q = Ω ×
V = {v(x, t) ∈ [0, 1] : v(·) is measurable on Q}.
Let n be an integer and v ∈ V, consider the following parabolic equation
(2.1)
wt (x, t) − Aw(x, t) = f (x, w(x, t), u(x))
1
+ v(x, t), (x, t) ∈ Q,
n
w = 0, w(x, 0) = y¯(x),
Γ
where y¯(x) is an optimal state in Problem E. From the standard parabolic
theory (see [10]), we know that, for each (u, v) ∈ U × V, there exists a
unique solution w = w(x, t; u, v) to problem (2.1) such that w ∈ Wp2,1 (Q)
(p > 1). We define the following cost functional
(2.2) Fn (u, v) =
Z
2
Q
[w(x, t) − B(x)] dx dt −
√
+ C0 n
Z
Ω
Z
Ω
ψ(x)[w(x, 1) − y¯(x)] dx
[u(x) − u
¯(x)]2 dx.
NECESSARY CONDITIONS FOR OPTIMAL CONTROL
243
Here, w(x, t) is the solution of problem (2.1) corresponding to u(x) and
v(x, t), ψ(x) is the solution of problem (1.7), (¯
y (x), u¯(x)) is an optimal
pair in Problem E, and constant C0 > 0 is to be determined later. Set
1
.
(2.3)
Un = u ∈ U : ku − u
¯k2,Ω ≤
n
An optimal control to the problem (2.1)–(2.2) is a pair (u, v) that minimizes the functional Fn .
Theorem 2.1. For each integer n, there exists an optimal control to
problem (2.1)–(2.2) in Un × V.
Proof. It is easy to see that Fn (u, v) is bounded below for any (u, v) ∈
Un × V. There thus exists a minimizing sequence {(uk , vk )} ⊂ Un × V,
such that
lim Fn (uk , vk ) = inf{Fn (u, v) : (u, v) ∈ Un × V}.
k→∞
For any k, equation (2.1) may be written as
where
wk t (x, t) − Awk (x, t) − ck (x, t) wk (x, t)
1
= f (x, 0, u(x)) + vk (x, t), (x, t) ∈ Q,
n
wk Γ = 0, wk (x, 0) = y¯(x),
ck (x, t) =
Z
1
fy (x, τ wk (x, t), uk (x)) dτ.
0
From the assumption (P2), we know that there exists constant C > 0,
such that
|ck (x, t)| ≤ C for all k.
It follows from the classical theory that
kwk kWp2,1 (Q) ≤ C,
p > 1.
There thus exists a subsequence {wki } ⊂ {wk }, and wn ∈ Wp2,1 (Q), such
that wki → w
¯n weakly in Wp2,1 (Q), and thus wki → w n in Lp (Q) by the
embedding theorem. Since kuki −uk2,Ω ≤ 1/n and kvki kp,Q ≤(measQ)1/p ,
there exist un , v n , and subsequences of {uki } and {vki }, denoted by
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HANG GAO AND MICHAEL Y. LI
themselves, such that uki → un weakly in L2 (Ω), and vki → v n weakly
in Lp (Q). Let q be such that 1/p + 1/q = 1. For any ϕ ∈ Lq (Q), we have
Z
Q
ϕ(x, t)[wki t (x, t) − Awki (x, t)] dx dt
1
ϕ(x, t) f (x, wki (x, t), uki (x)) + vki (x, t) dx dt.
=
n
Q
Z
Since f (x, y, u) is linear in u and continuous in y, we can take a limit in
the above equality as ki → ∞ to arrive at
Z
Q
ϕ(x, t)[w nt (x, t) − Aw n (x, t)] dx dt
1
ϕ(x, t) f (x, w n (x, t), un (x)) + v n (x, t) dx dt.
=
n
Q
Z
We also have w n |Γ = 0 and w n (x, 0) = y¯(x), by the H¨
older continuity
of wki and w n . Therefore, w n (x, t) is a solution to the problem (2.1)
corresponding to un (x) and v n (x, t). Next, we prove that (un , v n ) is an
optimal control for the problem (2.1)–(2.2). Let
Fn1 (u, v) =
and
Z
Q
[w(x, t) − B(x)]2 dx dt −
√
Fn2 (u, v) = C0 n
Z
Ω
Z
Ω
ψ(x)[w(x, t) − y¯(x)] dx
[u(x) − u(x)]2 dx.
Fn1 (u, v)
It is easy to see that
is weakly continuous. We will show that
Fn2 (u, v) is weakly lower semi-continuous. In fact, for any ` > 0, let
e` =
u∈U :
Z
2
Ω
[u(x) − u(x)] dx ≤ `
.
Suppose there exists a sequence {um } ⊂ e` such that um → u
˜ weakly in
L2 (Ω). Then,
k˜
u − uk22,Ω ≤ lim inf kum − uk22,Ω ≤ `,
m→∞
NECESSARY CONDITIONS FOR OPTIMAL CONTROL
245
namely u
˜ ∈ e` . This implies that the set e` is sequentially weakly closed.
From [2], we know that Fn2 (u, v) is weakly lower semi-continuous. Therefore,
Fn (un , v n ) =
Z
2
Q
[w n (x, t) − B(x)] dx dt −
√
+ C0 n
≤ lim
ki →∞
Z
−
Q
Z
Z
Ω
Z
Ω
ψ(x)[w n (x, 1) − y¯(x)] dx
(un (x) − u(x))2 dx
[wki (x, t) − B(x)]2 dx dt
Ω
ψ(x)[w ki (x, 1) − y¯(x)] dx
√
+ C0 n
Z
Ω
(uki (x) − u(x))2 dx
= inf{F (u, v) : (u, v) ∈ Un × V }.
This implies
Fn (un , v n ) = inf{F (u, v) : (u, v) ∈ Un × V },
completing the proof of Theorem 2.1.
Theorem
2.2. For some C0 > 0 in (2.2), un satisfies kun − uk2,Ω <
√
1/ n.
√
Proof. Suppose the contrary, namely, kun − uk2,Ω = 1/ n. Since
y¯(x, t) = y¯(x) satisfies
(
y¯t (x, t) − A¯
y (x, t) = f (x, y¯(x, t), u(x)),
y¯|Γ = 0,
y¯(x, 0) = y¯(x),
we have
(w n (x, t) − y¯)t − A(w n (x, t) − y¯) − cn (x, t)(w n (x, t) − y¯)
1
= f (x, y¯(x), un (x)) − f (x, y¯(x, t), u(x)) + v n (x, t),
n
(w n (x, t) − y¯)|Γ = 0, (w n (x, t) − y¯)(x, 0) = 0,
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HANG GAO AND MICHAEL Y. LI
where
cn (x, t) =
Z
1
0
fy (x, y¯(x) + τ (w n (x, t) − y¯(x)), un (x)) dτ,
and for each n,
|cn (x, t)| ≤ C1 .
Choose a constant L, such that L + C1 < −1. Let
hn (x, t) = eLt [wn (x, t) − y¯(x)].
Then
(2.4)
hn t (x, t) − Ahn (x, t) − (L + cn (x, t))hn (x, t)
= eLt [f (x, y¯(x), un (x)) − f (x, y¯(x, t), u(x))
+
hn Γ = 0,
1
v n (x, t)],
n
hn (x, 0) = 0.
Multiplying (2.4) by hn (x, t), and by integration by parts over Q, we
obtain
Z
Z
Z
1
h2n (x, 1)dx + λ
|∇hn (x, t)|2 dx dt +
h2n (x, t) dx dt
2 Ω
Q
Q
Z
≤
|fu (x, y¯(x), u(x)| |un (x) − u(x)| |hn (x, t)| dx dt
Q
+
1
n
Z
v n (x, t)|hn (x, t)| dx dt
Q
Z
1
≤ C (un (x) − u(x)) dx +
h2 (x, t) dx dt
4 Q n
Ω
Z
1
1
+ C+
h2 (x, t) dx dt.
n
4 Q n
Z
2
It follows from this inequality that
(2.5)
Z
Ω
[w n (x, 1) − y¯(x)]2 dx ≤
C∗
n
NECESSARY CONDITIONS FOR OPTIMAL CONTROL
247
and
Z
(2.6)
Q
[wn (x, t) − y¯(x)]2 dx dt ≤
C∗
.
n
Furthermore,
Z
C∗
≤ √
(x,
1)
−
y
¯
(x)]
dx
ψ(x)[w
,
n
n
Ω
(2.7)
and
(2.8)
Z
C∗
2 [w n (x, t) − y¯(x)][w n (x, t) − B(x)] dx dt ≤ √
.
n
Q
From the relation
Z
Z
[¯
y (x) − B(x)]2 dx dt
[¯
y (x) − B(x)]2 dx =
Q
Ω
=
Z
Q
y (x) − w n (x, t))(w n (x, t) − B(x))
{[¯
y(x) − wn (x, t)]2 + 2(¯
+ [w n (x, t) − B(x)]2 } dx dt
≤
Z
2C ∗
[w n (x, t) − B(x)]2 dx dt + √ ,
n
Q
we obtain
Fn (un , v n )
Z
=
[w n (x, t) − B(x)]2 dx dt
Q
−
Z
Ω
√
ψ(x)[w n (x, 1) − y¯(x)] dx + C0 n
1
≥ [¯
y (x) − B(x)]2 dx + (C0 − 3C ∗ ) √ .
n
Ω
Z
Z
Ω
[un (x) − u(x)]2 dx
Choose C0 such that C0 − 3C ∗ > 0, we obtain
Z
y (x) − B(x)]2 dx = Fn (u, 0).
(2.9)
Fn (un , v n ) > [¯
Ω
Since (u, 0) ∈ Un × V, inequality (2.9) indicates that (un , v n ) is not an
optimal control of the problem
√ (2.1)–(2.2) in Un × V. The contradiction
shows that kun − uk2,Ω < 1/ n, completing the proof.
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HANG GAO AND MICHAEL Y. LI
Remark 2.1. From the proofs of the Theorems 2.1 and 2.2, we have
that
(1)
lim kun − uk2,Ω = 0,
n→∞
(2)
lim (kwn − y¯k2,Ω + k∇(wn − y¯)k2,Ω ) = 0,
n→∞
and
(3)
lim Fn (un , v n ) = J(y, u).
n→∞
3 Optimality conditions for the approximating parabolic systems In this section, we derive necessary conditions for optimal control for the parabolic system (2.1)–(2.2).
From Theorem 2.2 we know that the optimal control function un is
an interior point of Un , that is, there exists ρ0 > 0, such that for any
ρ ∈ (0, ρ0 ) and u ∈ U,
uρ = un + ρ(u − un ) ∈ Un .
At the same time, for any v ∈ V,
vρ = v n + ρ(v − v n ) ∈ V.
Lemma 3.1. Suppose that wnρ (x, t) and wn (x, t) are the solutions of
problem (2.1) corresponding to (uρ , vρ ) and (un , v n ), respectively. Then,
(3.1)
kwnρ (x, t) − w n (x, t)k2,Q + k∇(wnρ (x, t) − wn (x, t))k2,Q ≤ ρ,
where C is a constant independent of ρ.
Proof. Let
hn (x, t) = eLt [wnρ (x, t) − w n (x, t)].
By the assumption of Lemma 3.1, we know that hn (x, t) satisfies
hn t (x, t) − Ahn (x, t) − (L + cn (x, t))hn (x, t)
Lt
= e f (x, wn (x, t), uρ (x)) − f (x, w n (x, t), un (x))
(3.2)
1
+ ρ(v − v n (x, t)) ,
n
hn Γ = 0, hn (x, 0) = 0.
NECESSARY CONDITIONS FOR OPTIMAL CONTROL
249
Multiplying (3.2) by hn (x, t), and by integration by parts over Q, we
obtain
Z
Z
Z
1
2
2
h2n (x, t) dx dt
|∇hn (x, t)| dx dt +
h (x, 1) dx + λ
2 Ω n
Q
Q
Z
Z
1
≤ C (uρ (x) − un (x))2 dx +
h2n (x, t) dx dt
4
Ω
Q
Z
1
1
h2 (x, t) dx dt,
+ Cρ2 +
n
4 Q n
which implies inequality (3.1).
Lemma 3.2. Suppose the function zn (x, t) satisfies the equation
(3.3)
Then,
(3.4)
zn t (x, t) − Azn (x, t) − fy (x, w n (x, t), un (x))zn (x, t)
= f (x, w n (x, t), u(x)) − f (x, w n (x, t), un (x))
+
zn Γ = 0,
1
(v − v n (x, t)),
n
zn (x, 0) = 0.
wnρ (x, t) = wn (x, t) + ρzn (x, t) + rρ (x, t),
and
(3.5)
krρ k2,Q + k∇rρ k2,Q = o(ρ).
Proof. Let (1/ρ)rρ (x, t) = (1/ρ)(wnρ (x, t) − w n (x, t)) − zn (x, t). By the
linearity of f on u and the definition of uρ , we have
1
[f (x, w n , uρ ) − f (x, wn , un )] = f (x, w n , u) − f (x, wn , un ).
ρ
Let rρ (x, t) = (wnρ (x, t) − w n (x, t)) − ρzn (x, t). From (3.2) and (3.3) we
obtain
rρ t (x, t) − Arρ (x, t) − cn (x, t)rρ (x, t)
= ρ [cn (x, t) − fy (x, w n (x, t), un (x))]zn (x, t),
rρ (x, t)Γ = 0, rρ (x, t)(x, 0) = 0,
250
HANG GAO AND MICHAEL Y. LI
where
cn (x, t) =
Z
1
0
fy (x, w n (x, t) + τ (wnρ (x, t) − wn (x, t)), uρ (x))dτ.
Similar to the proof of Lemma 3.1, we can show
krρ k2,Q + k∇rρ k2,Q ≤ C ρ k[cn − fy ( · , w n , un )]zn ]k2,Q .
From Lemma 3.1 we know that
lim [cn (x, t) − fy (x, w n (x, t), un (x))] = 0.
ρ→0
Thus (3.5) holds and Lemma 3.2 is proved.
Theorem 3.1. Suppose (un , v n ) is the optimal control for (2.1)–(2.2)
and w n (x, t) is the corresponding optimal state. Then, there exists a
function ψn (x, t) that satisfies
ψn (x, t) − Aψn (x, t) − fy (x, w n (x, t), un (x))ψn (x, t)
t
(3.6)
= −2[wn (x, t) − B(x)],
ψn |Γ = 0, ψn (x, 1) = ψ(x),
and a function ξn (x), kξn k2,Ω ≤ 2C0 , such that, for any (u, v) ∈ U × V,
the following inequality holds
(3.7)
Z
Q
ψ(x, t)[f (x, w n (x, t), un (x)) − f (x, w n (x, t), u(x))] dx dt
+
Z
Ω
ξn (x)[un (x) − u(x)] dx
1
+
n
Z
Q
ψ(x, t)[v n (x, t) − v(x, t)] dx dt ≥ 0.
Here ψ(x) is given by (1.7) and C0 , as in (2.2), is determined in
Theorem 2.2.
Proof. From Theorem 2.2 we know that un is an interior point of Un .
For any u ∈ U and v ∈ V, there exists ρ0 > 0, such that for ρ ∈ (0, ρ0 )
and
uρ = un + ρ(u − un ),
vρ = v n + ρ(v − v n ),
NECESSARY CONDITIONS FOR OPTIMAL CONTROL
251
we have (uρ , vρ ) ∈ Un × V. Therefore
0 ≤ Fn (uρ , vρ ) − Fn (un , v n ).
(3.8)
Direct calculation leads to
(3.9)
Fn (uρ , vρ ) − Fn (un , v n )
Z
=
{[wnρ (x, t) − B(x)]2 − [wn (x, t) − B(x)]2 } dx dt
Q
−
Z
Ω
ψ(x)[wnρ (x, t) − w n (x, t)] dx
√
+ C0 n
Z
Ω
[(uρ (x) − u(x))2 − (un (x) − u(x))2 ] dx
= I1 − I2 + I3 ,
where
(3.10)
I1 =
Z
Q
[wnρ (x, t) − w n (x, t)]
× [wnρ (x, t) + wn (x, t) − 2B(x)] dx dt
=
Z
Q
[wnρ (x, t) − w n (x, t)][2(w n (x, t) − B(x))
+ ρzn (x, t) + rρ (x, t)] dx dt
=
Z
Q
=−
[wnρ (x, t) − w n (x, t)][2(w n (x, t) − B(x))] dx dt + o(ρ)
Z
Q
[wnρ (x, t) − wn (x, t)] [ψn t (x, t) − Aψn (x, t)
− fy (x, w n (x, t), un (x))ψn (x, t)] dx dt + o(ρ)
=
Z
Ω
ψ(x)[wnρ (x, t)(x, 1) − wn (x, t)(x, 1)] dx
−ρ
Z
Q
ψn (x, t)[zn t (x, t) − Azn (x, t)
− fy (x, w n (x, t), un (x))zn (x, t)] dx dt + o(ρ)
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HANG GAO AND MICHAEL Y. LI
Z
= I2 + ρ
ψn (x, t)[f (x, w n (x, t), un (x))
Q
− f (x, wn (x, t), u(x))
1
(v n (x, t) − v(x, t))] dx dt + o(ρ),
n
+
and
(3.11)
√
I3 = C 0 n
Z
√
Ω
= ρ2C0 n
=ρ
Z
where
Ω
[uρ (x) − un (x)][uρ (x) + un (x) − 2u(x)]dx
Z
Ω
[u(x) − un (x)] [un (x) − u(x)]dx + o(ρ)
ξn (x)[un (x) − u(x)]dx + o(ρ),
√
ξn (x) = 2C0 n (u(x) − un (x)),
and thus
√
kξn k2,Ω = 2C0 n ku − un k2,Ω ≤ 2C0 .
From (3.7)–(3.11) we obtain
(3.12)
0≤ρ
Z
ψn (x, t)[f (x, w n (x, t), un (x))
Q
− f (x, wn (x, t), u(x))] dx dt
Z
+ρ
ξn (x)[un (x) − u(x)] dx
Ω
1
+ ρ
n
Z
Q
ψ(x, t)[v n (x, t) − v(x, t)] dx dt + o(ρ).
Dividing inequality (3.12) by ρ and then taking a limit as ρ → 0, we
obtain inequality (3.7).
4 Proofs of the main results
proof of Theorems 1.1 and 1.2.
In this section, we complete the
Proof of Theorem 1.1. Let ψn (x) and ψ(x) be the solutions of the problems (3.6) and (1.7), respectively. We prove that ψn → ψ as n → ∞.
NECESSARY CONDITIONS FOR OPTIMAL CONTROL
253
From equations (3.6) and (1.7), we can obtain
− (ψn (x, t) − ψ(x))t − A(ψn (x, t) − ψ(x))
= fy (x, w n (x, t), un (x))ψn (x, t)
(4.1)
− fy (x, y¯(x), u(x))ψ(x) − 2[w n (x, t) − B(x)],
(ψn − ψ)|Γ = 0,
(ψn − ψ)(x, 1) = 0.
Let hn (x, t) = e−Lt [ψn (x, t) − ψ(x)]. From (4.1), we have
− hn t − Ahn + (L − fy (x, w n , un ))hn
= e−Lt [fy (x, w n (x, t), un (x))
(4.2)
− fy (x, y¯(x), u(x))]ψ(x) − 2e−Lt[w n (x, t) − B(x)],
hn |Γ = 0,
hn (x, 1) = 0.
According to the assumption (P2), we can choose L such that
L − fy (x, w n (x, t), un (x)) ≥ 2
for all n. Multiplying (4.2) by hn (x, t), and by integration by parts over
Q, we have
(4.3)
1
2
Z
Ω
h2n (x, 1) dx + λ
1
≤
2
Z
Ω
Z
Q
|∇hn (x, t)|2 dx dt + 2
Z
Q
Q
h2n (x, t) dx dt
{e−Lt [fy (x, w n (x, t), un (x))
− fy (x, y¯(x), u(x))]ψ(x)}2 dx +
+
Z
1
2
Z
Q
h2n (x, t) dx dt
[e−Lt (w n (x, t) − y¯n (x))]2 dx dt +
1
2
Z
Q
h2n (x, t) dx dt.
√
This
√ relation and the fact that kun − uk2,Ω ≤ 1/ n and kwn − y¯k2,Q ≤
C/ n lead to
lim {khn k2,Q + k∇hn k2,Q } = 0.
n→∞
This implies limn→∞ ψn (x, t) = ψ(x).
254
HANG GAO AND MICHAEL Y. LI
Since kξn k2,Ω ≤ C, there exists a subsequence {ξni } and a ξ ∈ L2 (Ω),
such that kξk2,Ω ≤ C, and ξni → ξ weakly in L2 (Ω). Setting n = ni and
letting ni → ∞ in (3.7), we obtain
(4.4)
Z
Ω
[H(x, u(x)) − H(x, u(x))] dx ≥ 0,
where
H(x, u) = ψ(x)f (x, y¯(x), u) + ξ(x)u.
For any Lebesgue point x0 of the function H(x, u(x)) − H(x, u), we
choose the function
u,
x ∈ Bρ (x0 ) ⊂ Ω,
u(x) =
.
u(x), x ∈ Ω \ B .
ρ
Here, Bρ (x0 ) = {x : kx − x0 k < ρ}. From (4.4) we have
(4.5)
Z
Bρ
[H(x, u(x)) − H(x, u)] dx ≥ 0.
Dividing (4.5) by ρ and then letting ρ → 0, we obtain
H(x, u(x0 )) − H(x, u) ≥ 0.
Since the set of Lebesgue point of the function H(x, u) is dense in Ω, we
have
H(x, u(x0 )) = max H(x, u) a.e. x ∈ Ω,
u∈U
completing the proof of Theorem 1.1.
Now, we turn to the proof of Theorem 1.2.
Proof of Theorem 1.2. For any (y, u) ∈ P, we have J(y, u) ≥ 0. There
exists a minimizing sequence {(yk , uk )} ⊂ P. From [10], we know that,
kyk k2,p,Ω ≤ C for all k. There thus exist y¯ ∈ W02,p (Ω) and a subsequence
{yki } of {yk }, such that yki → y¯ weakly in W 2,p (Ω) and yki → y¯ in
Lp (Q). Let
hki (x) = f (x, yki (x), uki (x)).
NECESSARY CONDITIONS FOR OPTIMAL CONTROL
255
It is clear that khki kp,Ω ≤ C. Thus, there exist a subsequence of {hki },
denoted by itself, and a function h ∈ W 2,p (Ω), such that hki → h weakly
in Lp (Ω). Then, for any ϕ ∈ Lq (Ω), where 1/p + 1/q = 1, we have
Z
Z
ϕ(x)A¯
y (x) dx = − lim
ϕ(x)Ayki (x) dx
−
(4.6)
ki →∞
Ω
= lim
ki →∞
=
Z
Z
Ω
ϕ(x)hki (x) dx
Ω
ϕ(x)h(x) dx.
Ω
Let
Ek (x) = {f (x, yk (x), u) : u ∈ U }
and
E(x) = {f (x, y¯(x), u) : u ∈ U }.
Then, hki (x) ∈ Eki (x), a.e. x ∈ Ω and as ki → ∞, we have Eki (x) →
E(x), a.e. x ∈ Ω.
Next, we prove that h(x) ∈ E(x), a.e. x ∈ Ω. In fact, since f (x, y, ·)
is continuous on U and U is a closed set, E(x) is a closed set. Suppose
it is not ture that h(x) ∈ E(x), a.e. x ∈ Ω, then, there exists a function
ϕ(x), such that
Z
(4.7)
[h(x) − f (x, y¯(x), u(x))]ϕ(x)dx ≥ C > 0, ∀u ∈ U.
Ω
Since hki → h weakly in Lp (Ω), and from (4.7), we have
Z
(4.8)
[hki (x) − f (x, y¯(x), u(x))]ϕ(x)dx ≥ C/2 > 0,
Ω
∀u ∈ U,
for sufficiently large ki . On the other hand, taking u(x) = uki (x) in (4.8),
we have
Z
[hki (x) − f (x, y¯(x), uki (x))]ϕ(x) dx
Ω
Z
≤
|fy (x, y¯(x) + τ (yki (x) − y¯(x)), uki (x))|
Ω
× |yki (x) − y¯(x)| |ϕ(x)| dx
≤ Ckyki − y¯kp,Ω kϕkq,Ω .
256
HANG GAO AND MICHAEL Y. LI
Since yki → y¯ in Lp (Ω), we have
Z
[hki (x) − f (x, y¯(x), uki (x))]ϕ(x)dx ≤ C
2
Ω
for sufficiently large ki . This contradicts (4.8). Therefore h(x) ∈ E(x)
a.e. x ∈ Ω
From the Selection Theorem [6, 11], we know that there exists a
control function u ∈ U, such that
h(x) = f (x, y¯(x), u(x)).
It follows from (4.6) that (¯
y (x), u(x)) satisfies the equation
(
−A¯
y(x) = f (x, y¯(x), u(x)),
y¯|∂Ω = 0
and
J(¯
y , u) =
Z
Ω
[¯
y (x) − B(x)]2 dx
= lim
ki →∞
Z
Ω
[yki (x) − B(x)]2 dx ≤ J(y, u),
∀(y, u) ∈ P,
namely,
J(¯
y , u) = inf{J(y, u) : (y, u) ∈ P},
completing the proof of Theorem 1.2.
Acknowledgments The research of HG is partially supported by
the National Natural Sciences Foundation of China. The research of
MYL is supported in part by the National Science Foundation grant
DMS 0078250 and by the Natural Sciences and Engineering Research
Council of Canada.
REFERENCES
1. H. Aman and M. G. Crandall, On some existence theorems for semi-linear
elliptic equations, Indiana Univ. Math. J. 27 (1978), 779–790.
NECESSARY CONDITIONS FOR OPTIMAL CONTROL
257
2. M. S. Berger, Nonlinear and Functional Analysis, Academic Press, New York,
1977.
3. P. A. Binding, P. Drabek, and Y. X. Huang, On Neumann boundary value
problems for some quasilinear elliptic equations, Electronic J. Diff. Eqn. 1997
(1997), 1–11.
4. J. F. Bonnans and E. Casas, A boundary Pontragin’s principle for the optimal
control of state constrained elliptic systems, Int. Ser. Numer. Math. 107 (1992),
241–249.
5. E. Casas and J. Yong, Maximum principle for state-constrained optimal control
problems governed by quasilinear elliptic equation, Diff. Int. Eqn. 8 (1995),
1–18.
6. L. Cesari, Optimization Theory and Applications, Springer-Verlag, Now York,
1983.
7. H. O. Fattorini and H. Frankowska, Necessary conditions for infinite dimensional control problems, Math. Control Signal Systems 4 (1991), 41–67.
8. H. Gao, Optimality condition of a class of distributed parameter system, Chinese Sci. Bul. 41 (1996), 1501–1505.
9. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd Edition, Springer-Verlag, 1983.
10. O. A. Ladyzhenskaya, V. A. Solunikov and N. N. Uralceve, Linear and Quasilinear Equations of Parabolic Type, Trans. Math. Monographs Vol. 23, Amer.
Math. Soc., Providence, 1968.
11. X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems,
Birkhauser, Boston, 1995.
12. J. Yong, Necessary condition for minimax control problems of second order
elliptic partial differential equations, Kodai Math. J. 16 (1993), 469–486.
Correspondending author: Michael Y. Li
Department of Mathematical Sciences, University of Alberta,
Edmonton, Alberta, Canada T6G 2G1
E-mail address: [email protected]
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