Theoretical Mathematics & Applications, vol.3, no.4, 2013, 1-22
ISSN: 1792-9687 (print), 1792-9709 (online)
Scienpress Ltd, 2013
Optimal control of some degenerate
pseudo-parabolic variational
inequality involving state constraint
Youjun Xu1 , Xiaoyou Liu2 and Liangbong Li3
Abstract
Necessary conditions for optimal controls of some degenerate pseudo
- parabolic variational inequality are established by some well-known
penalty arguments involving finite co-dimensionality of certain sets.
Some analysis on the state equations are given by the theory of maccretive operators in a Hilbert space.
Mathematics Subject Classification: 35K70; 35Q27; 49K20
Keywords: Pseudo-parabolic inequality; Optimal control; Variational inequality; State constraint
1
School of Mathematics and
Hunan 421001, P.R. China.
2
School of Mathematics and
Hunan 421001, P.R. China.
3
School of Mathematics and
Hunan 421001, P.R. China.
Physics, University of South China, Hengyang,
E-mail: youjunxu@163.com
Physics, University of South China, Hengyang,
E-mail: 327319671@qq.com
Physics, University of South China, Hengyang,
E-mail: 1870778604@qq.com
Article Info: Received : September 19, 2013. Revised : November 10, 2013.
Published online : December 16, 2013.
2
1
Optimal control of some degenerate pseudo-parabolic variational inequality
Introduction
We shall study the optimal control problems governed by nonlinear pseudoparabolic equations of the form:

 dM y + Ay + β(y) 3 Bu a.e in Q,
dt
(1)
 M 1/2 y(0) = M 1/2 y0 ,
with the state constraint
F (y) ⊂ S.
The pay-off functional is given by
Z T
L(y, u) =
[g(t, y) + h(u)]dt,
(2)
(3)
0
where Q = Ω × (0, T ), Ω ⊂ RN , is a bounded domain with smooth boundary.
For the data in (1)-(3), we have the following assumptions.
(H1) M is a nonnegative selfadjoint operator in H = L
à 2 (Ω) with D(M ) ⊂
D(A + β).
(H2) V ⊂ H is a real Hilbert space such that V is dense in H and V ⊂ H ⊂ V 0
algebraically and topologically, where V 0 is the dual of V . Further, the injection
of V into H is compact.
A : V → V 0 is a linear continuous and symmetric operator from V to V 0
satisfying the coercivity condition
(Ay, y) ≥ w k y k2V +α | y |2H f or all y ∈ V,
(4)
where w > 0 and α ∈ R.
(H3) β is a maximal monotone graph in R × R with 0 ∈ β(0). Let φ(y) :
H → R = (−∞, +∞] be the lower semicontinuous convex function defined by
R
φ(y) = Ω j(y)dx, where j : R → R̄ is such that ∂j = β. Moreover,
(Ay, β² (y)) ≥ 0 ∀ y ∈ D(A), ² > 0,
(5)
where β² (r) = ²−1 (r − (1 + ²β)−1 r) for all ² > 0, r ∈ R. For every ξ ∈ β, there
exists a constant c > 0 such that
| ξ(u) |≤ c(1+ | u |p+1 ), 0 ≤ p ≤ 2/(N − 2) if N > 2; 0 ≤ p < +∞ if N = 1, 2.
(6)
Youjun Xu, Xiaoyou Liu and Liangbong Li
3
(H4) B is a linear continuous operator from a real Hilbert space U to H.
The norm and the scalar product of U will be denoted by | · |U and h·, ·i,
respectively. Let H be a real Hilbert space with the inner product (·, ·) and
the norm | · |.
(H5) Let Z be a Banach space with the dual Z ∗ strictly convex. S ⊂ Z
is a closed convex subset with finite codimensionality (cf.[22, 4, 5]). F :
L2 (0, T ; V ) → Z is of class C 1 .
(H6) The functional h : U → R is convex and lower semicontinuous (l. s. c),
such that
h(u) ≥ c1 | u |2U +c2 , ∀ u ∈ U,
where c1 > 0, c2 ∈ R.
(H7) g : [0, T ] × H ⊆ R+ is measurable in t, and for every δ > 0, there exists
Lδ > 0 independent of t such that g(t, 0) ∈ L∞ (0, T ) and
| g(t, y1 ) − g(t, y2 ) |≤ Lδ | y1 − y2 |H f or all t ∈ [0, T ], | y1 |H + | y2 |H ≤ δ.
Remark 1.1. From a perturbation theorem for m-accretive operators ([8]
Lemma 5) and (H2), (H3), we easily know that A+∂j is m-accretive in H ×H,
we may write (1) as

 dM y + ∂ϕ(y(t)) 3 Bu a.e t ∈ (0, T ),
dt
(7)
 M 1/2 y(0) = M 1/2 y0 ,
where ϕ(y) = 21 (Ay, y) + φ(y), y ∈ V.
Remark 1.2. M is not necessarily invertible and initial condition should not
be given in the form
y(0) = y0 .
Since the continuity with respect to t of the solution y(·, t) can not be expected.
As we know, by Barbu [19, 20] (see Chapter 4) and Theorem 1 of [17], we
easily obtain.
4
Optimal control of some degenerate pseudo-parabolic variational inequality
Lemma 1.3. Let (H1) − (H4) hold. Then, for any y0 ∈ D(M ) ∩ V, u ∈
L2 (0, T ; U ), (1) admits a unique solution y(x, t) satisfying
y ∈ L∞ (0, T ; D(M ) ∩ V ), M y, M 1/2 y ∈ AC([0, T ]; H),
(d/dt)M y, (d/dt)M 1/2 y ∈ L2 (0, T ; H).
Our optimal control problem is stated as follows.
Problem (P ). Find a (ȳ, ū) ∈ Aad such that
L(ȳ, ū) =
inf
(y,u)∈Aad
L(y, u).
(8)
Here Aad = {(y, u) ∈ (L∞ (0, T ; D(M ) ∩ V ) × L2 (0, T ; U ))| y is the solution
of (7) with (2) }. Any u(·) ∈ Aad satisfying (2) is called an optimal control,
and the corresponding ȳ(·) ≡ y(·; ū) is called an optimal state. The pair (ȳ, ū)
is called an optimal pair.
(1) cover cases of the following type: surface waves of long wavelength in
liquids, acoustic-gravity waves in compressible fluids, hydromagnetic waves in
cold plasma, acoustic waves in anharmonic crystals, etc. The wide applicability of these equations is the main reason why, during the last decades, they
have attracted so much attention from mathematicians. Recently, some optimal control problems governed by pseudoparabolic equations have already
been discussed. Linear optimal control problems for pseudoparabolic equations were considered by S. I. Lyashko [21, 15, 7, 18, 12, 13, 9]. However, these
problems studied in [15, 7, 18, 12, 13, 9] do not involve state constraints and
maximal monotone graph. On the other hand, optimal control governed by
some parabolic variational inequalities (cf. [11, 14, 10, 3, 23, 19, 20, 1]) have
already been discussed. Li and Yong [22] studied the maximal principle for optimal control governed by some nonlinear parabolic equations with two point
boundary (time variable) state constraints. In Cases’ work [2], the state constraint was considered, but the state equation did not involve monotone graph.
He [25] studied the optimal control problems involving some special maximal
monotone graph (Lipschitz continuous) with state constraint. Wang [4, 5] also
discussed the optimal control problem governed by the state equation involving some maximal monotone graph. Xu [24] have established the maximal
principle for optimal control to problem (P ) under the following conditions on
M and A + β:
Youjun Xu, Xiaoyou Liu and Liangbong Li
5
D(M ) ⊂ D(A + β) and M has the bounded inverse.
The present work in this paper is concerned with the optimal control problem governed by the pseudoparabolic equations where the operator M is not
necessarily invertible. The method in [4, 5, 19] can not be used directly. The
methods which be used primarily here are the theory of m-accretive operators
in a Hilbert space. Especially the properties of subdifferentials will play an
essential role in the approximating control process.
The plan of this paper is as follows. Section 2 gives an approximating
control process. In section 3, we state and prove the necessary conditions on
optimality for the problem (P ).
2
The approximating control process
Let (y ∗ , u∗ ) be optimal for the problem (P ). Then

 dM y∗ + Ay ∗ + ∂φ(y ∗ (t)) 3 Bu∗ a.e t ∈ (0, T ),
dt
 M 1/2 y ∗ (0) = M 1/2 y0 ,
(9)
with
F (y ∗ ) ∈ S,
and
L(y ∗ , u∗ ) = inf L(y, u) over (y, u) ∈ Aad .
Now consider the following approximating equations for (7):

 (² + M ) dy + C J M y = Bu a.e in Q,
² ²
dt
 y(0) = y0 ,
(10)
where C = ∂ϕ, C² = ²−1 (I − J²C ) and J²C = (I + ²C)−1 .
In virtue of (H1), for every fixed ² > 0, it is seen that the operator
P² (t) ≡ (² + M )−1 C² J²M is locally Lipschitz continuous in H and the mapping t → P² (t)y is strongly continuous on [0, T ] for every u ∈ H. Then we
easily obtain that any y0 ∈ D(M ) ∩ V , u ∈ L2 (0, T ; U ), (10) has a unique
solution W 1,2 ([0, T ]; H) ∩ C([0, T ]; H) ∩ L2 (0, T ; V ) and dM y/dt = M (dy/dt).
We also have the following result on (10).
6
Optimal control of some degenerate pseudo-parabolic variational inequality
Lemma 2.1. For ² > 0 given, un ∈ L2 (0, T ; U ), un → ũ weakly in L2 (0, T ; U ).
Then there exists some subsequence of {yn }, still denoted by itself, such that
yn → ỹ strongly in C([0, T ]; H), where ỹ, yn is the solutions of (10) corresponding to ũ and un , respectively.
Proof. Multiplying (10) by J²M yn (t) and using the selfadjointness of M , we see
²(yn0 (t), J²M yn (t)) +
d
| M²1/2 yn (t) |2 +2(C² J²M yn (t), J²M yn (t))
dt
= 2(Bun , J²M yn (t)).
From the identity J²M w = ²C² J²M w + J²C J²M w yields
d
d
| (J²M )1/2 yn (t) |2 + | M²1/2 yn (t) |2 +² | C² J²M yn (t) |2 +c | J²C J²M yn (t) |2
2dt
dt
1
1
≤ c | un |2U + ² | C² J²M yn (t) |2 + c | J²C J²M yn (t) |2 ,
4
4
²
where M² = ²−1 (I − J²M ). It is well known that M² is m-accretive on H and
Lipschitz continuous with 1/² as a Lipschitz constant and | M² y |≤| M 0 y |
with M² y → M 0 y as ² → 0 for every y ∈ D(M ). For an m-accretive operator
M , M y(y ∈ D(M )) is closed and convex; so M y has a unique element of the
least norm which is denoted by M 0 y. Integrating the above inequality from 0
to T , we see
² | (J²M )1/2 yn (t) |2C([0,T ];H) + | M²1/2 yn (t) |C([0,T ];H)
(11)
+ ²1/2 | C² J²M yn (t) |L2 (0,T ;H) + | J²C J²M yn |L2 (0,T ;H) ≤ c.
Multiplying (10) by J²M yn0 (t), we see
²(yn0 (t), J²M yn0 (t))+ | M²1/2 yn0 (t) |2 +
d
ϕ² (J²M yn (t)) ≤ (Bun , J²M yn0 (t)).
dt
Here we have used the fact (C² J²M yn (t), J²M yn0 (t)) =
²(yn0 (t), J²M yn0 (t))+ | M²1/2 yn0 (t) |2 +
d
ϕ (J M yn (t)),
dt ² ²
thus
d
ϕ² (J²M yn (t))
dt
1
≤ c | (J²M )1/2 Bun |2 + ² | (J²M )1/2 yn0 (t) |2
4
1
≤ c | un |2U + ² | (J²M )1/2 yn0 (t) |2 .
4
Integrating the above inequality from 0 to T , and ϕ² is Fréchet differentiable
on H(see [4]), we see
² | (J²M )1/2 yn0 (t) |L2 (0,T ;H) + | M²1/2 yn0 (t) |L2 (0,T ;H) +ϕ² (J²M yn (t)) ≤ c.
(12)
Youjun Xu, Xiaoyou Liu and Liangbong Li
7
On the other hand, from (4) and (H3), we get
c | J²C J²M yn (t) |2 ≤ ϕ(J²C J²M yn (t)) ≤ ϕ² (J²M yn (t)) f or any t ∈ [0, T ].
(13)
and
| J²M yn (t) |≤ ² | C² J²M yn (t)) | + | J²C J²M yn (t) | .
(14)
This implies
Z
yn0 (t) |L2 (0,T ;H) ≤
²|
T
²
0
(yn0 (t), J²M yn0 (t))dt + ²2 | M²1/2 yn0 (t) |L2 (0,T ;H) ≤ c. (15)
Multiplying (10) by M² J²M yn (t), we see
2²(yn0 (t), M² J²M yn (t)) +
d
| M² yn (t) |2 +2(C² J²M yn (t), M² J²M yn (t))
dt
= 2(Bun , M² J²M yn (t)),
(16)
thus
d
d
| M 1/2 J²M yn (t)) |2 + | M² yn (t) |2 ≤ c+ | M² yn (t) |2 ,
dt
dt
Integrating this inequality over (0, t], t ≤ T and using Gronwall’s inequality,
yields
| M² yn (t) |L2 (0,T ;H) ≤ c ∀ t ∈ [0, T ].
(17)
²
Hence we get
| yn (t) |≤ ² | M² yn (t) | + | J²M yn (t) |≤ c ∀ t ∈ [0, T ].
(18)
From (H2) and (H3), it follows that
| C² J²M yn (t) |L2 (0,T ;H) ≤ c,
then in view of (10) and the above inequality give
|M
d
yn (t) |L2 (0,T ;H) ≤ c,
dt
(19)
which implies
| M yn (t) |≤ c f or all t ∈ [0, T ],
and
|
d
yn (t) |L2 (0,T ;H) ≤ c.
dt
(20)
(21)
8
Optimal control of some degenerate pseudo-parabolic variational inequality
For every m, n > 0
²
d
d
| ym − yn |2 + | M 1/2 (ym − yn ) |2 +2(C² J²M (ym − yn ), ym − yn )
dt
dt
≤ c | um − un |2U +² | ym − yn |2 ,
by some calculations, we see
Z
2
1/2
2
t
(ym − yn )(t) | +
| (ym − yn ) |2 dτ
0
Z t
≤c
| um − un |2U dτ f or all t ∈ [0, T ].
² | (ym − yn )(t) | + | M
0
Hence {M 1/2 yn } and {yn } are Cauchy sequences in C([0, T ]; H) ∩ L2 (0, T ; H).
By (H2), then there exists a function ỹ ∈ C([0, T ]; D(M 1/2 )) such that as
n→∞
yn → ỹ strongly in C([0, T ]; H),
M 1/2 yn → M 1/2 ỹ strongly in C([0, T ]; H).
This completes the proof.
Next, we recall the approximation g ² of g and h² of h as follows. For the
details, we refer to [4, 5, 6, 19]. Let
Z
²
g (t, y) =
g(t, PN y − ²ΛN s)ρ(s)ds, ² > 0
RN
where ρ is a mollifier in RN , N = [²−1 ]. PN : L2 → XN is the projection of
L2 (Ω) on XN which is the finite dimensional space generated by {ei }N
i=1 , where
∞
2
N
{ei }i=1 is an orthonormal basis in L (Ω). ΛN : R → XN is the operator
P
defined by ΛN (s) = N
i=1 si ei , s = (s1 , · · ·, sN ).
Let h² : U → R be defined by
h² (u) = inf
n k u − v k2 2
L (0,T ;U )
2²
o
+ h(v) : v ∈ U , ² > 0.
Now we define the penalty L² : L2 (0, T ; U ) → R by
Z
T
L² (u) =
0
[g ² (t, y² ) + h² (u)]dt +
1
1
| u − u∗ |2L2 (0,T ;U ) + 1/2 [²1/2 + dS (F (y² ))]2 ,
2
2²
(22)
Youjun Xu, Xiaoyou Liu and Liangbong Li
9
where y² is the solution of (10). dS (F (y² )) denotes the distance of F (y² ) to S.
The approximating optimal control problems are as follows:
(P ² )
Minimize L² (u) over u ∈ L2 (0, T ; U ).
From Lemma 2.1, we easily obtain the following existence of optimal solutions for (P ² ) (see [4, 5, 19]).
Theorem 2.2. Problem (P ² ) has at least one optimal solution.
The following results are useful in discussing the approximating control
problems.
Lemma 2.3. Let u² → u weakly in L2 (0, T ; U ) as ² → 0. Then there exists a
subsequence {y² }²>0 , still denoted it self
y² → y strongly in C([0, T ]; H),
as ² → 0. where y² is the solutions of (10) corresponding to u² and y is the
solutions of (7) corresponding to u.
Proof. Rewrite (10) as follows:

 (² + M ) dy² (t) + C J M y (t) = Bu (t) a.e in Q,
² ²
²
²
dt
 y² (0) = y0 .
(23)
Multiplying (23) by J²M y² (t), we see
2²(y²0 (t), J²M y² (t)) +
d
| M²1/2 y² (t) |2 +2(C² J²M y² (t), J²M y² (t))
dt
= 2(Bu² , J²M y² (t)),
from the identity J²M y² (t) = ²C² J²M y² (t) + J²C J²M y² (t) and (H3) yield
(C² J²M y² (t), J²M y² (t)) ≥ ² | C² J²M y² (t) |2 +c | J²C J²M y² (t) |2 ,
thus,
d
d
| M²1/2 y² (t) |2 +² (y² (t), J²M y² (t)) + 2² | C² J²M y² (t) |2 +2c | J²C J²M y² (t) |2
dt
dt
1
M 1/2
≤ ² | (J² ) y² (t) |2 +c | (J²M )1/2 Bu² |2 .
4
10
Optimal control of some degenerate pseudo-parabolic variational inequality
Integrating the above inequality from 0 to t(t ∈ (0, T ]), we have
| M²1/2 y² (t) |C(0,T ;H) ≤ c,
(24)
²1/2 | C² J²M y² (t) |L2 (0,T ;H) ≤ c, | J²C J²M y² (t) |L2 (0,T ;H) ≤ c,
(25)
and
which implies
| J²M y² (t) |L2 (0,T ;H) ≤ c.
(26)
Multiplying (23) by J²M y²0 (t), we see
d
ϕ² (J²M yn (t))
dt
1
≤ (Bu² , J²M y²0 (t)) ≤ c | (J²M )1/2 Bu² |2 + ² | (J²M )1/2 y²0 |2 .
4
²(y²0 (t), J²M y²0 (t))+ | M²1/2 y²0 (t) |2 +
Here we have used the fact (C² J²M y² (t), J²M y²0 (t)) = dtd ϕ² (J²M y² (t)).
Integrating the above inequality over (0, T ], and ϕ² is Frechet differentiable
on H(see [4]) we see
Z T
1/2 0
M
| M² y² (t) |L2 (0,T ;H) ≤ c, ϕ² (J² y² (t)) ≤ c, ²
(y²0 (t), J²M y²0 (t))dt ≤ c. (27)
0
This implies
Z
²|
y²0 (t) |L2 (0,T ;H) ≤
T
²
0
(y²0 (t), J²M y²0 (t))dt + ²2 | M²1/2 y²0 (t) |L2 (0,T ;H) ≤ c. (28)
Multiplying (23) by M² J²M y² (t), we see
2²(y²0 (t), M² J²M y² (t)) +
d
| M² y² (t) |2 +2(C² J²M y² (t), M² J²M y² (t))
dt
= 2(Bu² , M² J²M y² (t)),
thus
²
d
d
| M 1/2 J²M y² (t)) |2 + | M² y² (t) |2 ≤ c | Bu² |2 + | M² y² (t) |2 ,
dt
dt
Integrating this inequality over (0, t], t ≤ T and Gronwall’s inequality yield
| M² y² (t) |≤ c.
(29)
| y² (t) |≤ ² | M² y² (t) | + | J²M y² (t) |≤ c ∀ t ∈ [0, T ].
(30)
Hence we get
11
Youjun Xu, Xiaoyou Liu and Liangbong Li
In view of (23) and (28), we finally obtain
| M y²0 (t) |L2 ([0,T ];H) ≤ c,
| M y² (t) |≤ c ∀ t ∈ [0, T ].
(31)
For every m, n > 0, we have
d
| M 1/2 (y²m − y²n ) |2 +(C²m J²Mm y²m
2dt
−C²n J²Mn y²n , y²m − y²n )
1
≤ c | B(u²m − u²n ) |2 + ² | y²m − y²n |2 ,
4
(²m y²0 m − ²n y²0 n , y²m − y²n ) +
Using the identities w = J²Mm w + ²m M²m w f or every w ∈ H, etc., we see
(C²m J²Mm y²m − C²n J²Mn y²n , y²m − y²n )
= (C²m J²Mm y²m − C²n J²Mn y²n , J²Cm J²Mm y²m − J²Cm J²Mn y²n )
+ (C²m J²Mm y²m − C²n J²Mn y²n , ²m C²m J²Mm y²m − ²n C²m J²Mn y²n )
+ (C²m J²Mm y²m − C²n J²Mn y²n , ²m M²m y²m − ²n M²n y²n )
≥ c | J²Cm J²Mm y²m − J²Cm J²Mn y²n |2 −c(²m + ²n )
≥ c | y²m − y²n |2 −c(²m + ²n ).
Here using the estimate
| y²m − y²n |≤| J²Mm y²m − J²Mn y²n | +²m | M²m y²m | +²n | M²n y²n | .
From (28) and (30) we see
Z
T
|
0
1/2
(²m y²0 m − ²n y²0 n , y²m − y²n )dt |≤ c(²1/2
m + ²n ).
Hence combining them, we get
Z t
Z
1/2
2
2
| M (y²m − y²n ) | +
| y²m − y²n | ds ≤ c
0
t
0
| (u²m − u²n ) |2U ds
1/2
+c(²1/2
m + ²n ).
Combining this with (H2), we know that {M 1/2 y²n } and {y²n } are Cauchy
sequences in C([0, T ]; H) ∩ L2 (0, T ; V ). Then there exists a function
y ∈ C([0, T ]; D(M 1/2 ))
such that as n → ∞, ²n → 0,
12
Optimal control of some degenerate pseudo-parabolic variational inequality
y²n → y strongly in C([0, T ]; H) ∩ L2 (0, T ; V ),
M 1/2 y²n → M 1/2 y strongly in C([0, T ]; H).
J²Mn y²n → y strongly in C([0, T ]; H),
(32)
²n y²0 n → 0 weakly in L2 (0, T ; H).
(33)
Note that
M 1/2 J²Mn y²n → M 1/2 y strongly in L2 (0, T ; H),
(34)
Indeed, we see
| M 1/2 J²Mn y²n (t) − M 1/2 y |2 (t) ≤ 2 | M 1/2 (J²Mn y²n (t) − y²n )(t) |2
+ | M 1/2 (y²n (t) − y(t)) |2
= −2²n (M²n y²n (t), M²n y²n (t) − M y²n (t))+ | M 1/2 (y²n (t) − y) |2
≤ c²n + | M 1/2 (y²n (t) − y) |2 → 0,
for all t ∈ [0, T ]. In virtue of (28), (30) and (31), weakly closedness of d/dt and
β, it is shown that
dy²n /dt → dy/dt weakly in L2 (0, T ; H),
M dy²n /dt → M dy/dt weakly in L2 (0, T ; H).
(35)
(36)
Therefore, y ∈ AC(0, T ; D(M )) and dy/dt ∈ L2 (0, T ; D(M )). We easily see
that y(t) ∈ D(C) a.e. t ∈ (0, T ) and there exists a function ξ ∈ L∞ (0, T ; H)
such that as ²n → 0,
C²n J²Mn y²n → ξ weakly star in L∞ (0, T ; H),
and ξ(t) ∈ Cy(= Ay + β(y)) a.e. t ∈ (0, T ). From (32), we have M 1/2 y(0) =
M 1/2 y0 . Thus letting ² → 0 in (23), it follows that

 dM y(t) + Ay + ξ(t) = Bu(t) a.e in Q,
dt
(37)
 M 1/2 y(0) = M 1/2 y0 .
Youjun Xu, Xiaoyou Liu and Liangbong Li
13
Lemma 2.4. Let y0 ∈ D(M ) ∩ V, u ∈ L2 (0, T ; U ), then y² → y strongly in
C([0, T ]; H) ∩ L2 (0, T ; V ) as ² → 0. where y² is the solutions of (10) corresponding to u and y be the solutions of (1) corresponding to u with the initial
condition M 1/2 y(0) = M 1/2 y0 . Furthermore,
| M 1/2 (y² − y) |C([0,T ];H) + | y² − y |L2 (0,T ;H) ≤ c²1/2 .
(38)
Proof. By the same argument in the proof of Lemma 2.3 we have y² →
y strongly in C([0, T ]; H). We have for all ² and λ,

 d (²y − λy ) + dM (y² (t)−yλ (t)) + C J M y (t) − C J M y (t) = 0 a.e in Q,
²
λ
² ²
²
λ λ λ
dt
dt
 y² (0) − yλ (0) = 0.
(39)
Multiplying (39) by y² (t) − yλ (t), we have
d | M 1/2 (y² (t) − yλ (t)) |2
dt
+2(C² J²M y² (t) − Cλ JλM yλ (t), y² (t) − yλ (t)) = 0.
2(²y²0 − λyλ0 , y² − yλ )(t) +
(40)
From (H2), (H3) and (30), the third term of the left hand side of (40), using
the identities w = J²M w + ²M² w f or every w ∈ H, etc., we see
(C² J²M y² − Cλ JλM yλ , y² − yλ )
= (C² J²M y² − Cλ JλM yλ , J²C J²M y² − J²C JλM yλ )
+ (C² J²M y² − Cλ JλM yλ , ²C² J²M y² − λC² JλM yλ )
+ (C² J²M y² − Cλ JλM yλ , ²M² y² − λMλ yλ )
≥ c | J²C J²M y² − J²C JλM yλ |2 −c(² + λ)
≥ c | y² − yλ |2 −c(² + λ).
From (28) and (30) we see
Z
T
|
0
(²y²0 − λyλ0 , y² − yλ )dt |≤ c(²1/2 + λ1/2 ).
Hence combining them, we get
Z T
1/2
2
| M (y² − yλ ) | +
| y² − yλ |2 ds ≤ c(²1/2 + λ1/2 ).
0
Letting λ → 0 in the above inequality, we get (38).
14
Optimal control of some degenerate pseudo-parabolic variational inequality
Lemma 2.5. Let u² be optimal for the problem (P ² ) and y² be the solution of
(10) corresponding to u² . For ² → 0, then
y² → y∗ strongly in C([0, T ]; H) ∩ L2 (0, T ; V ).
u² → u∗ strongly in L2 (0, T ; U ).
y² be the solutions of (10) corresponding to u² , y∗ be the solutions of (1) corresponding to u∗ with the initial condition M 1/2 y(0) = M 1/2 y0 .
Proof. For any ² > 0, we have
Z
T
L² (u² ) ≤ L² (u∗ ) =
[g ² (t, y² ) + h² (u∗ )]dt +
0
1
[²1/2 + dS (F (y² ))]2 .
1/2
2²
By Lemma 2.3, y² → y∗ strongly in C([0, T ]; H), We have
g ² (t, y² ) → g(t, y∗ ) f or all t ∈ [0, T ],
and
h² (u² ) → h(u∗ ).
So
Z
T
lim
²→0
Z
g (t, y² )dt =
0
Z
T
²
g(t, y∗ )dt, lim
²→0
0
Z
T
T
h² (u∗ )dt =
0
h(u∗ )dt.
0
Similarly, By (38) and (H5), we see
1
1
[²1/2 +dS (F (y² ))]2 ≤ [²1/2 + k F (y² )−F (y∗ ) kZ ]2 ≤ c²1/2 → 0 as ² → 0,
1/2
2²
2²
thus,
lim sup L² (u² ) ≤ L(u∗ ).
(41)
²→0
On the other hand, since {u² }²>0 is bounded in L2 (0, T ; U ), there exists u1 ∈
L2 (0, T ; U ) such that, on some subsequence {²}²>0 , still denoted by itself, as
² → 0,
u² → u1 weakly in L2 (0, T ; U ),
and so, by Lemma 11,
y² → y1 = y(u1 ) strongly in C([0, T ]; H).
15
Youjun Xu, Xiaoyou Liu and Liangbong Li
By (41), one can check easily that
1
[²1/2 + dS (F (y² ))]2 ≤ c.
2²1/2
Thus, dS (F (y² )) → 0 as ² → 0. Since S is closed and convex, F (y1 ) =
RT
lim F (y² ) ∈ S. Since the function u → 0 h(u)dt is weakly lower semicon²→0
tinuous on L2 (0, T ; U ), we see
lim inf L² (u² ) ≥ L(u1 ) ≥ L(u∗ ).
²→0
together with (41), yields
lim L² (u² ) = L(u∗ ).
²→0
Arguing as in the proof of Lemma 11, we have M 1/2 y² (0) → M 1/2 y∗ (0) in H.
Hence y1 = y∗ , u1 = u∗ . This completes the proof.
3
Necessary conditions on optimality
Let ∂g be the generalized gradient of y → g(t, y) and ∂h be the subdifferential of h (see [4, 5, 19]). Y ∗ = (H s (Ω))0 + V 0 is the dual of Y = H s (Ω) ∩ V
with s > N/2.
Firstly, we consider the Cauchy problem

 (² + M ) dp² − Ap − β̇ ² (y )p − [F 0 (y )]∗ ξ = λ ∇g ² (t, y ) in (0, T ),
²
² ²
²
²
²
²
dt
 p² (T ) = 0,
(42)
where β̇ = (β ) . β² = ² (I −(I +²β) ), β = −∞ [β² (r−² θ)−β² (−² θ)]ρ(θ)
+β² (0) and ρ is a C0∞ -mollifier on R.
²
² 0
−1
−1
²
R∞
2
2
Lemma 3.1. Problem (42) has a unique absolutely continuous function p² ∈
L2 (0, T ; V ) ∩ C([0, T ]; H) with p0² ∈ L2 (0, T ; V 0 ), such that
Z T
1/2
2
2
k p² (t) k2V dt ≤ c ∀ ² > 0, t ∈ [0, T ], (43)
² | p² (t) |2 + | M p² (t) |2 +
0
Z
| p² β̇ ² (y² ) | dxdt ≤ c ∀ ² > 0.
Q
(44)
16
Optimal control of some degenerate pseudo-parabolic variational inequality
Proof. From (H1)−(H3), β̇ ² (y² ) ≥ 0, it is see that C = (²+M )−1 (A+ β̇ ² (y² )) :
V → V 0 is demicontinuous monotone operator that satisfies
(Cω, ω) ≥ w k ω kp +c ∀ω ∈ V,
k Cω k∗ ≤ c(1+ k ω kp−1 ),
where w > 0 and p ≥ 2. It follows by Theorem 1.90 of [19] that (42) has a
unique solution p² ∈ L2 (0, T ; V ) ∩ C([0, T ]; H) with p0² ∈ L2 (0, T ; V 0 ). Multiplying (42) by p² (t) and integrating over [t, T ], we see
Z T
2
1/2
2
² | p² (t) | + | M p² (t) | +
k p² (s) k2V ds ≤ c.
t
Thus we obtain (43). Multiplying (42) by ζ(p² ) and integrating on Q, where
ζ is a smooth monotonically increasing approximation of the sign function such
that ζ(0) = 0. For instance
Z ∞
ζ = ζλ (r) =
(ζλ (r − λθ) − ζλ (−λθ))ρ(θ)dθ,
−∞
where ζλ (r) = r | r |−1 for | r |≥ λ, ζλ (r) = λ−1 r for | r |< λ, and ρ is a C0∞ −
mollifier. Then (Ap² (t), ζ(p² (t))) ≥ 0, therefore,
Z
Z
²
β̇ (y² )ζ(p² )p² dxdt ≤
| ∇y g ² (t, y² )ζ(p² ) | dxdt, ∀ ² > 0.
Q
Q
Then, letting ζ tend to the sign function, we get (44).
We state the main results of the necessary conditions on optimality as
follows.
Theorem 3.2. Suppose that (H1) − (H7) hold. Let (y∗ , u∗ ) be an optimal pair
of problem (P ). Then there exist the function p ∈ L∞ (0, T ; H) ∩ L2 (0, T ; V ) ∩
BV ([0, T ]; Y ∗ ), the measure µ ∈ (L∞ (Q))0 and λ0 ∈ R, ξ0 ∈ Z ∗ satisfying


d
M p − Ap − µ − [F 0 (y∗ )]∗ ξ0 ∈ L∞ (0, T ; H),
dt
d
M p(t)
dt
− Ap(t) − µ − [F 0 (y∗ )]∗ ξ0 ∈ λ0 ∂g(t, y∗ ) a.e. in (0, T ),
 M 1/2 p(T ) = 0,
(45)
(46)
hξ0 , w − F (y∗ )i ≤ 0 f or all w ∈ S,
(47)
B ∗ p ∈ λ0 ∂h(u∗ )(t), a.e. t ∈ (0, T ),
(48)
(λ0 , ξ0 ) 6= 0.
(49)
17
Youjun Xu, Xiaoyou Liu and Liangbong Li
Proof. Since (y² , u² ) is optimal for problem (P ² ), we see
L² (uρ² ) ≥ L² (u² ) f or any ρ > 0, v ∈ L2 (0, T ; V ),
Here uρ² = u² + ρv. Thus
L² (uρ² ) − L² (u² )
≥ 0.
ρ
(50)
By some calculation, we have
Z T ²
Z T
g (t, y²ρ ) − g ² (t, y² )
lim
=
(∇g ² (t, y² ), z² )dt,
ρ→0 0
ρ
0
where z² ∈ C([0, T ]; H) ∩ L2 (0, T ; V ) ∩ W 1,2 ([0, T ]; H) is the solution to the
linear equation

 (² + M ) dz + Az + β̇ ² (y )z = Bv in (0, T ),
²
dt
(51)
 z(0) = 0,
Thus, we also have
Z T
hZ T
i
²
λ²
h∇g (t, y² ), z² idt +
h∇h² (u² ), vidt + hξ² , F 0 (y² )z² i
0
0
Z T
≥
hu∗ − u² , vidt.
(52)
0
where
²1/2
λ² =
, ξ² =
dS (F (y² )) + ²1/2
(
∇dS (F (y ² )), if F (y ² ) ∈
/ S,
0
otherwise,
and ξ² ∈ ∂dS (F (y ² )). Since S is convex and closed, we see
(
1, if F (y ² ) ∈
/ S,
k ξ² kZ ∗ =
0
otherwise.
and
1 ≤ ϕ2² + k ξ² k2Z ∗ ≤ 2.
Thus, we see
λ² → λ0 , ξ² → ξ0 weakly in Z ∗ .
It follows from lemma 2.5 that y² → y∗ strongly in C([0, T ]; H) ∩ L2 (0, T ; V ).
18
Optimal control of some degenerate pseudo-parabolic variational inequality
Now, since {Ap² } is bounded in L2 (0, T ; V 0 ) and β̇ ² (y² )p² is bounded in
L1 (0, T ; L1 (Ω)), [F 0 (y² )]∗ ξ² is bounded in L2 (0, T ; V 0 ), we may infer that {p² } is
bounded in L1 (0, T ; L1 (Ω)+V 0 ) and so, {(²+M )p0² } is bounded in L1 (0, T ; Y ∗ ).
Since the injection of H into Y ∗ is compact and the set {p² } is bounded in H
for any t ∈ [0, T ]. By the same arguments as those in [19, 4, 5], there exists
p ∈ L∞ ([0, T ]; H) ∩ L2 (0, T ; V ) ∩ BV ([0, T ]; Y ∗ ) and µ ∈ (L∞ (Q))∗ such that,
on some subsequence ², still denoted itself
p² (t) → p strongly in Y ∗ , ∀ t ∈ [0, T ].
Here BV ([0, T ]; Y ∗ ) is the space of all Y ∗ -valued functions p : [0, T ] → Y ∗ with
bounded variation on [0, T ].
On the other hand, by (43) we see
p² → p weakly star in L∞ (0, T ; H), weakly in L2 (0, T ; V ).
(53)
Note that V ,→ H is compact, for every λ > 0 there is δ(λ) > 0 such that
| p² (t) − p(t) |2 ≤k p² (t) − p(t) kV +δ(λ) k p² (t) − p(t) kY ∗ ∀ t ∈ [0, T ].
This yields
p² → p strongly in L2 (0, T ; H),
(54)
p² (t) → p(t) weakly in H ∀ t ∈ [0, T ].
(55)
and
Moreover, by (44) we infer that there is µ ∈ (L∞ (Q))∗ such that, on some
generalized subsequence ²,
β̇ ² (y² )p² → µ weakly star in (L∞ (Q))∗ ,
(56)
∇g ² (t, y² ) → η weakly star in L∞ (0, T ; H))∗ , η(t) ∈ ∂g(t, y∗ ) a.e. t ∈ (0, T ).
(57)
2
Since F is continuously differentiable from L (0, T ; V ) to Z,
[F 0 (y² )]∗ ξ² → [F 0 (y∗ )]∗ ξ0 weakly L2 (0, T ; V 0 ).
From (43), we infer that | (² + M )1/2 p² (T ) |2 = h(² + M )p² (T ), p² (T )i = ² |
p² (T ) |22 + | M 1/2 p² (T ) |22 ≤ c. Together with (55), we obtain
(² + M )1/2 p² (T ) → M 1/2 p(T ) in H.
19
Youjun Xu, Xiaoyou Liu and Liangbong Li
Now letting ² → 0 in (42), it follows that


d
M p − Ap − µ − [F 0 (y∗ )]∗ ξ0 ∈ L∞ (0, T ; H),
dt
d
M p(t)
dt
− Ap(t) − µ − [F 0 (y∗ )]∗ ξ0 ∈ λ0 ∂g(t, y∗ ) a.e. in (0, T ),
 M 1/2 p(T ) = 0,
(58)
(59)
It follows from (51), (52) and (42) that
Z T
Z T
Z T
∗
−
hB p² , vidt + λ²
h∇h² (u² ), vidt ≥
hu∗ − u² , vidt,
0
0
0
2
for all v ∈ L (0, T ; V ). By lemma 2.5, u² → u∗ strongly in L2 (0, T ; U ), it
follows
Z T
Z T
h∇ζ(t), vidt, ζ(t) ∈ ∂h(u∗ ) a.e. in (0, T ), (60)
h∇h² (u² ), vidt →
0
0
2
for all v ∈ L (0, T ; V ). Thus,
Z T
Z T
∗
hζ(t), vidt ≥ 0 f or all v ∈ L2 (0, T ; V ).
hB p, vidt + λ0
−
0
0
Since ξ² ∈ ∂dS (F (y² )), we get hξ² , w − F (y² )i ≤ 0 for all w ∈ S.
Now we claim that (λ0 , ξ0 ) 6= 0. Indeed, if λ0 = 0, we have that {ξ² }²>0
is bounded in Z ∗ . By (H3), S has finite codimentionality, so dose S − F (y ∗ ).
Thus it follows that ξ² → ξ0 weakly in Z ∗ and
hξ0 , w − F (y ∗ )i ≤ 0 f or all w ∈ S.
(61)
Finally, if (λ0 , p) = 0, it follows from (59) that µ + [F 0 (y ∗ )]∗ ξ0 = 0. So in
the case that µ ∈
/ R([F 0 (y ∗ )]∗ ), we must have (λ0 , p) 6= 0. This together with
(58), (59) and (61) completes the proof.
Example 3.3. Consider the initial-boundary vary value controlled system


 −4 dy(x,t)
+ (I − 4)y(x, t) + β(y(x, t)) 3 Bu(x, t) in Ω × (0, T ),

dt

∂
y(x, t) = 0 on ∂Ω × (0, T ),
∂ν



 (−4)1/2 y(x, 0) = (−4)1/2 y0 in Ω,
where Ω ⊂ RN be a bounded domain with smooth boundary.
∂
y(x, t) = 0 a.e. on ∂Ω}, β(·)
∂ν
satisfies (H3). Thus, the results of Theorem 3.2 remain true.
y0 ∈ W = {y ∈ H 2 (Ω) :
(62)
20
Optimal control of some degenerate pseudo-parabolic variational inequality
Acknowledgements. The work is supported by the NSFC (10671211 and
11126170), and Hunan Provincial Natural Science Foundation of China (Grant
No. 11JJ4006) and the construct program in USC (Grant No. 5-2011-XQD008 and NHCXTD08).
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