Gen. Math. Notes, Vol. 4, No. 1, May 2011, pp.16-27
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2011
www.i-csrs.org
Available free online at http://www.geman.in
Variational Discretization and Mixed Methods for
Semilinear Parabolic Optimal Control Problem
Zuliang Lu
School of Mathematics and Statistics, Chongqing Three Gorges University,
Chongqing 404000, P.R.China
College of Civil Engineering and Mechanics, Xiangtan University,
Xiangtan 411105, P.R.China
E-mail: zulianglux@126.com
(Received:18-5-11/Accepted:16-5-11)
Abstract
In this paper we study the variational discretization and mixed finite element
methods for optimal control problem governed by semilinear parabolic equations.
The space discretization of the state variable is done using usual mixed finite elements. The state and the co-state are approximated by the lowest order RaviartThomas mixed finite element spaces and the control is not discreted. Then we derive
a priori error estimates both for the coupled state and the control approximation.
Keywords: A priori error estimates, semilinear parabolic optimal control problem, variational discretization, mixed finite element methods
1
Introduction
Optimal control problems governed by semilinear parabolic equations is an important problem in engineering applications. The finite element method was undoubtedly the most widely used numerical method in computing optimal control
problems. There have been extensive studies in convergence of the finite element
approximation of optimal control problems. For optimal control problems governed
by linear elliptic equations, a priori error estimates of the standard finite element discretization were established long ago, see, for example, Falk [10]. The authors presented error estimates of finite element approximations of state constrained convex
Variational Discretization and Mixed Methods for...
17
parabolic boundary control problems in [1]. Then, Malanowski in [21] established
a priori error estimates for the finite element approximations to convex constrained
optimal control systems. In [2] the authors considered the finite element approximation of a distributed optimal control problem governed by a semilinear elliptic
partial differential equation, where pointwise constraints on the control were given.
Casas studied the numerical approximation of distributed semilinear optimal control problems and proved that the L2 -error estimates were of order o(h), which was
optimal according to the C 0,1 (Ω̄)-regularity of the optimal control in [3]. While
the a priori error analysis for finite element discretization of optimal control problems governed by elliptic equations was discussed in many publications, see, e.g.,
[13, 26], there were only few published results on this topic for parabolic problems.
Meidner and Vexler proposed a priori error estimates for space-time finite element
discretization of parabolic optimal control problems without control constraints in
[22]. The space discretization of the state variable was done using usual conforming finite elements, whereas the time discretization was based on discontinuous
Galerkin methods. Some recent progress in a priori error estimates can be found
in [15, 24], but there were only few published results on this topic for nonlinear
optimal control problems.
In many control problems, the objective functional contains gradient of the state
variables. Thus the accuracy of gradient is important in numerical approximation
of the state equations. Mixed finite element methods are appropriate for the state
equations in such cases since both the scalar variable and its flux variable can be
approximated to the same accuracy by using such methods, see, for example, [16].
However, there was only very limited research work on analyzing such elements for
optimal control problems. Recently, we have derived a priori error estimates, a posteriori error estimates and superconvergence for quadratic optimal control problems
using mixed finite element methods in [4, 5, 6, 7, 8, 18, 19, 20, 27].
In [14], the author first presents the variational discretization concept for optimal
control problems with control constraints, with implicitly utilizes the first order
optimality conditions and the discretization of the state and adjoint equations for the
discretization of the control instead of discretizing the space of admissible controls.
m,p
In this paper, we adopt the standard notation
for Sobolev spaces on Ω
P W α (Ω)
p
with a norm k · km,p given by k v km,p =
k D v kpLp (Ω) , a semi-norm | · |m,p
|α|≤m
P
p
p
α
given by | v |m,p =
k D v kLp (Ω) . We set W0m,p (Ω) = {v ∈ W m,p (Ω) :
|α|=m
v |∂Ω = 0}. For p = 2, we denote H m (Ω) = W m,2 (Ω), H0m (Ω) = W0m,2 (Ω), and
k · km =k · km,2 , k · k=k · k0,2 . We denote by Ls (0, T ; W m,p (Ω)) the Banach space
of all Ls integrable functions from J into W m,p (Ω) with norm k v kLs (J;W m,p (Ω)) =
R
1s
T
s
||v||W m,p (Ω) dt for s ∈ [1, ∞), and the standard modification for s = ∞.
0
The details can be found in [17].
In this paper we study a priori error estimates of the variational discretiza-
18
Zuliang Lu
tion and mixed finite element methods for optimal control problem governed by
semilinear parabolic equations. We focus our attention on the following semilinear
parabolic optimal control problem:
Z T
1
2
2
2
(1)
k p~ − p~d k + k y − yd k + k u k dt
min
u(t)∈K⊂U
2 0
subject to the state equation
yt (x, t) + div~p(x, t) + φ(y(x, t)) = f (x, t) + Bu(x, t), x ∈ Ω, t ∈ J, (2)
p~(x, t) = −A(x)∇y(x, t), x ∈ Ω,
(3)
y(x, t) = 0, x ∈ ∂Ω, t ∈ J, y(x, 0) = y0 (x), x ∈ Ω,
(4)
where the bounded open set Ω ⊂ R2 , is 2 regular convex polygon with boundary
∂Ω, J = (0, T ], f ∈ L2 (J; L2 (Ω)), and U = L2 (J; L2 (Ω)). For any R > 0 the
function φ(·) ∈ W 2,∞ (−R, R), φ0 (y) ∈ L2 (Ω) for any y ∈ L2 (J; H 1 (Ω)), and
φ0 (y) ≥ 0. Here, A(x) ∈ H 1 (Ω) and K denotes the admissible set of the control
variable, defined by
K = u(x, t) ∈ L2 (J; L2 (Ω)) : u(x, t) ≥ 0 a.e. x ∈ Ω, t ∈ J .
(5)
The outline of this paper is as follows. In Section 2, we construct the variational discretization and mixed finite element discretization for the optimal control
problem (1)-(4). In Section 3, we derive a priori error estimates for the variational
discretization and fully discrete mixed finite element approximation of the semilinear parabolic optimal control problem. Finally, we analyze the conclusion in section
4.
2
Variational Discretization and Mixed Methods
First, we introduce the co-state parabolic equation
−zt − div(A(∇z + p~ − p~d )) + φ0 (y)z = y − yd ,
x ∈ Ω,
(6)
with the conditions
z(x, t) = 0, x ∈ ∂Ω, t ∈ J;
z(x, T ) = 0, x ∈ Ω.
Next, we assume that the two given functions p~d , yd are continuously differentiable
with respect to t, moreover, yd ∈ L2 (J; H 2 (Ω)), p~d ∈ (L2 (J; H 2 (Ω)))2 .
We now describe the variational discretization and mixed finite element approximation of semilinear parabolic optimal control problem (1)-(4). Let V~ =
H(div) = {~v ∈ (L2 (Ω))2 , div~v ∈ L2 (Ω)} endowed with the norm given by
k ~v kH(div) = (k~v k20,Ω + kdiv~v k20,Ω )1/2 . We denote W = L2 (Ω).
19
Variational Discretization and Mixed Methods for...
We recast (1)-(4) as the following weak form: find (~p, y, u) ∈ V~ × W × K such
that
Z T
1
2
2
2
(7)
min
k~p − p~d k + ky − yd k + kuk dt
u∈K⊂U
2 0
(A−1 p~, ~v ) − (y, div~v ) = 0, ∀~v ∈ V~ ,
(8)
(yt , w) + (div~p, w) + (φ(y), w) = (f + Bu, w), ∀w ∈ W,
(9)
y(x, 0) = y0 (x), ∀x ∈ Ω.
(10)
It is well known (see, e.g., [17]) that the optimal control problem (7)-(10) has
a solution (~p, y, u), and that a triplet (~p, y, u) is the solution of (7)-(10) if and only
if there is a co-state (~q, z) ∈ V~ × W such that (~p, y, ~q, z, u) satisfies the following
optimality conditions:
(A−1 p~, ~v ) − (y, div~v ) = 0, ∀~v ∈ V~ ,
(yt , w) + (div~p, w) + (φ(y), w) = (f + Bu, w), ∀w ∈ W,
y(x, 0) = y0 (x), ∀x ∈ Ω,
−1
(A ~q, ~v ) − (z, div~v ) = −(~p − p~d , ~v ), ∀~v ∈ V~ ,
0
−(zt , w) + (div~q, w) + (φ (y)z, w) = (y − yd , w), ∀w ∈ W,
z(x, T ) = 0, ∀x ∈ Ω,
Z T
(B ∗ z + u, ũ − u)U dt ≥ 0, ∀ũ ∈ K,
(11)
(12)
(13)
(14)
(15)
(16)
(17)
0
where B ∗ is the adjoint operator of B and (·, ·)U is the inner product of U . In the
rest of the paper, we shall simply write the product as (·, ·) whenever no confusion
should be caused.
We also assume that both parabolic equations (2) and (6) have sufficiently regularity and u ∈ L2 (J; W 1,∞ (Ω)), y, z ∈ L2 (J; H 2 (Ω)), p~, ~q ∈ (L2 (J; H 2 (Ω)))2 .
Let Th be regular triangulation of Ω. They are assumed to satisfy the angle
condition which means that there is a positive constant C such that
C −1 h2τ ≤ |τ | ≤ Ch2τ ,
∀τ ∈ Th ,
where |τ | is the area of τ and hτ is the diameter of τ . Let h = max hτ . In addition
C or c denotes a general positive constant independent of h.
Let V~h ×Wh ⊂ V~ ×W denote the Raviart-Thomas space [25] of the lowest order
associated with the triangulation Th of Ω, namely, V~ (τ ) = {~v ∈ P02 (τ )+x·P0 (τ )},
W (τ ) = P0 (τ ), ∀τ ∈ Th , where Pk denotes the space of polynomials of total degree
at most k, x = (x1 , x2 ) which treated as a vector, and
V~h := {~vh ∈ V~ : ∀τ ∈ Th , ~vh |τ ∈ V~ (τ )},
Wh := {wh ∈ W : ∀τ ∈ Th , wh |τ ∈ W (τ )}.
20
Zuliang Lu
The mixed finite element discretization of (7)-(10) is as follows: compute (~ph , yh ,
uh ) ∈ V~h × Wh × K such that
Z T
1
2
2
2
min
k~ph − p~d k + kyh − yd k + kuh k dt
uh ∈K
2 0
(A−1 p~h , ~vh ) − (yh , div~vh ) = 0, ∀~vh ∈ V~h ,
(18)
(19)
(yht , wh ) + (div~ph , wh ) + (φ(yh ), wh ) = (f + Buh , wh ), ∀wh ∈ Wh , (20)
yh (x, 0) = Y (x, 0), ∀x ∈ Ω,
(21)
where Y (x, 0) is the elliptic mixed methods projection into the finite dimensional
space Wh of the initial data function y0 (x).
The optimal control problem (18)-(21) again has a solution (~ph , yh , uh ), and
that a triplet (~ph , yh , uh ) is the solution of (18)-(21) if and only if there is a co-state
(~qh , zh ) ∈ V~h × Wh such that (~ph , yh , ~qh , zh , uh ) satisfies the following optimality
conditions:
(A−1 p~h , ~vh ) − (yh , div~vh ) = 0,
(yht , wh ) + (div~ph , wh ) + (φ(yh ), wh ) = (f + Buh , wh ),
yh (x, 0) = Y (x, 0), ∀x ∈ Ω,
−1
(A ~qh , ~vh ) − (zh , div~vh ) = −(~ph − p~d , ~vh ),
−(zht , wh ) + (div~qh , wh ) + (φ0 (yh )zh , wh ) = (yh − yd , wh ),
zh (x, T ) = 0, ∀x ∈ Ω,
∗
(B zh + uh , ũh − uh ) ≥ 0,
(22)
(23)
(24)
(25)
(26)
(27)
(28)
where ~v ∈ V~h , w ∈ Wh , ũ ∈ K.
We now consider the time discretization of the difference methods. Let 4t > 0,
N = T /4t ∈ Z, and tn = n4t, n ∈ Z. Also, let
ψ n = ψ n (x) = ψ(x, tn ),
dt ψ n =
ψ n − ψ n−1
.
4t
We define for 1 ≤ p < ∞ the discrete time dependent norms
|||ψ|||Lp (J;H s (Ω)) :=
N
X
! p1
4tkψ n kps
,
n=1
and the standard modification for p = ∞.
Then we define the fully discrete finite element solution (~pnh , yhn , ~qhn−1 , zhn−1 , unh ) sat-
Variational Discretization and Mixed Methods for...
21
isfies
(A−1 p~nh , ~v ) − (yhn , div~v ) = 0,
(dt yhn , w) + (div~pnh , w) + (φ(yhn ), w) = (f + Bunh , w),
yh0 (x) = Y (x, 0), ∀x ∈ Ω,
(A−1 ~qhn−1 , ~v ) − (zhn−1 , div~v ) = −(~pnh − p~d , ~v ),
−(dt zhn , w) + (div~qhn−1 , w) + (φ0 (yhn )zhn−1 , w) = (yhn − yd , w),
zhN (x) = 0, ∀x ∈ Ω,
(B ∗ zhn + unh , ũ − unh ) ≥ 0,
(29)
(30)
(31)
(32)
(33)
(34)
(35)
where ~v ∈ V~h , w ∈ Wh , ũ ∈ K.
For ϕ ∈ Wh , we shall write
φ(ϕ) − φ(ρ) = −φ̃0 (ϕ)(ρ − ϕ) = −φ0 (ρ)(ρ − ϕ) + φ̃00 (ϕ)(ρ − ϕ)2 ,
(36)
where
0
Z
1
φ0 (ϕ + s(ρ − ϕ))ds,
φ̃ (ϕ) =
φ̃00 (ϕ) =
Z0 1
(1 − s)φ00 (ρ + s(ϕ − ρ))ds
0
are bounded functions in Ω̄ [23].
3
A Priori Error Estimates
In the rest of the paper, we shall use some intermediate variables. For any control
function ũ ∈ K, we first define the state solution (~p(ũ), y(ũ), ~q(ũ), z(ũ)) associated
with ũ that satisfies
(A−1 p~(ũ), ~v ) − (y(ũ), div~v ) = 0, ∀~v ∈ V~ ,
(yt (ũ), w) + (div~p(ũ), w) + (φ(y(ũ)), w) = (f + B ũ, w), ∀w ∈ W,
y(ũ)(x, 0) = y0 (x), ∀x ∈ Ω,
(A−1 ~q(ũ), ~v ) − (z(ũ), div~v ) = −(~p(ũ) − p~d , ~v ), ∀~v ∈ V~ ,
(1)
(2)
(3)
(4)
−(zt (ũ), w) + (div~q(ũ), w) + (φ (y(ũ))z(ũ), w) = (y(ũ) − yd , w), ∀w ∈ W,(5)
z(ũ)(x, T ) = 0, ∀x ∈ Ω.
(6)
0
Then, we define the discrete time state solution (~pn (ũ), y n (ũ), ~qn−1 (ũ), z n−1 (ũ)) of
22
Zuliang Lu
the system (1)-(6) associated with ũ ∈ K that satisfies
(A−1 p~n (ũ), ~v ) − (y n (ũ), div~v ) = 0, ∀~v ∈ V~ ,
(7)
(ytn (ũ), w) + (div~pn (ũ), w) + (φ(y n (ũ)), w) = (f + B ũ, w), ∀w ∈ W, (8)
y 0 (ũ)(x) = y0 (x), ∀x ∈ Ω,
(9)
(A−1 ~qn−1 (ũ), ~v ) − (z n−1 (ũ), div~v ) = −(~pn (ũ) − p~d , ~v ), ∀~v ∈ V~ ,
(10)
n−1
0 n
n−1
n
(11)
−(zt (ũ), w) + (div~q (ũ), w) + (φ (y (ũ))z (ũ), w)
n
N
= (y (ũ) − yd , w), ∀w ∈ W, z (ũ)(x) = 0, ∀x ∈ Ω.
(12)
According to the assumption on the domain Ω, we can easily observe that Ω is
2 regular. The domain Ω is said to be 2 regular if the Dirichlet problem
Lλ ϕ = −div(A(x)∇ϕ) + λϕ = F, x ∈ Ω,
ϕ = 0, x ∈ ∂Ω,
(13)
(14)
is uniquely solvable for F ∈ L2 (Ω) and if kϕk2 ≤ kF k0 for all F ∈ L2 (Ω).
~ n (ũ), Z n (ũ))
For any ũ ∈ K, we define an elliptic projection (P~ n (ũ), Y n (ũ), Q
of the solution of the differential problem into the finite dimensional space V~h × Wh
~
to be the map (P~ (ũ), Y (ũ), Q(ũ),
Z(ũ)) : {0, t1 , t2 , ..., tn = T } → V~h × Wh given
by
(A−1 (~pn (ũ) − P~ n (ũ)), ~v ) − (y n (ũ) − Y n (ũ), div~v ) = 0, ∀~v ∈ V~h , (15)
(div(~pn (ũ) − P~ n (ũ)), w) + λ(y n (ũ) − Y n (ũ), w) = 0, ∀w ∈ Wh , (16)
~ n (ũ)), ~v ) − (z n (ũ) − Z n (ũ), div~v ) = 0, ∀~v ∈ V~h , (17)
(A−1 (~qn (ũ) − Q
~ n (ũ)), w) + λ(z n (ũ) − Z n (ũ), w) = 0, ∀w ∈ Wh .
(div(~qn (ũ) − Q
(18)
Let λ > 0, such that λ is sufficiently large so that the bilinear form associated with
Lλ (·) is coercive over H01 (Ω). In fact, let λ be chosen so that [9]:
(19)
(A−1 ξ, ξ) + λ(η, η) ≥ C kξk20 + kηk20 , ∀ξ ∈ V~ , ∀η ∈ W.
The projection (15)-(18) is associated with the operator Lλ . Let
τ1n = y n (uh ) − Y n (uh ),
τ2n = z n (uh ) − Z n (uh ),
σ1n = p~n (uh ) − P~ n (uh ),
~ n (uh ).
σ n = ~qn (uh ) − Q
2
(20)
(21)
Estimates for τ1n , τ2n , σ1n , and σ2n are given in [11]. We state them here without a
proof.
Lemma 3.1 For t ∈ J and for h sufficiently small, there is a positive constant C
independent of h such that
kσ1n k0 + kτ1n k0 + kτ1n k0,∞ ≤ Ch,
kσ2n k0 + kτ2n k0 + kτ2n k0,∞ ≤ Ch,
kdivσ1n k0 + kdivσ2n k0 ≤ Ch.
(22)
(23)
(24)
23
Variational Discretization and Mixed Methods for...
n
n
Estimates for τ1tn , τ2tn , σ1t
, and σ2t
are given in [12]. We state them here without a
proof.
Lemma 3.2 For t ∈ J and for h sufficiently small, there is a positive constant C
independent of h such that
n
kσ1t
k0 + kτ1tn k0 + kτ1tn k0,∞ ≤ Ch,
n
k0 + kτ2tn k0 + kτ2tn k0,∞ ≤ Ch,
kσ2t
n
n
kdivσ1t
k0 + kdivσ2t
k0 ≤ Ch.
(25)
(26)
(27)
With the aid of Lemmas 3.1-3.2, we can also derive the following error estimates:
Theorem 3.3 There is a positive constant C > 0, independent of h, such that
|||~p(uh ) − p~h |||L∞ (J;H(div)) + |||y(uh ) − yh |||L∞ (J;L2 (Ω)) ≤ C(4t + h), (28)
|||~q(uh ) − ~qh |||L∞ (J;H(div)) + |||z(uh ) − zh |||L∞ (J;L2 (Ω)) ≤ C(4t + h). (29)
Set some intermediate errors:
en1 = p~n − p~n (uh ),
en2 = ~qn − ~qn (uh ),
r1n = y n − y n (uh ),
r2n = z n − z n (uh ).
(30)
(31)
From (11)-(16) and (7)-(12), we derive the following error equations:
(A−1 en1 , ~v ) − (r1n , div~v ) = 0, ∀~v ∈ V~h ,
(ytn − dt y n (uh ), w) + (diven1 , w) + (φ̃0 (y n )r1n , w)
= (B(un − unh ), w), ∀w ∈ Wh ,
(A−1 en−1
, ~v ) − (r2n−1 , div~v ) = −(en , ~v ), ∀~v ∈ V~h ,
2
1
− dt z (uh ), w) + (dive2n−1 , w) + (φ0 (y n )r2n−1 , w)
+(φ̃00 (y n )r1n z n−1 (uh ), w) = (r1n , w), ∀w ∈ Wh .
−(ztn
(32)
(33)
(34)
n
(35)
Theorem 3.4 There is a constant C > 0, independent of h and 4t, such that
|||~p − p~(uh )|||L∞ (J;H(div)) + |||y − y(uh )|||L∞ (J;L2 (Ω))
≤ C(4t + h + |||u − uh |||L2 (J;L2 (Ω)) ),
|||~q − ~q(uh )|||L∞ (J;H(div)) + |||z − z(uh )|||L∞ (J;L2 (Ω))
≤ C(4t + h + |||u − uh |||L2 (J;L2 (Ω)) ).
(36)
(37)
Proof. Part I. Choose ~v = en1 and w = r1n as the test functions and add the two
relations of (32)-(33), then we obtain that
(A−1 en1 , en1 ) + (φ̃0 (y n )r1n , r1n ) = (B(un − unh ), r1n ) − (ytn − dt y n (uh ), r1n ).
24
Zuliang Lu
By using δ-Cauchy inequality, we can find an estimate as follows
ken1 k20 + kr1n k20 ≤ C (4t)2 + h2 + kun − unh k20 + δkr1n k20 ,
(38)
for any small δ > 0. This leads to
ken1 k0 + kr1n k0 ≤ C(4t + h + kun − unh k0 ).
(39)
Now, take w = diven1 as a test function in (33), then we get
kdiven1 k20 = (B(un − unh ), diven1 )
−(ytn − dt y n (uh ), diven1 ) − (φ̃0 (yhn )r1n , diven1 )
≤ Ckytn − dt y n (uh )k20 + Ckun − unh k20 + Ckr1n k20 + δkdiven1 k20 ,
(40)
then, using the estimate (39), we have
kdiven1 k0 ≤ Ckytn − dt y n (uh )k0 + Ckun − unh k0 + Ckr1n k0
≤ C(4t + h + kun − unh k0 ).
(41)
Then (36) follows from (38) and (41).
Part II. Similarly, choose ~v = en−1
and w = r2n−1 as the test functions and add the
2
two relations of (34)-(35), then we obtain that
(A−1 en−1
, en−1
) + (φ0 (y n )r2n−1 , r2n−1 ) = (r1n , r2n−1 ) + (ztn − dt z n (uh ), r2n−1 )
2
2
−(en1 , en−1
) − (φ̃00 (y n )z n−1 (uh )r1n , r2n−1 ).
2
Then, using δ-Cauchy inequality, we can find an estimate as follows
ken−1
k20 + kr2n−1 k20 ≤ C ((4t)2 + h2 + kun − unh k20 )
2
+δ(kr2n−1 k20 + ken−1
k20 ),
2
(42)
or equivalently,
ken−1
k0 + kr2n−1 k0 ≤ C(4t + h + kun − unh k0 ).
2
(43)
Taking w = diven−1
as a test function in (35) and using δ-Cauchy inequality, then
2
we get
k20 = (r1n , dive2n−1 ) − (φ0 (y n )r2n−1 , dive2n−1 )
kdiven−1
2
) − (φ̃00 (y n )z n−1 (uh )r1n , diven−1
)
+(ztn − dt z n (uh ), diven−1
2
2
n−1 2
n−1 2
n
n
2
n 2
≤ Ckzt − dt z (uh )k0 + Ckr1 k0 + Ckr2 k0 + δkdive2 k0 ,
(44)
then, using the estimate (39) and (43), we verify that
kdiven−1
k0 ≤ C(4t + h + kun − unh k0 ).
2
(45)
This implies (37).
Now we combine the bounds given by Theorems 3.3-3.4 to come up with the
following main results.
Variational Discretization and Mixed Methods for...
25
Theorem 3.5 Let (~p, y, ~q, z, u) ∈ (V~ × W )2 × K and (~ph , yh , ~qh , zh , uh ) ∈ (V~h ×
Wh )2 × K be the solutions of (11)-(17) and (22)-(28), respectively. Assume that
∀n = [0, 1, · · · , N ], B ∗ z n + un ∈ H 1 (Ω). Then, we have
|||u − uh |||L2 (J;L2 (Ω)) ≤ C(4t + h),
|||~p − p~h |||L∞ (J;H(div)) + |||y − yh |||L∞ (J;L2 (Ω)) ≤ C(4t + h),
|||~q − ~qh |||L∞ (J;H(div)) + |||z − zh |||L∞ (J;L2 (Ω)) ≤ C(4t + h).
4
(46)
(47)
(48)
Conclusions
In this paper, we derive a priori error estimates of the variational discretization and
mixed finite element methods for semilinear parabolic optimal control problem.
Our priori error estimates for the optimal control problems governed by semilinear
paraboli equations by the variational discretization and fully discrete mixed finite
element methods seem to be new.
Acknowledgements
This work was supported by National Nature Science Foundation under Grant 10971074
and Hunan Provincial Innovation Foundation For Postgraduate CX2009B119.
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