3 A Priori Error Estimates

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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: [email protected] (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 el- ements. The state and the co-state are approximated by the lowest order Raviart- Thomas 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 prob- lem, variational discretization, mixed finite element methods

1 Introduction

Optimal control problems governed by semilinear parabolic equations is an im- portant problem in engineering applications. The finite element method was un- doubtedly 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 dis- cretization were established long ago, see, for example, Falk [10]. The authors pre- sented error estimates of finite element approximations of state constrained convex

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 approx- imation 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 con- trol problems and proved that theL²-error estimates were of ordero(h), which was optimal according to the C^0,1( ¯Ω)-regularity of the optimal control in [3]. While the a priori error analysis for finite element discretization of optimal control prob- lems 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 conform- ing 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 pos- teriori 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.

In this paper, we adopt the standard notationW^m,p(Ω)for Sobolev spaces onΩ with a normk · k_m,p given byk v k^p_m,p= P

|α|≤m

k D^αv k^p_Lp(Ω), a semi-norm| · |_m,p given by | v |^p_m,p= P

|α|=m

k D^αv k^p_Lp(Ω). We set W₀^m,p(Ω) = {v ∈ W^m,p(Ω) : v |∂Ω= 0}. Forp = 2, we denoteH^m(Ω) = W^m,2(Ω), H₀^m(Ω) = W₀^m,2(Ω), and k · k_m=k · k_m,2, k · k=k · k_0,2. We denote byL^s(0, T;W^m,p(Ω))the Banach space of allL^sintegrable functions fromJ intoW^m,p(Ω)with normk v k_L^s_(J;W^m,p_(Ω))= RT

0 ||v||^s_Wm,p(Ω)dt¹_s

for s ∈ [1,∞), and the standard modification for s = ∞.

The details can be found in [17].

In this paper we study a priori error estimates of the variational discretiza-

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:

u(t)∈K⊂Umin 1

2 Z T

0

k~p−~p_dk² +ky−y_d k² +kuk² dt

(1) subject to the state equation

y_t(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 setΩ ⊂ R², is2regular convex polygon with boundary

∂Ω, J = (0, T], f ∈ L²(J;L²(Ω)), and U = L²(J;L²(Ω)). For anyR > 0 the function φ(·) ∈ W^2,∞(−R, R), φ⁰(y) ∈ L²(Ω) for any y ∈ L²(J;H¹(Ω)), and φ⁰(y) ≥ 0. Here, A(x) ∈ H¹(Ω) and K denotes the admissible set of the control variable, defined by

K =

u(x, t)∈L²(J;L²(Ω)) : u(x, t)≥0a.e. x∈Ω, t∈J . (5) The outline of this paper is as follows. In Section 2, we construct the varia- tional 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 semilin- ear 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

−z_t−div(A(∇z+~p−~p_d)) +φ⁰(y)z=y−y_d, 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,y_dare continuously differentiable with respect tot, moreover,y_d∈L²(J;H²(Ω)),~p_d∈(L²(J;H²(Ω)))².

We now describe the variational discretization and mixed finite element ap- proximation 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~vk_H(div) = (k~vk²_0,Ω+kdiv~vk²_0,Ω)^1/2. We denoteW =L²(Ω).

We recast (1)-(4) as the following weak form: find(~p, y, u)∈V~ ×W×Ksuch that

u∈K⊂Umin 1

2 Z T

0

k~p−~pdk²+ky−ydk²+kuk² dt

(7) (A⁻¹~p, ~v)−(y,div~v) = 0, ∀~v ∈V ,~ (8) (y_t, 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 ,~ (11) (y_t, w) + (div~p, w) + (φ(y), w) = (f+Bu, w), ∀w∈W, (12) y(x,0) =y₀(x), ∀x∈Ω, (13) (A⁻¹~q, ~v)−(z,div~v) =−(~p−~p_d, ~v), ∀~v ∈V ,~ (14)

−(z_t, w) + (div~q, w) + (φ⁰(y)z, w) = (y−y_d, w), ∀w∈W, (15) z(x, T) = 0, ∀x∈Ω, (16) Z T

0

(B^∗z+u,u˜−u)_Udt≥0, ∀˜u∈K, (17) whereB^∗ is the adjoint operator of B and(·,·)_U is the inner product ofU. 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 regu- larity andu∈L²(J;W^1,∞(Ω)),y, z∈L²(J;H²(Ω)),p, ~~ q∈(L²(J;H²(Ω)))².

Let T_h be regular triangulation of Ω. They are assumed to satisfy the angle condition which means that there is a positive constantCsuch that

C⁻¹h²_τ ≤ |τ| ≤Ch²_τ, ∀τ ∈T_h,

where|τ|is the area ofτ andhτ is the diameter ofτ. Leth = maxhτ. In addition Corcdenotes a general positive constant independent ofh.

LetV~_h×W_h ⊂V~×W denote the Raviart-Thomas space [25] of the lowest order associated with the triangulationThofΩ, namely,V~(τ) = {~v ∈P₀²(τ) +x·P0(τ)}, W(τ) =P₀(τ),∀τ ∈T_h, whereP_kdenotes the space of polynomials of total degree at mostk,x= (x₁, x₂)which treated as a vector, and

V~h :={~vh ∈V~ : ∀τ ∈Th, ~vh|τ ∈V~(τ)}, W_h :={w_h ∈W : ∀τ ∈T_h, w_h|_τ ∈W(τ)}.

The mixed finite element discretization of (7)-(10) is as follows: compute(~p_h, y_h, u_h)∈V~_h×W_h×K such that

uminh∈K

1 2

Z T

0

k~p_h−~p_dk²+ky_h−y_dk²+ku_hk² dt

(18) (A⁻¹~p_h, ~v_h)−(y_h,div~v_h) = 0, ∀~v_h ∈V~_h, (19) (y_ht, w_h) + (div~p_h, w_h) + (φ(y_h), w_h) = (f +Bu_h, w_h), ∀w_h ∈W_h, (20)

y_h(x,0) = Y(x,0), ∀x∈Ω, (21)

whereY(x,0)is the elliptic mixed methods projection into the finite dimensional spaceW_h of the initial data functiony₀(x).

The optimal control problem (18)-(21) again has a solution (~p_h, y_h, u_h), and that a triplet(~p_h, y_h, u_h)is the solution of (18)-(21) if and only if there is a co-state (~q_h, z_h) ∈ V~_h ×W_h such that (~p_h, y_h, ~q_h, z_h, u_h)satisfies the following optimality conditions:

(A⁻¹~p_h, ~v_h)−(y_h,div~v_h) = 0, (22) (y_ht, w_h) + (div~p_h, w_h) + (φ(y_h), w_h) = (f +Bu_h, w_h), (23) y_h(x,0) =Y(x,0), ∀x∈Ω, (24) (A⁻¹~q_h, ~v_h)−(z_h,div~v_h) =−(~p_h−~p_d, ~v_h), (25)

−(z_ht, w_h) + (div~q_h, w_h) + (φ⁰(y_h)z_h, w_h) = (y_h−y_d, w_h), (26) zh(x, T) = 0, ∀x∈Ω, (27) (B^∗z_h+u_h,u˜_h−u_h)≥0, (28) where~v ∈V~_h,w∈W_h,u˜∈K.

We now consider the time discretization of the difference methods. Let4t >0, N =T /4t∈Z, andtⁿ=n4t,n∈Z. Also, let

ψⁿ =ψⁿ(x) =ψ(x, tⁿ), d_tψⁿ= ψⁿ−ψⁿ⁻¹ 4t . We define for1≤p <∞the discrete time dependent norms

|||ψ|||_L^p_(J;H^s_(Ω)) :=

N

X

n=1

4tkψⁿk^p_s

!¹_p ,

and the standard modification forp=∞.

Then we define the fully discrete finite element solution(~pⁿ_h, y_hⁿ, ~q_hⁿ⁻¹, z_hⁿ⁻¹, uⁿ_h)sat-

isfies

(A⁻¹~pⁿ_h, ~v)−(y_hⁿ,div~v) = 0, (29) (d_ty_hⁿ, w) + (div~pⁿ_h, w) + (φ(y_hⁿ), w) = (f +Buⁿ_h, w), (30) y_h⁰(x) = Y(x,0), ∀x∈Ω, (31) (A⁻¹~qⁿ⁻¹_h , ~v)−(z_hⁿ⁻¹,div~v) =−(~pⁿ_h−~p_d, ~v), (32)

−(dtz_hⁿ, w) + (div~q_hⁿ⁻¹, w) + (φ⁰(y_hⁿ)z_hⁿ⁻¹, w) = (y_hⁿ−yd, w), (33) z_h^N(x) = 0, ∀x∈Ω, (34) (B^∗z_hⁿ+uⁿ_h,u˜−uⁿ_h)≥0, (35) where~v ∈V~_h,w∈W_h,u˜∈K.

Forϕ∈W_h, we shall write

φ(ϕ)−φ(ρ) =−φ˜⁰(ϕ)(ρ−ϕ) = −φ⁰(ρ)(ρ−ϕ) + ˜φ⁰⁰(ϕ)(ρ−ϕ)², (36) where

φ˜⁰(ϕ) = Z 1

0

φ⁰(ϕ+s(ρ−ϕ))ds, φ˜⁰⁰(ϕ) =

Z 1

0

(1−s)φ⁰⁰(ρ+s(ϕ−ρ))ds are bounded functions inΩ¯ [23].

3 A Priori Error Estimates

In the rest of the paper, we shall use some intermediate variables. For any control functionu˜∈K, we first define the state solution(~p(˜u), y(˜u), ~q(˜u), z(˜u))associated withu˜that satisfies

(A⁻¹~p(˜u), ~v)−(y(˜u),div~v) = 0, ∀~v ∈V ,~ (1) (y_t(˜u), w) + (div~p(˜u), w) + (φ(y(˜u)), w) = (f +Bu, w),˜ ∀w∈W, (2) y(˜u)(x,0) = y₀(x), ∀x∈Ω, (3) (A⁻¹~q(˜u), ~v)−(z(˜u),div~v) = −(~p(˜u)−~pd, ~v), ∀~v ∈V ,~ (4)

−(z_t(˜u), w) + (div~q(˜u), w) + (φ⁰(y(˜u))z(˜u), w) = (y(˜u)−y_d, w), ∀w∈W,(5) z(˜u)(x, T) = 0, ∀x∈Ω. (6) Then, we define the discrete time state solution(~pⁿ(˜u), yⁿ(˜u), ~qⁿ⁻¹(˜u), zⁿ⁻¹(˜u))of

the system (1)-(6) associated withu˜∈K that satisfies

(A⁻¹~pⁿ(˜u), ~v)−(yⁿ(˜u),div~v) = 0, ∀~v ∈V ,~ (7) (y_tⁿ(˜u), w) + (div~pⁿ(˜u), w) + (φ(yⁿ(˜u)), w) = (f +Bu, w),˜ ∀w ∈W, (8) y⁰(˜u)(x) = y0(x), ∀x∈Ω, (9) (A⁻¹~qⁿ⁻¹(˜u), ~v)−(zⁿ⁻¹(˜u),div~v) =−(~pⁿ(˜u)−~p_d, ~v), ∀~v ∈V ,~ (10)

−(z_tⁿ(˜u), w) + (div~qⁿ⁻¹(˜u), w) + (φ⁰(yⁿ(˜u))zⁿ⁻¹(˜u), w) (11)

= (yⁿ(˜u)−y_d, w),∀w∈W, z^N(˜u)(x) = 0, ∀x∈Ω. (12) According to the assumption on the domainΩ, we can easily observe thatΩis 2regular. The domainΩis said to be2regular if the Dirichlet problem

L_λϕ=−div(A(x)∇ϕ) +λϕ=F, x ∈Ω, (13)

ϕ= 0, x∈∂Ω, (14)

is uniquely solvable forF ∈L²(Ω)and ifkϕk₂ ≤ kFk₀ for allF ∈L²(Ω).

For anyu˜ ∈ K, we define an elliptic projection(P~ⁿ(˜u), Yⁿ(˜u), ~Qⁿ(˜u), Zⁿ(˜u)) of the solution of the differential problem into the finite dimensional spaceV~h×Wh

to be the map(P~(˜u), Y(˜u), ~Q(˜u), Z(˜u)) : {0, t¹, t², ..., tⁿ =T} → V~_h×W_h given by

(A⁻¹(~pⁿ(˜u)−P~ⁿ(˜u)), ~v)−(yⁿ(˜u)−Yⁿ(˜u),div~v) = 0, ∀~v ∈V~_h, (15) (div(~pⁿ(˜u)−P~ⁿ(˜u)), w) +λ(yⁿ(˜u)−Yⁿ(˜u), w) = 0, ∀w∈W_h, (16) (A⁻¹(~qⁿ(˜u)−Q~ⁿ(˜u)), ~v)−(zⁿ(˜u)−Zⁿ(˜u),div~v) = 0, ∀~v ∈V~h, (17) (div(~qⁿ(˜u)−Q~ⁿ(˜u)), w) +λ(zⁿ(˜u)−Zⁿ(˜u), w) = 0, ∀w∈W_h. (18) Letλ > 0, such thatλis sufficiently large so that the bilinear form associated with L_λ(·)is coercive overH₀¹(Ω). In fact, letλbe chosen so that [9]:

(A⁻¹ξ, ξ) +λ(η, η)≥C kξk²₀+kηk²₀

, ∀ξ∈V ,~ ∀η∈W. (19) The projection (15)-(18) is associated with the operatorL_λ. Let

τ₁ⁿ=yⁿ(u_h)−Yⁿ(u_h), σ₁ⁿ=~pⁿ(u_h)−P~ⁿ(u_h), (20) τ₂ⁿ=zⁿ(u_h)−Zⁿ(u_h), σⁿ₂ =~qⁿ(u_h)−Q~ⁿ(u_h). (21) Estimates forτ₁ⁿ,τ₂ⁿ,σⁿ₁, andσⁿ₂ are given in [11]. We state them here without a proof.

Lemma 3.1 Fort ∈Jand forhsufficiently small, there is a positive constantC independent ofhsuch that

kσ₁ⁿk₀+kτ₁ⁿk₀+kτ₁ⁿk0,∞≤Ch, (22) kσ₂ⁿk₀+kτ₂ⁿk₀+kτ₂ⁿk0,∞≤Ch, (23) kdivσ₁ⁿk₀ +kdivσ₂ⁿk₀ ≤Ch. (24)

Estimates forτ_1tⁿ, τ_2tⁿ, σ_1tⁿ, andσ_2tⁿ are given in [12]. We state them here without a proof.

Lemma 3.2 Fort ∈Jand forhsufficiently small, there is a positive constantC independent ofhsuch that

kσⁿ_1tk₀+kτ_1tⁿk₀+kτ_1tⁿk0,∞ ≤Ch, (25) kσⁿ_2tk₀+kτ_2tⁿk₀+kτ_2tⁿk_0,∞ ≤Ch, (26) kdivσ_1tⁿk₀+kdivσ_2tⁿk₀ ≤Ch. (27) With the aid of Lemmas 3.1-3.2, we can also derive the following error esti- mates:

Theorem 3.3 There is a positive constantC >0, independent ofh, such that

|||~p(uh)−~ph|||_L^∞_(J;H(div))+|||y(uh)−yh|||_L^∞_(J;L²_(Ω))≤C(4t+h), (28)

|||~q(u_h)−~q_h|||_L^∞_(J;H(div))+|||z(u_h)−z_h|||_L^∞_(J;L²_(Ω)) ≤C(4t+h). (29) Set some intermediate errors:

eⁿ₁ =~pⁿ−p~ⁿ(u_h), rⁿ₁ =yⁿ−yⁿ(u_h), (30) eⁿ₂ =~qⁿ−~qⁿ(u_h), rⁿ₂ =zⁿ−zⁿ(u_h). (31) From (11)-(16) and (7)-(12), we derive the following error equations:

(A⁻¹eⁿ₁, ~v)−(r₁ⁿ,div~v) = 0, ∀~v ∈V~_h, (32) (yⁿ_t −d_tyⁿ(u_h), w) + (diveⁿ₁, w) + ( ˜φ⁰(yⁿ)r₁ⁿ, w) (33)

= (B(uⁿ−uⁿ_h), w), ∀w∈W_h,

(A⁻¹eⁿ⁻¹₂ , ~v)−(rⁿ⁻¹₂ ,div~v) = −(eⁿ₁, ~v), ∀~v ∈V~_h, (34)

−(z_tⁿ−d_tzⁿ(u_h), w) + (diveⁿ⁻¹₂ , w) + (φ⁰(yⁿ)r₂ⁿ⁻¹, w)

+( ˜φ⁰⁰(yⁿ)r₁ⁿzⁿ⁻¹(u_h), w) = (r₁ⁿ, w), ∀w∈W_h. (35) Theorem 3.4 There is a constantC > 0, independent ofhand4t, such that

|||~p−~p(u_h)|||_L^∞_(J;H(div))+|||y−y(u_h)|||_L^∞_(J;L²_(Ω))

≤C(4t+h+|||u−u_h|||_L²_(J;L²_(Ω))), (36)

|||~q−~q(u_h)|||_L^∞_(J;H(div))+|||z−z(u_h)|||_L^∞_(J;L²_(Ω))

≤C(4t+h+|||u−u_h|||_L²_(J;L²_(Ω))). (37) Proof. Part I.Choose~v = eⁿ₁ andw = r₁ⁿ as the test functions and add the two relations of (32)-(33), then we obtain that

(A⁻¹eⁿ₁, eⁿ₁) + ( ˜φ⁰(yⁿ)rⁿ₁, rⁿ₁) = (B(uⁿ−uⁿ_h), r₁ⁿ)−(y_tⁿ−d_tyⁿ(u_h), r₁ⁿ).

By usingδ-Cauchy inequality, we can find an estimate as follows keⁿ₁k²₀ +kr₁ⁿk²₀ ≤C (4t)²+h²+kuⁿ−uⁿ_hk²₀

+δkrⁿ₁k²₀, (38) for any smallδ >0. This leads to

keⁿ₁k₀+krⁿ₁k₀ ≤C(4t+h+kuⁿ−uⁿ_hk₀). (39) Now, takew= diveⁿ₁ as a test function in (33), then we get

kdiveⁿ₁k²₀ = (B(uⁿ−uⁿ_h),diveⁿ₁)

−(y_tⁿ−dtyⁿ(uh),diveⁿ₁)−( ˜φ⁰(y_hⁿ)rⁿ₁,diveⁿ₁)

≤Cky_tⁿ−d_tyⁿ(u_h)k²₀+Ckuⁿ−uⁿ_hk²₀+Ckrⁿ₁k²₀+δkdiveⁿ₁k²₀, (40) then, using the estimate (39), we have

kdiveⁿ₁k₀ ≤Cky_tⁿ−d_tyⁿ(u_h)k₀+Ckuⁿ−uⁿ_hk₀+Ckr₁ⁿk₀

≤C(4t+h+kuⁿ−uⁿ_hk0). (41) Then (36) follows from (38) and (41).

Part II.Similarly, choose~v =eⁿ⁻¹₂ andw=r₂ⁿ⁻¹as the test functions and add the two relations of (34)-(35), then we obtain that

(A⁻¹eⁿ⁻¹₂ , eⁿ⁻¹₂ ) + (φ⁰(yⁿ)r₂ⁿ⁻¹, r₂ⁿ⁻¹) = (rⁿ₁, r₂ⁿ⁻¹) + (z_tⁿ−dtzⁿ(uh), rⁿ⁻¹₂ )

−(eⁿ₁, eⁿ⁻¹₂ )−( ˜φ⁰⁰(yⁿ)zⁿ⁻¹(u_h)r₁ⁿ, rⁿ⁻¹₂ ).

Then, usingδ-Cauchy inequality, we can find an estimate as follows keⁿ⁻¹₂ k²₀+kr₂ⁿ⁻¹k²₀ ≤C((4t)²+h²+kuⁿ−uⁿ_hk²₀)

+δ(krⁿ⁻¹₂ k²₀+keⁿ⁻¹₂ k²₀), (42) or equivalently,

keⁿ⁻¹₂ k₀+krⁿ⁻¹₂ k₀ ≤C(4t+h+kuⁿ−uⁿ_hk₀). (43) Takingw = diveⁿ⁻¹₂ as a test function in (35) and usingδ-Cauchy inequality, then we get

kdiveⁿ⁻¹₂ k²₀ = (r₁ⁿ,diveⁿ⁻¹₂ )−(φ⁰(yⁿ)rⁿ⁻¹₂ ,diveⁿ⁻¹₂ ) +(z_tⁿ−d_tzⁿ(u_h),diveⁿ⁻¹₂ )−( ˜φ⁰⁰(yⁿ)zⁿ⁻¹(u_h)r₁ⁿ,diveⁿ⁻¹₂ )

≤Ckz_tⁿ−d_tzⁿ(u_h)k²₀ +Ckrⁿ₁k²₀+Ckr₂ⁿ⁻¹k²₀ +δkdiveⁿ⁻¹₂ k²₀, (44) then, using the estimate (39) and (43), we verify that

kdiveⁿ⁻¹₂ k₀ ≤C(4t+h+kuⁿ−uⁿ_hk₀). (45) This implies (37).

Now we combine the bounds given by Theorems 3.3-3.4 to come up with the following main results.

Theorem 3.5 Let(~p, y, ~q, z, u)∈(V~ ×W)²×Kand(~p_h, y_h, ~q_h, z_h, u_h)∈(V~_h× W_h)² ×K be the solutions of (11)-(17) and (22)-(28), respectively. Assume that

∀n = [0,1,· · · , N],B^∗zⁿ+uⁿ∈H¹(Ω). Then, we have

|||u−u_h|||_L²_(J;L²_(Ω)) ≤C(4t+h), (46)

|||~p−~p_h|||_L^∞_(J;H(div))+|||y−y_h|||_L^∞_(J;L²_(Ω)) ≤C(4t+h), (47)

|||~q−~q_h|||_L^∞_(J;H(div))+|||z−z_h|||_L^∞_(J;L²_(Ω)) ≤C(4t+h). (48)

4 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.

References

[1] W. Alt and U. Machenroth, Convergence of finite element approximations to state constrained convex parabolic boundary control problems, SIAM J. Con- trol Optim., 27(1989), 718-736.

[2] N. Arada, E. Casas and F. Tr¨oltzsch, Error estimates for the numerical ap- proximation of a semilinear elliptic control problem, Comp. Optim. Appl., 23(2002), 201-229.

[3] E. Casas, Using piecewise linear functions in the numerical approximation of semilinear elliptic control problems, Adv. Comp. Math., 26(2007), 137-153.

[4] Y. Chen and W.B. Liu, Posteriori error estimates for mixed finite elements of a quadratic optimal control problem, Recent Progress Comp. Appl. PDEs, 2(2002), 123-134.

[5] Y. Chen and W.B. Liu, Error estimates and superconvergence of mixed finite elements for quadratic optimal control, Internat. J. Numer. Anal. Modeling, 3(2006), 311-321.

[6] Y. Chen and W.B. Liu, A posteriori error estimates for mixed finite element so- lutions of convex optimal control problems, J. Comp. Appl. Math., 211(2008), 76-89.

[7] Y. Chen and Z. Lu, Error estimates for parabolic optimal control problem by fully discrete mixed finite element methods, Finite Elem. Anal. Des., 46(2010), 957-965.

[8] Y. Chen and Z. Lu, Error estimates of fully discrete mixed finite element meth- ods for semilinear quadratic parabolic optimal control problems, Comput.

Methods Appl. Mech. Engrg., 199(2010), 1415-1423.

[9] L. Evans, Partial differential equations, Amer. Math. Soc., Providence, RI, (1998).

[10] F.S. Falk, Approximation of a class of optimal control problems with order of convergence estimates, J. Math. Anal. Appl., 44(1973), 28-47.

[11] S.M.F. Garcia, Improved error estimates for mixed finite-element approxima- tions for nonlinear parabolic equations: The continutious-time case, Numer.

Methods Partial Differential Equations, 10(1994), 129-147.

[12] S.M.F. Garcia, Improved error estimates for mixed finite-element approxima- tions for nonlinear parabolic equations: The discrete-time case,Numer. Meth- ods Partial Differential Equations, 10(1994), 149-169.

[13] M.D. Gunzburger and S.L. Hou, Finite dimensional approximation of a class of constrained nonlinear control problems, SIAM J. Control Optim., 34(1996), 1001-1043.

[14] M. Hinze, A variational discretization concept in control constrained optimiza- tion: the linear-quadratic case, J. Comput. Optim. Appl., 30(2010), 45-63.

[15] G. Knowles, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim., 20(1982), 414-427.

[16] Y. Kwon and F.A. Milner,L^∞-error estimates for mixed methods for semilin- ear second-order elliptic equations, SIAM J. Numer. Anal., 25(1988), 46-53.

[17] J.L. Lions,Optimal Control of Systems Governed by Partial Differential Equa- tions, Springer, Berlin, (1971).

[18] Z. Lu and Y. Chen, A posteriori error estimates of triangular mixed finite el- ement methods for semilinear optimal control problems, Adv. Appl. Math.

Mech., 1(2) (2009), 242-256.

[19] Z. Lu and Y. Chen, L^∞-error estimates of triangular mixed finite element methods for optimal control problem govern by semilinear elliptic equation, Numer. Anal. Appl., 12(2) (2009),74-86.

[20] Z. Lu, Y. Chen and H. Zhang, A priori error estimates of mixed finite element methods for nonlinear quadratic optimal control problems, Lobachevskii J.

Math., 29(3) (2008), 164-174.

[21] K. Malanowski, Convergence of approximations vs. regularity of solutions for convex, control constrained, optimal control systems,Appl. Math. Optim., 8(2003), 69-95.

[22] D. Meidner and B. Vexler, A priori error estimates for space-time finite ele- ment discretization of parabolic optimal control problems. I. problems without control constraints,SIAM J. Control Optim., 47(2008), 1150-1177.

[23] F.A. Miliner, Mixed finite element methods for quasilinear second-order ellip- tic problems,Math. Comp., 44(1985), 303-320.

[24] P. Neittaanmaki and D. Tiba, Optimal Control of Nonlinear Parabolic Sys- tems: Theory, Algorithms and Applications, Marcell Dekker, New York, (1994).

[25] P.A. Raviart and J.M. Thomas, A mixed finite element method for 2nd order elliptic problems, math. aspects of the finite element method,Lecture Notes in Math., Springer, Berlin, 606(1977), 292-315.

[26] D. Tiba and F. Troltzsch, Error estimates for the discretization of state con- strained convex control problems, Numer. Funct. Anal. Optim., 17(1996), 1005-1028.

[27] X. Xing and Y. Chen, Error estimates of mixed methods for optimal control problems governed by parabolic equations,Internat. J. Numer. Methods Eng., 75(6) (2008), 735-754.