A Posteriori Error Estimates for a Semidiscrete Parabolic Integrodifferential Control on Multimeshes

Abstract

We extend the existing techniques to study semidiscrete adaptive finite element approximation schemes for a constrained optimal control problem governed by parabolic integrodifferential equations. The control problem involves time accumulation and the control constrain is given in an integral obstacle sense. We first prove the uniqueness and existence of the solution of this optimal control problem. We then derive the upper a posteriori error estimators for both the state and the control approximation, which are useful indicators in adaptive multimesh finite element approximation schemes.

1. Introduction

For the last twenty years great progress has been made on standard finite element approximation of optimal control problems governed by elliptic or parabolic equations; see, for example, [1–9], although it is impossible to give even a very brief review here. More specifically, research on finite element approximation of parabolic optimal control problems can be found in, for example, [10, 11]. Systematic introduction of the finite element method for PDEs and optimal control problems can be found in, for example, [5, 12, 13].

In many real modeling applications it is important to consider time accumulation. Parabolic integrodifferential equations and their control of this nature appear in heat conduction in materials with memory, population dynamics, and viscoelasticity; see, for example, Friedman and Shinbrot [14], Heard [15], and Renardy et al. [16]. This calls for more studies on the finite element approximation on integrodifferential equations. For instance, finite element methods for parabolic integrodifferential equations with a smooth kernel have been discussed in, for example, Cannon et al. [17], Sloan and Thomée [18], and Yanik and Fairweather [19].

Recently adaptive finite element method has been widely studied and applied in practical control computations. In order to obtain a numerical solution of acceptable accuracy for the optimal control problem, the finite element meshes have to be refined according to a mesh refinement scheme. Adaptive finite element approximation uses a posteriori indicators to guide the mesh refinement procedure. Only the area where the error indicator is larger will be refined so that a higher density of nodes is distributed over the area where the solution is difficult to approximate. In this sense efficiency and reliability of adaptive finite element approximation rely very much on the error indicator used.

Adaptive finite element schemes have been widely studied for some optimal control problems governed by elliptic and parabolic PDEs as well. Some of recent accounts of progress can be found, for example, in [11, 20, 21]. Although there exists so much progress in adaptive finite element elliptic and parabolic control problems, it is much more complicated to study and implement adaptive multimeshes computational schemes for parabolic integrodifferential control problems. More specificity, there has been a lack of a posteriori error estimates for any parabolic integrodifferential control problem, in spite of the fact that such control problems are widely encountered in practical engineering applications and scientific computations as we discussed above.

In this work, we extend the results and the techniques used in [20–22] to a quadratic optimal control problem governed by a linear parabolic integrodifferential equation, which can be generalized to the control problems with more general objective functions. The semidiscrete finite element scheme for this problem is presented. We derive the upper a posteriori error estimates for the semidiscrete finite element approximation for the case where the control constraints are given in an integral sense: 𝑈_𝑎𝑑={𝑣∈𝑋;∫_Ω𝑈𝑣⩾0,𝑡∈[0,𝑇]}.

We are interested in the following optimal control problems: min_{𝑢∈𝑈𝑎𝑑⊂𝑋}𝐽𝑢,𝑦(𝑢)=^𝑇₀𝑔𝑦+ℎ(𝑢)𝑑𝑡 (1.1) subject to 𝑦_𝑡+𝛼𝑦+^𝑡₀𝛾()𝑦(𝜏)(],[],𝑦𝑡,𝜏𝑑𝜏=𝑓+𝐵𝑢,inΩ×0,𝑇𝑦=0,on𝜕Ω×0,𝑇_𝑡=0=𝑦⁰,inΩ, (1.2) where Ω and Ω_𝑈 are bounded open sets in 𝑅^𝑛(𝑛⩾2) with the Lipschitz boundary 𝜕Ω and 𝜕Ω_𝑈, 𝑦⁰∈𝐻¹₀(Ω), 𝑓∈𝐿²(0,𝑇;𝐿²(Ω)), 𝑈=𝐿²(Ω_𝑈), 𝑋=𝐿²(0,𝑇;𝑈), 𝑈_𝑎𝑑⊆𝑋 is a closed convex subset; 𝛼 is a linear strongly elliptic self-adjoint partial differential operator of second order with coefficients depending smoothly on spacial variables, 𝛾(𝑡,𝜏) is an arbitrary second-order linear partial differential operator, with coefficients depending smoothly on both of time and spacial variables in the closure of their respective domains, and 𝐵 is a suitable linear and continuous operator.

Here we assume 𝑔(⋅) is a convex functional which is continuously differential on 𝐿²(Ω), and ℎ(⋅) is a strictly convex continuously differential functional on 𝑈. We further assume that ℎ(𝑢)→+∞ as ‖𝑢‖_𝑈→+∞ and that 𝑔(⋅) is bounded below. Details will be specified later.

The plan of the paper is as follows: in Section 2, we give the weak formulation and prove the existence and uniqueness of the solution for this optimal control problem. In Section 3, we will give a brief review on the finite element methods and optimality conditions and construct the semidiscrete finite element approximation schemes for the optimal control problem. In Section 4, the upper a posteriori error bounds in 𝐿²(0,𝑇;𝐻¹(Ω))-norm are derived for the control problem. In Section 5, we obtain the a posteriori error bounds in 𝐿²(0,𝑇;𝐿²(Ω))-norm for the control problem.

2. Existence and Uniqueness of the Solution of the Model Problem

In this paper, we adopt the standard notation 𝑊^𝑚,𝑞(Ω) for Sobolev spaces on Ω with norm ‖⋅‖_𝑚,𝑞,Ω, and seminorm |⋅|_𝑚,𝑞,Ω. We set 𝑊₀^𝑚,𝑞(Ω)={𝑤∈𝑊^𝑚,𝑞(Ω)∶𝑤|_𝜕Ω=0}. We denote 𝑊^𝑚,2(Ω)(𝑊₀^𝑚,2(Ω)) by 𝐻^𝑚(Ω)(𝐻^𝑚₀(Ω)), with norm ‖⋅‖_𝑚,Ω, and seminorm |⋅|_𝑚,Ω. In addition 𝑐 or 𝐶 denotes a general positive constant independent of ℎ.

We denote by 𝐿^𝑠(0,𝑇;𝑊^𝑚,𝑞(Ω)) the Banach space of all 𝐿^𝑠 integrable functions from (0,𝑇) into 𝑊^𝑚,𝑞(Ω) with norm ‖𝑣‖_𝐿^𝑠_(0,𝑇;𝑊^𝑚,𝑞_(Ω))=(∫^𝑇₀‖𝑣‖^𝑠_𝑊^𝑚,𝑞_(Ω)𝑑𝑡)^1/𝑠 for 𝑠∈[1,∞) and the standard modification for 𝑠=∞. Similarly, one can define the spaces 𝐻¹(0,𝑇;𝑊^𝑚,𝑞(Ω)) and 𝐶^𝑘(0,𝑇;𝑊^𝑚,𝑞(Ω)). The details can be found in [23]. To fix idea, we will take the state space 𝑊=𝐿²(0,𝑇;𝑉) with 𝑉=𝐻¹₀(Ω) and the control space 𝑋=𝐿²(0,𝑇;𝑈). Let the observation space be 𝑌=𝐿²(0,𝑇;𝐻) with 𝐻=𝐿²(Ω).

Let 𝑓₁,𝑓₂=_Ω𝑓₁𝑓₂,∀𝑓₁,𝑓₂()∈𝐻×𝐻,𝑢,𝑣_𝑈=_Ω𝑈()𝑎()=()()=𝛾()𝑢𝑣,∀𝑢,𝑣∈𝑈×𝑈,𝑧,𝜔𝛼𝑧,𝜔,𝑐𝑡,𝜏;𝑧,𝜔𝑡,𝜏𝑧,𝜔,∀𝑧,𝑤∈𝑉×𝑉. (2.1)

Assume that there are constants 𝑐 and 𝐶, such that for all 𝑡, 𝜏∈[0,𝑇]: (𝑎)𝑎(𝑧,𝑧)⩾𝑐‖𝑧‖²_1,Ω(𝑏𝑎(‖𝑧‖,∀𝑧∈𝑉,)|𝑧,𝑤)|⩽𝐶_1,Ω‖𝑤‖_1,Ω(𝑐𝑐(‖𝑧‖,∀𝑧,𝑤∈𝑉,)|𝑡,𝜏;𝑧,𝑤)|⩽𝐶_1,Ω‖𝑤‖_1,Ω,∀𝑧,𝑤∈𝑉. (2.2)

We will assume the following convexity conditions: ℎ^′(𝑢)−ℎ^′(𝑣),𝑢−𝑣⩾𝑐‖𝑢−𝑣‖²_0,Ω𝑈,∀𝑢,𝑣∈𝐿²Ω_𝑈, (2.3) that is to say ℎ(⋅) is uniformly convex.

Noting that 𝑔(⋅) is convex, it is easy to see that 𝑔^′(𝑢)−𝑔^′(𝑣),𝑢−𝑣⩾0,∀𝑢,𝑣∈𝐻¹(Ω). (2.4)

Also, we have that |𝐵𝑣,𝑤|⩽𝑐‖𝑣‖_0,Ω𝑈‖𝑤‖_0,Ω,∀𝑣∈𝐿²Ω_𝑈,𝑢∈𝐻¹(Ω) (2.5) because 𝐵 is a bounded linear operator.

Therefore the above-mentioned control problem (1.1)-(1.2) can be restated as follows (OCP): min_{𝑢∈𝑈𝑎𝑑}𝐽𝑢,𝑦(𝑢)=^𝑇₀𝑔𝑦+ℎ(𝑢)𝑑𝑡,𝜕𝑦+𝜕𝑡,𝑤+𝑎𝑦,𝑤^𝑡₀𝑐(𝜏)(],𝑦𝑡,𝜏;𝑦,𝑤𝑑𝜏=𝑓+𝐵𝑢,𝑤,∀𝑤∈𝑉,𝑡∈0,𝑇_𝑡=0=𝑦⁰, (2.6) where 𝐵 is a linear bounded operator from 𝐿²(Ω_𝑈) to 𝐿²(Ω) and independent with 𝑡.

From Yanik and Fairweather [19], we know that the above state equation has at least one solution 𝑦∈𝑊(0,𝑇)={𝑤∈𝐿²(0,𝑇;𝐻¹(Ω)),𝑤^′_𝑡∈𝐿²(0,𝑇;𝐻⁻¹(Ω))}. For the existence and uniqueness of the solution of the system (2.6), we have the following lemmas.

Lemma 2.1. For the optimal control problem (2.6), there exists the unique solution (𝑢,𝑦), such that 𝑢∈𝑋 and 𝑦∈𝐿^∞(0,𝑇;𝐿²(Ω))∩𝐿²(0,𝑇;𝐻¹₀(Ω)) and 𝜕𝑦/𝜕𝑡∈𝐿²(0,𝑇;𝐻⁻¹(Ω)).

Proof. Assume that {(𝑢^𝑛,𝑦^𝑛)}^∞_𝑛=1 is a minimization sequence for the problem (2.6). It follows that {𝑢^𝑛}^∞_𝑛=1 are bounded in 𝐿²(0,𝑇;𝐿²(Ω_𝑈)). Therefore there is a subsequence of {𝑢^𝑛}^∞_𝑛=1 (still denoted by {𝑢^𝑛}^∞_𝑛=1) such that 𝑢^𝑛 converges to 𝑢^∗ weakly in 𝐿²(0,𝑇;𝐿²(Ω_𝑈)). It is clear that for the subsequence 𝑢^𝑛𝜕𝑦^𝑛𝑦𝜕𝑡,𝑤+𝑎^𝑛,𝑤+^𝑡₀𝑐𝑡,𝜏;𝑦^𝑛(𝜏),𝑤(𝑡)𝑑𝜏=𝑓+𝐵𝑢^𝑛,𝑤,∀𝑤∈𝑉,𝑡∈(0,𝑇]. (2.7)
By taking 𝑤=𝑦^𝑛 and integrating time from 0 to 𝑡 in (2.7), and applying Gronwall's inequality, we have ^𝑇₀𝑦^𝑛2_1,Ω𝑑𝜏⩽𝐶𝑦⁰²_0,Ω+^𝑇₀𝑓²_−1,Ω+𝑢^𝑛2_0,Ω𝑈𝑒^𝐶𝑇⩽𝐶^∗<∞,max_{0⩽𝑡⩽𝑇}𝑦^𝑛(𝑡)²_0,Ω⩽𝐶𝑦⁰²_0,Ω+^𝑇₀𝑓²_−1,Ω+𝑢^𝑛2_0,Ω𝑈+^𝑇₀^𝑡₀𝑦^𝑛(𝜏)²_1,Ω𝑑𝜏𝑑𝑡⩽𝐶. (2.8)
This infers that 𝑢^𝑛∈𝐿²(0,𝑇;𝐿²(Ω_𝑈)) and 𝑦^𝑛∈𝐿^∞(0,𝑇;𝐿²(Ω))∩𝐿²(0,T;𝐻¹(Ω)), and 𝑢^𝑛→𝑢weaklyin𝐿²0,𝑇;𝐿²Ω_𝑈,𝑦^𝑛→𝑦weaklyin𝐿^∞0,𝑇;𝐿²(Ω)𝐿²0,𝑇;𝐻¹(Ω),𝑦^𝑛(𝑇)→𝑦(𝑇)weaklyin𝐿²(Ω). (2.9)
Now let us integrate time from 0 to 𝑇 in (2.7) and take limits as 𝑛→∞. Clearly we have 𝑦(𝑇),𝑤(𝑇)−𝑦⁰,𝑤(0)−^𝑇₀𝑦,𝑤^′_𝑡𝑑𝑡+^𝑇₀𝑎𝑦,𝑤𝑑𝑡+^𝑇₀^𝑡₀𝑐(𝜏)(𝑡)=𝑡,𝜏;𝑦,𝑤𝑑𝜏𝑑𝑡_𝑇0𝑓+𝐵𝑢,𝑤𝑑𝑡,∀𝑤∈𝑊(0,𝑇). (2.10)
Therefore ^𝑇₀𝜕𝑦𝜕𝑡,𝑤𝑑𝑡+^𝑇₀𝑎𝑦,𝑤𝑑𝑡+^𝑇₀^𝑡₀𝑐(𝜏)(𝑡)=𝑡,𝜏;𝑦,𝑤𝑑𝜏𝑑𝑡_𝑇0𝑓+𝐵𝑢,𝑤𝑑𝑡,∀𝑤∈𝑊(0,𝑇). (2.11)
Furthermore, we have 𝜕𝑦𝜕𝑡_𝐿²_(0,𝑇;𝐻⁻¹_(Ω))=sup_{𝑤∈𝐿20,𝑇;𝐻10(Ω)}∫^𝑇₀𝜕𝑦/𝜕𝑡,𝑤𝑑𝑡‖𝑤‖_𝐿²_(0,𝑇;𝐻¹_0(Ω))⩽𝐶𝑦⁰²_0,Ω+^𝑇₀𝑓²_−1,Ω+‖𝑢‖²_0,Ω𝑈𝑑𝑡𝑒^𝐶𝑇. (2.12)
It follows that 𝜕𝑦/𝜕𝑡∈𝐿²(0,𝑇;𝐻⁻¹(Ω)).
Since 𝑔(⋅) is a convex function on space 𝐿²(0,𝑇;𝐿²(Ω)) and ℎ(⋅) is a strictly convex function on 𝑈_𝑎𝑑, we have ^𝑇₀𝑔𝑦+ℎ(𝑢)𝑑𝑡⩽lim_𝑛→∞^𝑇₀𝑔𝑦^𝑛+ℎ𝑢^𝑛𝑑𝑡. (2.13)
It follows that (𝑢,𝑦) is one solution of (2.6). Because 𝐽(𝑢,𝑦(𝑢)) is a strictly convex function on 𝑈_𝑎𝑑, we have that the solution for the minimization problem (2.6) is unique. The proof of Lemma 2.1 is completed.

Remark 2.2. Here we suppose that the operator 𝛼 is independent of time variable 𝑡. The above results can also be applied to the case 𝛼=𝛼(𝑥,𝑡) provided that suitable conditions for the operator 𝛼 are to be imposed.

3. Finite Element Approximation of Control

In this section, we firstly state the optimality conditions and set up the finite element approximation for optimal control problems governed by parabolic integrodifferential equation.

It follows from [24] that we can similarly deduce the following optimality conditions of the problem (2.6).

Theorem 3.1. A pair (𝑦,𝑢)∈(𝐿²(0,𝑇;𝐻¹₀(Ω))∩𝐿^∞(0,𝑇;𝐿²(Ω)))×𝐿²(0,𝑇;𝐿²(Ω_𝑈)) is the solution of the optimal control problem (2.6), if and only if there exists a costate 𝑝∈𝐿²(0,𝑇;𝐻¹₀(Ω))∩𝐿^∞(0,𝑇;𝐿²(Ω)) such that the triple (𝑦,𝑝,𝑢) satisfies the following optimality conditions: 𝜕𝑦+𝜕𝑡,𝑤+𝑎𝑦,𝑤^𝑡₀𝑐(𝜏)(𝑡)(],𝑦𝑡,𝜏;𝑦,𝑤𝑑𝜏=𝑓+𝐵𝑢,𝑤,∀𝑤∈𝑉,𝑡∈0,𝑇_𝑡=0=𝑦⁰,−𝑞,𝜕𝑝+𝜕𝑡+𝑎𝑞,𝑝^𝑇_𝑡𝑐𝜏,𝑡;𝑞(𝑡),𝑝(𝜏)𝑑𝜏=𝑦−𝑧_𝑑[),𝑝,𝑞,∀𝑞∈𝑉,𝑡∈0,𝑇_𝑡=𝑇=0,_𝑇0ℎ^′(𝑢)+𝐵^∗𝑝,𝑣−𝑢_𝑈𝑑𝑡⩾0,∀𝑣∈𝑈_𝑎𝑑, (3.1)(3.2)(3.3) where 𝐵^∗ is the adjoint operator of 𝐵.

In the following, we construct the semidiscrete finite element approximation of the control problem (2.6) by approximating the optimality conditions.

Let Ω^ℎ be a polygonal approximation to Ω with boundary 𝜕Ω^ℎ. For simplicity, we assume that Ω is a convex polygon so that Ω=Ω^ℎ. Let 𝑇^ℎ be a partitioning of Ω^ℎ into disjoint regular 𝑛-simplices 𝜏, so that Ω^ℎ=⋃_{𝜏∈𝑇ℎ}𝜏. Each element has at most one face on 𝜕Ω^ℎ, and 𝜏 and 𝜏^′ have either only one common vertex or a whole edge or face if 𝜏 and 𝜏^′∈𝑇^ℎ. As usual, ℎ denotes the diameter of the triangulation 𝑇^ℎ.

Associated with 𝑇^ℎ is a finite-dimensional subspace 𝑆^ℎ of 𝐶(Ω^ℎ), such that 𝜒|_𝜏 are polynomials of 𝑚(𝑚⩾1) order for all 𝜒∈𝑆^ℎ and 𝜏∈𝑇^ℎ. Let 𝑉^ℎ=𝑆^ℎ∩𝐻¹₀(Ω). Note that we do not impose a continuity requirement. It is easy to see that 𝑉^ℎ⊂𝑉, 𝑊^ℎ⊂𝑊.

Let 𝑇^ℎ_𝑈 be a partitioning of Ω^ℎ_𝑈 into disjoint regular 𝑛-simplices 𝜏_𝑈, so that Ω^ℎ_𝑈=⋃_{𝜏𝑈∈𝑇ℎ𝑈}𝜏_𝑈. For simplicity, we again assume that Ω_𝑈 is a convex polygon so that Ω_𝑈=Ω^ℎ_𝑈. 𝜏_𝑈 and 𝜏^′_𝑈 have either only one common vertex or a whole edge or face if 𝜏_𝑈 and 𝜏^′_𝑈∈𝑇^ℎ_𝑈. Let ℎ_𝜏(ℎ_𝜏𝑈) denote the maximum diameter of the element 𝜏(𝜏_𝑈) in 𝑇^ℎ(𝑇^ℎ_𝑈).

Associated with 𝑇^ℎ_𝑈 is another finite-dimensional subspace 𝑈^ℎ of 𝐿²(Ω^ℎ_𝑈), such that 𝜒|_𝜏𝑈 are polynomials of order 𝑚(𝑚⩾0) for all 𝜒∈𝑈^ℎ and 𝜏_𝑈∈𝑇^ℎ_𝑈. Here there is no requirement for the continuity. Let 𝑋^ℎ=𝐿²(0,𝑇;𝑈^ℎ). It is easy to see that 𝑋^ℎ⊂𝑋. Let ℎ_𝜏(ℎ_𝜏𝑈) denote the maximum diameter of the element 𝜏(𝜏_𝑈) in 𝑇^ℎ(𝑇^ℎ_𝑈).

In this paper we only consider the piecewise constant finite element space for the approximation of the control for the reason of the limited regularity of the optimal control 𝑢 in general. For ease of exposition, in this paper we assume that 𝑈^ℎ_𝑎𝑑⊂𝑈_𝑎𝑑∩𝑋^ℎ.

In order to derive a posteriori error estimates of residual type, we need the following important lemmas.

Lemma 3.2 (see [12]). Let 𝜋_ℎ be the standard Lagrange interpolation operator. For 𝑚=0 or 1, 𝑞>𝑛/2 and 𝑣∈𝑊^2,𝑞(Ω), 𝑣−𝜋_ℎ𝑣_𝑚,𝑞,Ω⩽𝐶ℎ^2−𝑚|𝑣|_2,𝑞,Ω. (3.4)

Lemma 3.3 (see [25]). Let 𝜋_ℎ be the average interpolation operator defined in [25]. For 𝑚=0 or 1, 1⩽𝑞⩽∞ and for all 𝑣∈𝑊^1,𝑞(Ω^ℎ)𝑣−𝜋_ℎ𝑣_{𝑚,𝑞,𝜏}⩽_{𝜏′⋂𝜏≠∅}𝐶ℎ_𝜏^1−𝑚|𝑣|_{1,𝑞,𝜏′}. (3.5)

Lemma 3.4 (see [26]). For all 𝑣∈𝑊^1,𝑞(Ω), 1⩽𝑞<∞‖𝑣‖_{0,𝑞,𝜕𝜏}⩽𝐶ℎ_𝜏^−1/𝑞‖𝑣‖_0,𝑞,𝜏+ℎ_𝜏^1−1/𝑞|𝑣|_1,𝑞,𝜏. (3.6)

Then a possible semidiscrete finite element approximation of (OCP) is thus defined by (OCP)^ℎ: min_{𝑢ℎ∈𝑈ℎ𝑎𝑑}𝐽𝑢_ℎ,𝑦_ℎ=^𝑇₀𝑔𝑦_ℎ+ℎ𝑢_ℎ𝑑𝑡 (3.7) subject to 𝜕𝑦_ℎ𝜕𝑡,𝑤_ℎ+𝑎𝑦_ℎ,𝑤_ℎ+^𝑡₀𝑐𝑡,𝜏;𝑦_ℎ(𝜏),𝑤_ℎ(𝑡)𝑑𝜏=𝑓+𝐵𝑢_ℎ,𝑤_ℎ,∀𝑤_ℎ∈𝑉^ℎ(],𝑦,𝑡∈0,𝑇_ℎ𝑡=0=𝑦⁰_ℎ, (3.8) where 𝑦_ℎ∈𝑊^ℎ, 𝑦⁰_ℎ∈𝑉^ℎ is the approximation of 𝑦⁰.

Since this is a finite dimensional linear control problem and the reduced objective function is convex, we can easily prove that the above problem (3.7)-(3.8) has a unique solution (𝑦_ℎ,𝑢_ℎ)∈𝑊^ℎ×𝑈^ℎ_𝑎𝑑.

By applying [24] again we can show that a pair (𝑦_ℎ,𝑢_ℎ)∈𝑊^ℎ×𝑈^ℎ_𝑎𝑑 is a solution of (3.7)-(3.8), if and only if there exists a costate 𝑝_ℎ∈𝑊^ℎ such that the triple (𝑦_ℎ,𝑝_ℎ,𝑢_ℎ) satisfies the following optimality conditions, which we will label (OCP−OPT)^ℎ: 𝜕𝑦_ℎ𝜕𝑡,𝑤_ℎ+𝑎𝑦_ℎ,𝑤_ℎ+^𝑡₀𝑐𝑡,𝜏;𝑦_ℎ(𝜏),𝑤_ℎ(𝑡)𝑑𝜏=𝑓+𝐵𝑢_ℎ,𝑤_ℎ,∀𝑤_ℎ∈𝑉^ℎ,𝑦_ℎ𝑡=0=𝑦⁰_ℎ,−𝑞_ℎ,𝜕𝑝_ℎ𝑞𝜕𝑡+𝑎_ℎ,𝑝_ℎ+^𝑇_𝑡𝑐𝜏,𝑡;𝑞_ℎ,𝑝_ℎ(𝜏)𝑑𝜏=𝑦_ℎ−𝑧_𝑑,𝑞_ℎ,∀𝑞_ℎ∈𝑉^ℎ,𝑝_ℎ𝑡=𝑇=0,_𝑇0ℎ^′𝑢_ℎ+𝐵^∗𝑝_ℎ,𝑣_ℎ−𝑢_ℎ𝑈𝑑𝑡⩾0,∀𝑣_ℎ∈𝑈^ℎ_𝑎𝑑. (3.9)(3.10)(3.11)

The optimality conditions in (3.9)–(3.11) are the semidiscrete approximation to the problem (3.1)–(3.3).

4. A Posteriori Error Estimates for First-Order Derivatives

Adaptive finite element approximation has been found very useful in computing optimal control, as mentioned in Introduction. It uses an a posteriori error indicator to guide the mesh refinement procedure. Furthermore it has been recently found that for constrained control problems, different adaptive meshes are often needed for the control and the states; see [27]. Using different adaptive meshes for the control and the state allows very coarse meshes to be used in solving the state and costate equations. Thus much computational work can be saved since one of the major computational loads is to solve the state and costate equations repeatedly. In this section, we derive the upper a posteriori error estimates for the optimal control problem allowing different meshes to be used for the states and the control.

For simplicity, we will only consider the case of quadratic objective functionals as follows: 𝐽𝑢,𝑦=^𝑇₀𝑔𝑦(𝑢1+ℎ)𝑑𝑡=2^𝑇₀𝑦−𝑧_𝑑²_0,Ω+𝛽2^𝑇₀‖𝑢‖²_0,Ω𝑈. (4.1)

Here 𝑔𝑦=12^𝑇₀𝑦−𝑧_𝑑²_0,Ω,ℎ(𝑢)=𝛽2^𝑇₀‖𝑢‖²_0,Ω𝑈, (4.2) where 𝛾 is a positive regularity constant. By differential theory, we have ℎ^′(𝑢)(𝑣)=𝛽𝑢,𝑣. (4.3)

Then the inequality (3.3) and (3.11) in optimality condition can be restated as follows: ^𝑇₀𝛽𝑢+𝐵^∗𝑝,𝑣−𝑢_𝑈𝑑𝑡⩾0,∀𝑣∈𝑈_𝑎𝑑,_𝑇0𝛽𝑢_ℎ+𝐵^∗𝑝_ℎ,𝑣_ℎ−𝑢_ℎ𝑈𝑑𝑡⩾0,∀𝑣_ℎ∈𝑈^ℎ_𝑎𝑑. (4.4)(4.5)

In this paper, we consider the integration obstacle type control constraint: 𝑈_𝑎𝑑=𝑣∈𝑋;_Ω𝑈𝑣⩾0,𝑡∈[0,𝑇]. (4.6)

By computation and from [28], we know that the solutions of inequality (4.4) and (4.5) yield 𝛽𝑢=−𝐵^∗∫𝑝+max0,^Ω𝑈𝐵^∗𝑝∫_Ω𝑈1,𝛽𝑢_ℎ=𝑃_ℎ−𝐵^∗𝑝_ℎ∫+max0,^Ω𝑈𝐵^∗𝑝_ℎ∫_Ω𝑈1, (4.7) where 𝑃_ℎ is the 𝐿²-projection from 𝐿²(Ω) to 𝑈^ℎ.

And we will consider the special case of the differential operator of 𝛼 and 𝛾: 𝛼𝑦=−div𝐴∇𝑦,𝛾(𝑡,𝜏)𝑦=−div𝐶(𝑡,𝜏)∇𝑦, (4.8) where 𝐴=𝐴(𝑥)=(𝑎_𝑖,𝑗(⋅))_𝑛×𝑛∈(𝐶^∞(Ω))^𝑛×𝑛, such that there are constants 𝑐>0 satisfying 𝑋^𝑡𝐴𝑋⩾𝑐‖𝑋‖²_𝑅^𝑛,∀𝑋∈𝑅^𝑛 (4.9) and 𝐶=𝐶(𝑥,𝑡,𝜏)=(𝑐_𝑖,𝑗(𝑥,𝑡,𝜏))_𝑛×𝑛∈(𝐶^∞(0,𝑇;𝐿²(Ω))^𝑛×𝑛).

In this case, we have the following bilinear form 𝑎()=()=(())=(),𝑐()=𝛾()=((𝐶()))=(𝐶()).𝑧,𝑤𝛼𝑧,𝑤−div𝐴∇𝑧,𝑤𝐴∇𝑧,∇𝑤𝑡,𝜏;𝑧,𝑤𝑡,𝜏𝑧,𝑤−div𝑡,𝜏∇𝑧,𝑤𝑡,𝜏∇𝑧,∇𝑤 (4.10)

4.1. Main Results

We first state the main results of this section.

Theorem 4.1. Let (𝑦,𝑢) and (𝑦_ℎ,𝑢_ℎ) be the solutions of (𝑂𝐶𝑃) and (𝑂𝐶𝑃)^ℎ, respectively. Let 𝑝 and 𝑝_ℎ be the solution of the costate equations (3.2) and (3.10). Then there hold the a posteriori error estimates 𝑢−𝑢_ℎ²_{𝐿20,𝑇;𝐿2Ω𝑈}+𝑦−𝑦_ℎ²_{𝐿∞0,𝑇;𝐿2(Ω)}+𝑦−𝑦_ℎ²_{𝐿20,𝑇;𝐻1(Ω)}+𝑝−𝑝_ℎ²_{𝐿∞0,𝑇;𝐿2(Ω)}+𝑝−𝑝_ℎ²_{𝐿20,𝑇;𝐻1(Ω)}⩽𝐶⁶_𝑖=1𝜂²_𝑖, (4.11) where 𝜂²₁,…,𝜂²₆ is defined as follows: 𝜂²₁=^𝑇_{0𝜏𝑈𝜏𝑈}𝐵^∗𝑝_ℎ−𝑃_ℎ𝐵^∗𝑝_ℎ²𝜂𝑑𝑡,₂₂=^𝑇_0𝜏ℎ²_𝜏𝜏𝜕𝑝𝜕𝑡^ℎ+div𝐴^∗∇𝑝_ℎ+^𝑇_𝑡div𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)𝑑𝜏+𝑦_ℎ−𝑧_𝑑²𝜂𝑑𝜏𝑑𝑡,₂₃=^𝑇_0𝜏ℎ_𝑙𝜕𝜏𝐴^∗∇𝑝_ℎ⋅𝑛+^𝑇_𝑡𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)⋅𝑛𝑑𝜏²𝜂𝑑𝑙𝑑𝑡,₂₄=^𝑇_0𝜏ℎ²_𝜏𝜏𝜕𝑦𝜕𝑡^ℎ−div𝐴∇𝑦_ℎ−^𝑡₀div𝐶(𝑡,𝜏)∇𝑦_ℎ(𝜏)𝑑𝜏−𝑓−𝐵𝑢_ℎ²𝜂𝑑𝑡,₂₅=^𝑇_0𝜏ℎ_𝑙𝜕𝜏𝐴∇𝑦_ℎ⋅𝑛+^𝑡₀𝐶(𝑡,𝜏)∇𝑦_ℎ(𝜏)⋅𝑛𝑑𝜏²𝜂𝑑𝑙𝑑𝑡,₂₆=𝑦^ℎ₀−𝑦₀²_𝐿2(Ω), (4.12) where 𝑙 is a face of an element 𝜏, ℎ_𝑙 is the maximum diameter of 𝑙, [∇𝑝_ℎ⋅𝑛] and [∇𝑦_ℎ⋅𝑛] are the normal derivative jumps over the interior face 𝑙, defined by ∇𝑝_ℎ⋅𝑛_𝑙=∇𝑝_ℎℎ_𝜏1𝑙−∇𝑝_ℎ𝜏2𝑙⋅𝑛,∇𝑦_ℎ⋅𝑛_𝑙=∇𝑦_ℎ𝜏1𝑙−∇𝑦_ℎ𝜏2𝑙⋅𝑛, (4.13) where 𝑛 is the unit normal vector on 𝑙=𝜏¹_𝑙∩𝜏²_𝑙 outwards 𝜏¹_𝑙. For later convenience, one defines [∇𝑝_ℎ⋅𝑛]_𝑙=0 and [∇𝑦_ℎ⋅𝑛]_𝑙=0 when 𝑙⊂𝜕Ω.

In the following subsections, we will prove Theorem 4.1. To this end, we first give some lemmas in the following subsection. The proofs of Theorem 4.1 are put in the last subsection.

4.2. Some Lemmas

We have the following.

Lemma 4.2. Let (𝑦,𝑢) and (𝑦_ℎ,𝑢_ℎ) be the solutions of (𝑂𝐶𝑃) and (𝑂𝐶𝑃)^ℎ, respectively. Let 𝑝 and 𝑝_ℎ be the solution of the costate equations (3.2) and (3.10). Then 𝑢−𝑢_ℎ²_{𝐿20,𝑇;𝐿2Ω𝑈}⩽𝐶𝜂²₁+𝐶𝑝_ℎ−𝑝𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)}, (4.14) where 𝑝(𝑢_ℎ) is defined by the following system: 𝜕𝑦𝑢𝜕𝑡^ℎ,𝜔+𝑎𝑦𝑢_ℎ,𝜔+^𝑡₀𝑐𝑡,𝜏;𝑦𝑢_ℎ(𝜏),𝜔(𝑡)𝑑𝜏=𝑓+𝐵𝑢_ℎ𝑦𝑢,𝜔,∀𝜔∈𝑉,_ℎ(0)=𝑦^ℎ₀(𝑥)−𝜕,𝑥∈Ω,𝑞,𝑝𝑢𝜕𝑡^ℎ+𝑎𝑞,𝑝𝑢_ℎ+^𝑇_𝑡𝑐𝜏,𝑡;𝑞(𝑡),𝑝𝑢_ℎ(𝜏)𝑑𝜏=𝑦𝑢_ℎ−𝑧_𝑑,𝑞,∀𝑞∈𝑉. (4.15)(4.16)(4.17)

Proof. It follows from (4.4) that we have 𝛽𝑢,𝑢−𝑢_ℎ𝑈⩽−𝐵^∗𝑝,𝑢−𝑢_ℎ𝑈. (4.18)
Then by (4.5) and (4.18) 𝛽𝑢−𝑢_ℎ²_{𝐿20,𝑇;𝐿2Ω𝑈}=^𝑇₀𝛽𝑢,𝑢−𝑢_ℎ𝑈−𝛽𝑢_ℎ,𝑢−𝑢_ℎ𝑈⩽𝑑𝑡_𝑇0−𝐵^∗𝑝,𝑢−𝑢_ℎ𝑈−𝛽𝑢_ℎ,𝑢−𝑢_ℎ𝑈𝑑𝑡=^𝑇₀−𝐵^∗𝑝+𝛽𝑢_ℎ,𝑢−𝑢_ℎ𝑈𝑑𝑡=−^𝑇₀𝐵^∗𝑝_ℎ+𝛽𝑢_ℎ,𝑢−𝑣_ℎ𝑈𝑑𝑡−^𝑇₀𝐵^∗𝑝_ℎ+𝛽𝑢_ℎ,𝑣_ℎ−𝑢_ℎ𝑈+𝑑𝑡_𝑇0𝐵^∗𝑝_ℎ−𝐵^∗𝑝𝑢_ℎ,𝑢−𝑢_ℎ𝑈𝑑𝑡+^𝑇₀𝐵^∗𝑝𝑢_ℎ−𝐵^∗𝑝,𝑢−𝑢_ℎ𝑈𝑑𝑡⩽inf_{𝑣ℎ∈𝑈ℎ𝑎𝑑}^𝑇₀𝐵^∗𝑝_ℎ+𝛽𝑢_ℎ,𝑣_ℎ−𝑢_𝑈+𝑑𝑡_𝑇0𝐵^∗𝑝_ℎ−𝑝𝑢_ℎ,𝑢−𝑢_ℎ𝑈𝑑𝑡+^𝑇₀𝐵^∗𝑝𝑢_ℎ−𝑝,𝑢−𝑢_ℎ𝑈𝑑𝑡=𝐼₁+𝐼₂+𝐼₃. (4.19)
Since 𝑃_ℎ is the 𝐿²-projection, then, for any 𝑣∈𝑈_𝑎𝑑, we have _Ω𝑈𝑃_ℎ𝑣−𝑣𝜙=0,∀𝜙∈𝑋^ℎ,𝑡∈(0,𝑇]. (4.20)
Note that ∫_Ω𝑈𝑣⩾0 and ∫_Ω𝑈(𝑃_ℎ𝑣−𝑣)=0. Then we have ∫_Ω𝑈𝑃_ℎ𝑣⩾0, thus 𝑃_ℎ𝑣∈𝑈^ℎ_𝑎𝑑. So that we can take 𝑣_ℎ=𝑃_ℎ𝑢 in 𝐼₁ and by (4.7), we have 𝐼₁⩽^𝑇₀𝐵^∗𝑝_ℎ+𝛽𝑢_ℎ,𝑃_ℎ𝑢−𝑢_𝑈=𝑑𝑡_{𝑇0𝜏𝑈𝜏𝑈}𝑃_ℎ−𝐵^∗𝑝_ℎ∫+max0,^Ω𝑈𝐵^∗𝑝_ℎ∫_Ω𝑈1+𝐵^∗𝑝_ℎ𝑃_ℎ⩽𝑢−𝑢𝑑𝑡_{𝑇0𝜏𝑈𝜏𝑈}−𝑃_ℎ𝐵^∗𝑝_ℎ+𝐵^∗𝑝_ℎ𝑃_ℎ𝑢−𝑢𝑑𝑡=^𝑇_{0𝜏𝑈𝜏𝑈}−𝑃_ℎ𝐵^∗𝑝_ℎ+𝐵^∗𝑝_ℎ𝑃_ℎ𝑢−𝑢_ℎ−𝑢−𝑢_ℎ(𝛿)𝑑𝑡⩽𝐶^𝑇_{0𝜏𝑈𝜏𝑈}−𝑃_ℎ𝐵^∗𝑝_ℎ+𝐵^∗𝑝_ℎ²𝑑𝑡+𝛿𝑢−𝑢_ℎ²_{𝐿20,𝑇;𝐿2Ω𝑈}=𝐶𝜂²₁+𝛿𝑢−𝑢_ℎ²_{𝐿20,𝑇;𝐿2Ω𝑈},𝐼₂⩽𝐶𝑝_ℎ−𝑝𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)}+𝛿𝑢−𝑢_ℎ²_{𝐿20,𝑇;𝐿2Ω𝑈}. (4.21)(4.22)
From (4.15) and (3.1), we have for 𝑡∈(0,𝑇]𝜕𝑢𝜕𝑡𝑦−𝑦_ℎ,𝜔+𝑎𝑦−𝑦𝑢_ℎ,𝑤+^𝑡₀𝑐𝑡,𝜏;𝑦−𝑦𝑢_ℎ𝜏)(𝑡)=𝐵(,𝜔𝑑𝜏𝑢−𝑢_ℎ,𝜔,∀𝜔∈𝑉 (4.23) and from (4.17) and (3.2) −𝜕𝑞,𝑢𝜕𝑡𝑝−𝑝_ℎ+𝑎𝑞,𝑝−𝑝𝑢_ℎ+^𝑇_𝑡𝑐𝜏,𝑡;𝑞(𝑡),𝑝−𝑝𝑢_ℎ𝜏)=𝑢(𝑑𝜏𝑦−𝑦_ℎ,𝑞,∀𝑞∈𝑉. (4.24)
Then from (4.23), (4.24) and integrating by part 𝐼₃=^𝑇₀𝑝𝑢_ℎ−𝑝,𝐵𝑢−𝑢_ℎ𝑈=𝑑𝑡_𝑇0𝜕𝑢𝜕𝑡𝑦−𝑦_ℎ,𝑝𝑢_ℎ−𝑝+𝑎𝑦−𝑦𝑢_ℎ,𝑝𝑢_ℎ+−𝑝_𝑡0𝑐𝑡,𝜏;𝑦−𝑦𝑢_ℎ(𝜏),𝑝𝑢_ℎ𝑡)=−𝑝(𝑑𝜏𝑑𝑡_𝑇0−𝑦−𝑦𝑢_ℎ,𝜕𝑢𝜕𝑡𝑝−𝑝_ℎ+𝑎𝑦−𝑦𝑢_ℎ,𝑝𝑢_ℎ+−𝑝_𝑇𝑡𝑐𝜏,𝑡;𝑦−𝑦𝑢_ℎ(𝑡),𝑝𝑢_ℎ(𝜏)=−𝑝𝑑𝜏𝑑𝑡_𝑇0−𝑦−𝑦𝑢_ℎ,𝑦−𝑦𝑢_ℎ𝑑𝑡⩽0. (4.25)
Following from (4.21)–(4.25), let 𝛿 be small enough: 𝑢−𝑢_ℎ²_{𝐿20,𝑇;𝐿2Ω𝑈}⩽𝐶𝜂²₁+𝐶𝑝_ℎ−𝑝𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)}. (4.26)
The proof of Lemma 4.2 is completed.

Lemma 4.3. Let (𝑦,𝑢) and (𝑦_ℎ,𝑢_ℎ) be the solutions of (𝑂𝐶𝑃) and (𝑂𝐶𝑃)^ℎ, respectively. Let 𝑝 and 𝑝_ℎ be the solution of the costate equations (3.2) and (3.10). Then there hold the a posteriori error estimates 𝑦_ℎ−𝑦𝑢_ℎ²_{𝐿∞0,𝑇;𝐿2(Ω)}+𝑦_ℎ−𝑦𝑢_ℎ²_{𝐿20,𝑇;𝐻1(Ω)}+𝑝_ℎ−𝑝𝑢_ℎ²_{𝐿∞0,𝑇;𝐿2(Ω)}+𝑝_ℎ−𝑝𝑢_ℎ²_{𝐿20,𝑇;𝐻1(Ω)}⩽𝐶⁶_𝑖=2𝜂²_𝑖. (4.27)

Proof. Let 𝐸𝑢_ℎ𝜕,𝑣=−𝑣,𝑝𝜕𝑡^ℎ−𝑝𝑢_ℎ+𝑎𝑣,𝑝_ℎ−𝑝𝑢_ℎ+^𝑇_𝑡𝑐𝜏,𝑡;𝑣(𝑡),𝑝_ℎ−𝑝𝑢_ℎ(𝜏)𝑑𝜏 (4.28) and 𝜋_ℎ be the average interpolation operator defined as in [27] and 𝑒=𝑝_ℎ−𝑝(𝑢_ℎ). Then from (3.10) and (4.17) −𝑞_ℎ,𝜕𝑝𝜕𝑡^ℎ−𝑝𝑢_ℎ+𝑎𝑞_ℎ,𝑝_ℎ−𝑝𝑢_ℎ+^𝑇_𝑡𝑐𝜏,𝑡;𝑞_ℎ(𝑡),𝑝_ℎ−𝑝𝑢_ℎ𝜏)=𝑦(𝑑𝜏_ℎ−𝑦𝑢_ℎ,𝑞_ℎ,∀𝑞_ℎ∈𝑉^ℎ. (4.29)
So we have −𝜕𝑣,𝑝𝜕𝑡^ℎ−𝑝𝑢_ℎ+𝑎𝑣,𝑝_ℎ−𝑝𝑢_ℎ+^𝑇_𝑡𝑐𝜏,𝑡;𝑣(𝑡),𝑝_ℎ−𝑝𝑢_ℎ(𝜏)𝑑𝜏=−𝑣−𝜋_ℎ𝜕𝑣,𝑝𝜕𝑡^ℎ+𝑎𝑣−𝜋_ℎ𝑣,𝑝_ℎ+^𝑇_𝑡𝑐𝜏,𝑡;𝑣−𝜋_ℎ𝑣(𝑡),𝑝_ℎ(𝜏+)𝑑𝜏𝑣−𝜋_ℎ𝜕𝑣,𝑝𝑢𝜕𝑡^ℎ−𝑎𝑣−𝜋_ℎ𝑣,𝑝𝑢_ℎ−^𝑇_𝑡𝑐𝜏,𝑡;𝑣−𝜋_ℎ𝑣(𝑡),𝑝𝑢_ℎ(𝜏)𝑑𝜏+𝑦_ℎ−𝑦𝑢_ℎ,𝜋_ℎ𝑣⩽_𝜏𝜏ℎ²_𝜏−𝜕𝑝𝜕𝑡^ℎ−div𝐴^∗∇𝑝_ℎ−^𝑇_𝑡div𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)𝑑𝜏−𝑦_ℎ+𝑧_𝑑²+_𝜏𝜕𝜏ℎ_𝑙𝐴^∗∇𝑝_ℎ⋅𝑛+^𝑇_𝑡𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)⋅𝑛𝑑𝜏^21/2‖𝑣‖_1,Ω+𝑦_ℎ−𝑦𝑢_ℎ,𝑣. (4.30)
By letting 𝑣=𝑝_ℎ−𝑝(𝑢_ℎ) in (4.30) and from (2.2), we have −12𝑑𝑝𝑑𝑡^ℎ−𝑝𝑢_ℎ²_0,Ω+𝑐𝑝_ℎ−𝑝𝑢_ℎ²_1,Ω⩽_𝜏𝜏ℎ²_𝜏−𝜕𝑝𝜕𝑡^ℎ−div𝐴^∗∇𝑝_ℎ−^𝑇_𝑡div𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)𝑑𝜏−𝑦_ℎ+𝑧_𝑑²+_𝜏𝜕𝜏ℎ_𝑙𝐴^∗∇𝑝_ℎ⋅𝑛+^𝑇_𝑡𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)⋅𝑛𝑑𝜏^21/2𝑝_ℎ−𝑝𝑢_ℎ1,Ω+𝑦_ℎ−𝑦𝑢_ℎ,𝑝_ℎ−𝑝𝑢_ℎ−^𝑇_𝑡𝑐𝜏,𝑡;𝑝_ℎ−𝑝𝑢_ℎ(𝑡),𝑝_ℎ−𝑝𝑢_ℎ(𝜏)𝑑𝜏. (4.31)
Integrating time from 𝑡 to 𝑇 in (4.31) and by Schwartz inequality, Lemmas 3.3 and 3.4, we have 12𝑝_ℎ−𝑝𝑢_ℎ²_0,Ω+𝑐^𝑇_𝑡𝑝_ℎ−𝑝𝑢_ℎ²_1,Ω⩽𝑑𝜏_𝑇𝑡𝜏ℎ²_𝜏𝜏𝜕𝑝𝜕𝑡^ℎ+div𝐴^∗∇𝑝_ℎ+^𝑇_𝜏div𝐶^∗(𝑠,𝜏)∇𝑝_ℎ(𝑠)𝑑𝑠+𝑦_ℎ−𝑧_𝑑²+𝑑𝜏_𝑇𝑡𝜏ℎ_𝑙𝜕𝜏𝐴^∗∇𝑝_ℎ⋅𝑛+^𝑇_𝜏𝐶^∗(𝑠,𝜏)∇𝑝_ℎ(𝑠)⋅𝑛𝑑𝑠²𝑑𝜏+𝛿^𝑇_𝑡𝑝_ℎ−𝑝𝑢_ℎ²_1,Ω𝑑𝜏+𝐶^𝑇_𝑡𝑦_ℎ−𝑦𝑢_ℎ²_0,Ω𝑑𝜏+𝐶^𝑇_𝑡^𝑇_𝜏𝑝_ℎ−𝑝𝑢_ℎ(𝑠)²_1,Ω𝑑𝑠𝑑𝜏. (4.32)
Letting 𝛿 be small enough, we obtain ^𝑇_𝑡𝑝_ℎ−𝑝𝑢_ℎ²_1,Ω𝑑𝜏⩽𝐶^𝑇_𝑡𝜏ℎ²_𝜏𝜏𝜕𝑝𝜕𝑡^ℎ+div𝐴^∗∇𝑝_ℎ+^𝑇_𝜏div𝐶^∗(𝑠,𝜏)∇𝑝_ℎ(𝑠)𝑑𝑠+𝑦_ℎ−𝑧_𝑑²𝑑𝜏+𝐶^𝑇_𝑡𝜏ℎ_𝑙𝜕𝜏𝐴^∗∇𝑝_ℎ⋅𝑛+^𝑇_𝜏𝐶^∗(𝑠,𝜏)∇𝑝_ℎ(𝑠)⋅𝑛𝑑𝑠²𝑑𝜏+𝐶^𝑇_𝑡𝑦_ℎ−𝑦𝑢_ℎ²_0,Ω𝑑𝜏+𝐶^𝑇_𝑡^𝑇_𝜏𝑝_ℎ−𝑝𝑢_ℎ(𝑠)²_1,Ω𝑑𝑠𝑑𝜏. (4.33)
By Gronwall inequality and (4.30)–(4.33) 𝑝_ℎ−𝑝𝑢_ℎ²_{𝐿20,𝑇;𝐻1(Ω)}⩽𝐶𝜂²₂+𝐶𝜂²₃+𝐶𝑦_ℎ−𝑦𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)}. (4.34)
Similarly, we have 𝑝_ℎ−𝑝𝑢_ℎ²_{𝐿∞0,𝑇;𝐿2(Ω)}⩽𝐶𝜂²₂+𝜂²₃+𝑦_ℎ−𝑦𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)}+𝐶^𝑇₀^𝑇_𝑡𝑝_ℎ−𝑝𝑢_ℎ(𝜏)²_1,Ω𝑑𝜏𝑑𝑡⩽𝐶𝜂²₂+𝐶𝜂²₃+𝐶𝑦_ℎ−𝑦𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)}. (4.35)
Then from (4.30), (4.34), and (4.35) 𝜕𝑝𝜕𝑡^ℎ−𝑝𝑢_{ℎ𝐿2(0,𝑇;𝐻−1(Ω))}=sup_{𝑣∈𝐿20,𝑇;𝐻10(Ω)}∫^𝑇₀(𝜕/𝜕𝑡)𝑝_ℎ−𝑝𝑢_ℎ,𝑣‖𝑣‖_𝐿²_(0,𝑇;𝐻¹_(Ω))=sup_{𝑣∈𝐿20,𝑇;𝐻10(Ω)}∫^𝑇₀−𝐸𝑢_ℎ,𝑣+𝑎𝑣,𝑝_ℎ−𝑝𝑢_ℎ+∫^𝑇_𝑡𝑐𝜏,𝑡;𝑣(𝑡),𝑝_ℎ−𝑝𝑢_ℎ(𝜏)𝑑𝜏𝑑𝑡‖𝑣‖_𝐿²_(0,𝑇;𝐻¹_(Ω))≤𝐶𝜂₂+C𝜂₃+𝐶𝑦_ℎ−𝑦𝑢_{ℎ𝐿2(0,𝑇;𝐿2(Ω))}+𝐶𝑝_ℎ−𝑝𝑢_{ℎ𝐿2(0,𝑇;𝐻1(Ω))}≤𝐶𝜂₂+𝐶𝜂₃+𝐶𝑦_ℎ−𝑦𝑢_{ℎ𝐿2(0,𝑇;𝐿2(Ω))}. (4.36)
Similarly by analysis of ‖𝑦_ℎ−𝑦(𝑢_ℎ)‖_{𝐿2(0,𝑇;𝐻1(Ω))}, we let 𝑄𝑢_ℎ=𝜕,𝑣𝜕_𝑡𝑦_ℎ−𝑦𝑢_ℎ,𝑣+𝑎𝑦_ℎ−𝑦𝑢_ℎ,𝑣+^𝑡₀𝑐𝑡,𝜏;𝑦_ℎ−𝑦𝑢_ℎ(𝜏),𝑣(𝑡)𝑑𝜏. (4.37)
By (3.9) and (4.15) 𝜔_ℎ,𝜕𝑦𝜕𝑡^ℎ−𝑦𝑢_ℎ+𝑎𝑦_ℎ−𝑦𝑢_ℎ,𝜔_ℎ+^𝑡₀𝑐𝑡,𝜏;𝑦_ℎ−𝑦𝑢_ℎ(𝜏),𝜔_ℎ(𝑡)𝑑𝜏=0,∀𝜔_ℎ∈𝑉^ℎ. (4.38)
So 𝜕𝑦𝜕𝑡^ℎ−𝑦𝑢_ℎ,𝑣+𝑎𝑦_ℎ−𝑦𝑢_ℎ,𝑣+^𝑡₀𝑐𝑡,𝜏;𝑦_ℎ−𝑦𝑢_ℎ𝜏)(𝑡)=𝜕(,𝑣𝑑𝜏𝑦𝜕𝑡^ℎ,𝑣−𝜋_ℎ𝑣+𝑎𝑦_ℎ,𝑣−𝜋_ℎ𝑣+^𝑡₀𝑐𝑡,𝜏;𝑦_ℎ(𝜏),𝑣−𝜋_ℎ𝑣(𝑡)𝑑𝜏−𝑓+𝐵𝑢_ℎ,𝑣−𝜋_ℎ𝑣⩽_𝜏𝜏ℎ²_𝜏𝜕𝑦𝜕𝑡^ℎ−div𝐴∇𝑦_ℎ−^𝑡₀div𝐶(𝑡,𝜏)∇𝑦_ℎ𝑑𝜏−𝑓−𝐵𝑢_ℎ²+_𝜏𝜕𝜏ℎ_𝑙𝐴∇𝑦_ℎ⋅𝑛+^𝑡₀𝐶(𝑡,𝜏)∇𝑦_ℎ⋅𝑛𝑑𝜏^21/2‖𝑣‖_1,Ω. (4.39)
By letting 𝑣=𝑦_ℎ−𝑦(𝑢_ℎ) and applying Swartz inequality, we have 12𝑑𝑦𝑑𝑡^ℎ−𝑦𝑢_ℎ²_0,Ω+𝑐𝑦_ℎ−𝑦𝑢_ℎ²_1,Ω⩽_𝜏𝜏ℎ²_𝜏𝜕𝑦𝜕𝑡^ℎ−div𝐴∇𝑦_ℎ−^𝑡₀div𝐶(𝑡,𝜏)∇𝑦_ℎ𝑑𝜏−𝑓−𝐵𝑢_ℎ²+_𝜏𝜕𝜏ℎ_𝑙𝐴∇𝑦_ℎ⋅𝑛+^𝑡₀𝐶(𝑡,𝜏)∇𝑦_ℎ⋅𝑛𝑑𝜏²+𝛿𝑦_ℎ−𝑦𝑢_ℎ²_1,Ω−^𝑡₀𝑐𝑡,𝜏;𝑦_ℎ−𝑦𝑢_ℎ(𝜏),𝑦_ℎ−𝑦𝑢_ℎ(𝑡)𝑑𝜏. (4.40)
By integrating time from 0 to 𝑡 in (4.40) 𝑦_ℎ−𝑦𝑢_ℎ²_0,Ω+𝑐^𝑡₀𝑦_ℎ−𝑦𝑢_ℎ²_1,Ω𝑑𝜏⩽𝐶^𝑡_0𝜏ℎ²_𝜏𝜏𝜕𝑦𝜕𝑡^ℎ−div𝐴∇𝑦_ℎ−^𝜏₀div𝐶(𝜏,𝑠)∇𝑦_ℎ𝑑𝑠−𝑓−𝐵𝑢_ℎ²+𝑑𝜏_𝑡0𝜏ℎ_𝑙𝜕𝜏𝐴∇𝑦_ℎ⋅𝑛+^𝜏₀𝐶(𝜏,𝑠)∇𝑦_ℎ⋅𝑛𝑑𝑠²𝑑𝑡+𝛿^𝑡₀𝑦_ℎ−𝑦𝑢_ℎ²_1,Ω𝑑𝜏+𝐶^𝑡₀^𝜏₀𝑦_ℎ−𝑦𝑢_ℎ²_1,Ω𝑑𝑠𝑑𝜏+𝐶𝑦₀−𝑦^ℎ₀²_0,Ω. (4.41)
Since 𝛿 is small, then from (4.41) and Gronwall inequality we have ^𝑡₀𝑦_ℎ−𝑦𝑢_ℎ²_1,Ω𝑑𝑡⩽𝐶^𝑡_0𝜏ℎ²_𝜏𝜏𝜕𝑦𝜕𝑡^ℎ−div𝐴∇𝑦_ℎ−^𝜏₀div𝐶(𝜏,𝑠)∇𝑦_ℎ𝑑𝑠−𝑓−𝐵𝑢_ℎ²𝑑𝜏+𝐶^𝑡_0𝜏ℎ_𝑙𝜕𝜏𝐴∇𝑦_ℎ⋅𝑛+^𝜏₀𝐶(𝜏,𝑠)∇𝑦_ℎ⋅𝑛𝑑𝑠²𝑑𝑡+𝐶𝑦₀−𝑦^ℎ₀²_0,Ω (4.42) and we obtain 𝑦_ℎ−𝑦𝑢_ℎ²_{𝐿20,𝑇;𝐻1(Ω)}⩽𝐶𝜂²₄+𝜂²₅+𝜂²₆,𝑦_ℎ−𝑦𝑢_ℎ²_{𝐿∞0,𝑇;𝐿2(Ω)}⩽𝐶𝜂²₄+𝜂²₅+𝜂²₆. (4.43)
In the same way of getting (4.36), we also have 𝜕𝑦𝜕𝑡^ℎ−𝑦𝑢_ℎ²_{𝐿20,𝑇;𝐻−1(Ω)}⩽𝐶𝜂²₄+𝜂²₅+𝜂²₆. (4.44)
Then the desired results (4.27) follow from (4.34)–(4.36) and (4.43)–(4.44).

4.3. Proof of Theorem 4.1

By Lemmas 4.2 and 4.3, we prove Theorem 4.1 in the following.

Proof. By using the triangle inequality, Lemmas 4.2 and 4.3, and from (4.29) and (4.38), [17–19], using the following stability results 𝑦−𝑦𝑢_ℎ²_{𝐿∞0,𝑇;𝐿2(Ω)}+𝑦−𝑦𝑢_ℎ²_{𝐿20,𝑇;𝐻1(Ω)}⩽𝐶𝑢−𝑢_ℎ²_{𝐿20,𝑇;𝐿2Ω𝑈},𝑝−𝑝𝑢_ℎ²_{𝐿∞0,𝑇;𝐿2(Ω)}+𝑝−𝑝𝑢_ℎ²_{𝐿20,𝑇;𝐻1(Ω)}⩽𝐶𝑦−𝑦𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)}⩽𝐶𝑢−𝑢_ℎ²_{𝐿20,𝑇;𝐿2Ω𝑈}, (4.45)
we can easily obtain (4.11). The proof of Theorem 4.1 is completed.

5. A Posteriori Error Estimates in Integral

Often we need sharper a posteriori error estimates. Then we need deriving the a posteriori error estimates in 𝐿²(0,𝑇;𝐿²(Ω))-norm. To this end we first state the main results in this paper.

5.1. Main Results

We have the following results.

Theorem 5.1. Let (𝑦,𝑢) and (𝑦_ℎ,𝑢_ℎ) be the solutions of (𝑂𝐶𝑃) and (𝑂𝐶𝑃)^ℎ, respectively. Let 𝑝 and 𝑝_ℎ be the solution of the costate equations (3.2) and (3.10). Then there hold the a posteriori error estimates 𝑢−𝑢_ℎ²_{𝐿20,𝑇;𝐿2Ω𝑈}+𝑦−𝑦_ℎ²_{𝐿20,𝑇;𝐿2(Ω)}+𝑝−𝑝_ℎ²_{𝐿20,𝑇;𝐿2(Ω)}⩽𝐶𝜂²₁+⁵_𝑖=2𝜉²_𝑖+𝜂²₆, (5.1) where 𝜂²₁,𝜂²₆ is defined in Theorem 4.1 and 𝜉²₂=^𝑇_0𝜏ℎ⁴_𝜏𝜏𝜕𝑝𝜕𝑡^ℎ+div𝐴^∗∇𝑝_ℎ+^𝑇_𝑡div𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)𝑑𝜏+𝑦_ℎ−𝑧_𝑑²𝜉𝑑𝑡,₂₃=^𝑇_0𝜏ℎ³_𝑙𝜕𝜏𝐴^∗∇𝑝_ℎ⋅𝑛+^𝑇_𝑡𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)⋅𝑛𝑑𝜏²𝜉𝑑𝑙𝑑𝑡,₂₄=^𝑇_0𝜏ℎ⁴_𝜏𝜏𝜕𝑦𝜕𝑡^ℎ−div𝐴∇𝑦_ℎ−^𝑡₀div𝐶(𝑡,𝜏)∇𝑦_ℎ(𝜏)𝑑𝜏−𝑓−𝐵𝑢_ℎ²𝜉𝑑𝑡,₂₅=^𝑇_0𝜏ℎ³_𝑙𝜕𝜏𝐴∇𝑦_ℎ⋅𝑛+^𝑡₀𝐶(𝑡,𝜏)∇𝑦_ℎ(𝜏)⋅𝑛𝑑𝜏²𝑑𝑙𝑑𝑡. (5.2)

In order to prove Theorem 5.1, we need the following dual equations and lemmas.

5.2. Dual Equations and Some Lemmas

For given 𝐹∈𝐿²(0,𝑇;𝐿²(Ω)) and the following equation: 𝜕−𝜕𝑡𝜙−div>𝐴∇𝜙^𝑡₀𝐶()(𝜏)()(],𝜙div𝑡,𝜏∇𝜙𝑑𝜏=𝐹,𝑥,𝑡∈Ω×0,𝑇_𝜕Ω(],𝜙()=0,𝑡∈0,𝑇𝑥,0=0,𝑥∈Ω, (5.3)

we have its dual equation: −𝜕𝐴𝜕𝑡𝜓−div^∗∇𝜓−^𝑇_𝑡div𝐶^∗()(𝜏)()(],𝜓𝜏,𝑡∇𝜓𝑑𝜏=𝐹,𝑥,𝑡∈Ω×0,𝑇_𝜕Ω(],𝜓()=0,𝑡∈0,𝑇𝑥,𝑇=0,𝑥∈Ω. (5.4)

In order to derive a posteriori error estimates in 𝐿²-norm, it is necessary to have some stability results of the dual equations. From [17–19, 29], we have the following stability results.

Lemma 5.2. Assume that 𝜙 and 𝜓 are the solution of (5.3) and (5.4), respectively. Then 𝜙_{𝐿∞(0,𝑇;𝐿2(Ω))}⩽𝐶‖𝐹‖_𝐿²_(0,𝑇;𝐿²_(Ω)),∇𝜙_{𝐿2(0,𝑇;𝐿2(Ω))}⩽𝐶‖𝐹‖_𝐿²_(0,𝑇;𝐿²_(Ω)),𝐷²𝜙_{𝐿2(0,𝑇;𝐿2(Ω))}⩽𝐶‖𝐹‖_𝐿²_(0,𝑇;𝐿²_(Ω)),𝜕𝜙𝜕𝑡_𝐿²_(0,𝑇;𝐿²_(Ω))⩽𝐶‖𝐹‖_𝐿²_(0,𝑇;𝐿²_(Ω)),𝜓_{𝐿∞(0,𝑇;𝐿2(Ω))}⩽𝐶‖𝐹‖_𝐿²_(0,𝑇;𝐿²_(Ω)),∇𝜓_{𝐿2(0,𝑇;𝐿2(Ω))}⩽𝐶‖𝐹‖_𝐿²_(0,𝑇;𝐿²_(Ω)),𝐷²𝜓_{𝐿2(0,𝑇;𝐿2(Ω))}⩽𝐶‖𝐹‖_𝐿²_(0,𝑇;𝐿²_(Ω)),𝜕𝜓𝜕𝑡_𝐿²_(0,𝑇;𝐿²_(Ω))⩽𝐶‖𝐹‖_𝐿²_(0,𝑇;𝐿²_(Ω)), (5.5) where 𝐷²𝜙=𝜕²𝜙/𝜕𝑥_𝑖𝜕𝑥_𝑗, 1⩽𝑖, 𝑗⩽𝑛, and 𝐷²𝜓 is defined similarly.

It follows from Lemmas 4.2 and 5.2 that we have the following upper bounds.

Lemma 5.3. Let (𝑦,𝑢) and (𝑦_ℎ,𝑢_ℎ) be the solutions of (𝑂𝐶𝑃) and (𝑂𝐶𝑃)^ℎ, respectively. Let 𝑝 and 𝑝_ℎ be the solution of the costate equations (3.2) and (3.10). Then there hold the a posteriori error estimates 𝑦_ℎ−𝑦𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)}+𝑝_ℎ−𝑝𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)}⩽𝐶⁵_𝑖=2𝜉²_𝑖+𝜂²₆. (5.6)

Proof. It follows from Lemma 4.2 that we only need to estimate ‖𝑝_ℎ−𝑝(𝑢_ℎ)‖²_{𝐿2(0,𝑇;𝐿2(Ω))}.
Let 𝜙 be the solution of (5.3) with 𝐹=𝑝_ℎ−𝑝(𝑢_ℎ) and 𝜙_𝐼=𝜋_ℎ𝜙 be the interpolation of 𝜙 in Lemma 3.2.
It follows from (5.3), (4.29) and by integrating by parts 𝑝_ℎ−𝑝𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)}=^𝑇₀𝐹(𝑡),𝑝_ℎ−𝑝𝑢_ℎ𝑡)=(𝑑𝑡_𝑇0−𝜕𝑝𝜕𝑡^ℎ−𝑝𝑢_ℎ,𝜙+𝑎𝜙,𝑝_ℎ−𝑝𝑢_ℎ+^𝑡₀𝑐𝑡,𝜏;𝜙(𝜏),𝑝_ℎ−𝑝𝑢_ℎ(𝑡=)𝑑𝜏𝑑𝑡_𝑇0−𝜕𝑝𝜕𝑡^ℎ,𝜙−𝜙_𝐼+𝑎𝜙−𝜙_𝐼,𝑝_ℎ+^𝑇_𝑡𝑐𝜏,𝑡;𝜙−𝜙_𝐼(𝑡),𝑝_ℎ(𝜏)+𝜕𝑑𝜏𝑝𝑢𝜕𝑡^ℎ,𝜙−𝜙_𝐼−𝑎𝜙−𝜙_𝐼,𝑝𝑢_ℎ−^𝑇_𝑡𝑐𝜏,𝑡;𝜙−𝜙_𝐼(𝑡),𝑝𝑢_ℎ(𝜏)−𝜕𝑑𝜏𝑝𝜕𝑡^ℎ−𝑝𝑢_ℎ,𝜙_𝐼+𝑎𝜙_𝐼,𝑝_ℎ−𝑝𝑢_ℎ+^𝑇_𝑡𝑐𝜏,𝑡;𝜙_𝐼(𝑡),𝑝_ℎ−𝑝𝑢_ℎ(𝜏)=𝑑𝜏𝑑𝑡_{𝑇0𝜏𝜏}𝜕𝑝𝜕𝑡^ℎ+div𝐴^∗∇𝑝_ℎ+^𝑇_𝑡div𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)𝑑𝜏+𝑦_ℎ−𝑧_𝑑𝜙−𝜙_𝐼+𝑑𝑡_{𝑇0𝜏𝜕𝜏}𝐴^∗∇𝑝_ℎ⋅𝑛+^𝑇_𝑡𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)⋅𝑛𝑑𝜏𝜙−𝜙_𝐼𝑑𝑙𝑑𝑡+^𝑇₀𝑦_ℎ−𝑦𝑢_ℎ,𝜙𝑑𝑡=𝐽₁+𝐽₂+𝐽₃. (5.7)
By Lemmas 3.2, 3.4, and 5.2, we obtain 𝐽₁⩽𝐶(𝛿)^𝑇_0𝜏ℎ⁴_𝜏𝜏𝜕𝑝𝜕𝑡^ℎ+div𝐴^∗∇𝑝_ℎ+^𝑇_𝑡div𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)𝑑𝜏+𝑦_ℎ−𝑧_𝑑²𝑑𝑡+𝛿^𝑇₀𝜙²_2,Ω𝑑𝑡⩽𝐶(𝛿)𝜉²₂+𝛿𝑝_ℎ−𝑝𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)},𝐽₂⩽𝐶(𝛿)^𝑇_0𝜏ℎ³_𝑙𝜕𝜏𝐴^∗∇𝑝_ℎ⋅𝑛+^𝑇_𝑡𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)⋅𝑛𝑑𝜏²𝑑𝑙𝑑𝑡+𝛿^𝑇₀𝜙²_0,Ω(𝛿)𝜉𝑑𝑡⩽𝐶²₃+𝛿𝑝_ℎ−𝑝𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)}. (5.8)
By Schwartz inequality 𝐽₃⩽𝐶(𝛿)𝑦_ℎ−𝑦𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)}+𝛿𝑝_ℎ−𝑝𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)}. (5.9)
Letting 𝛿 be small enough, it follows from (5.7)–(5.9) 𝑝_ℎ−𝑝𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)}⩽𝐶³_𝑖=2𝜉²_𝑖+𝐶𝑦_ℎ−𝑦𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)}. (5.10)
Similarly, let 𝜓 be the solution of (5.4) with 𝐹=𝑦_ℎ−𝑦(𝑢_ℎ) and 𝜓_𝐼=𝜋_ℎ𝜙 be the interpolation of 𝜙 in Lemma 3.2.
From (4.38) and by integrating by parts, we have that 𝑦_ℎ−𝑦𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)}=^𝑇₀𝐹(𝑡),𝑦_ℎ−𝑦𝑢_ℎ(𝑡)=𝑑𝑡_𝑇0𝜕𝑦𝜕𝑡^ℎ−𝑦𝑢_ℎ,𝜓+𝑎𝑦_ℎ−𝑦𝑢_ℎ,𝜓+^𝑇_𝑡𝑐𝜏,𝑡;𝑦_ℎ−𝑦𝑢_ℎ(𝑡)(𝜏)+𝑦,𝜓𝑑𝜏𝑑𝑡_ℎ0−𝑦₀,𝜓(0)=^𝑇₀𝜕𝑦𝜕𝑡^ℎ,𝜓−𝜓_𝐼+𝑎𝑦_ℎ,𝜓−𝜓_𝐼+^𝑡₀𝑐𝑡,𝜏;𝑦_ℎ(𝜏),𝜓−𝜓_𝐼(𝑡)−𝑑𝜏𝑑𝑡_𝑇0𝜕𝑦𝑢𝜕𝑡^ℎ,𝜓−𝜓_𝐼+𝑎𝑦𝑢_ℎ,𝜓−𝜓_𝐼+^𝑡₀𝑐𝑡,𝜏;𝑦𝑢_ℎ(𝜏),𝜓−𝜓_𝐼(𝑡)𝑑𝜏𝑑𝑡+𝑦^ℎ₀−𝑦₀(0)=,𝜓_{𝑇0𝜏𝜏}𝜕𝑦𝜕𝑡^ℎ−div𝐴∇𝑦_ℎ−^𝑡₀div𝐶(𝜏,𝑡)∇𝑦_ℎ(𝜏)𝑑𝜏−𝑓−𝐵𝑢_ℎ𝜓−𝜓_𝐼+𝑑𝑡_{𝑇0𝜏𝜕𝜏}𝐴∇𝑦_ℎ⋅𝑛−^𝑡₀𝐶(𝑡,𝜏)∇𝑦_ℎ(𝜏)⋅𝑛𝑑𝜏𝜓−𝜓_𝐼+𝑦𝑑𝑙𝑑𝑡_ℎ0−𝑦₀,𝜓(0)=𝐷₁+𝐷₂+𝐷₃. (5.11)
Similarly, it follows from Lemmas 3.2, 3.4, and 5.2 that 𝐷₁⩽𝐶(𝛿)^𝑇_0𝜏ℎ⁴_𝜏𝜏𝜕𝑦𝜕𝑡^ℎ−div𝐴∇𝑦_ℎ−^𝑡₀div𝐶(𝜏,𝑡)∇𝑦_ℎ(𝜏)𝑑𝜏−𝑓−𝐵𝑢_ℎ²𝑑𝑡+𝛿^𝑇₀𝜓²_2,Ω𝑑𝑡⩽𝐶𝜉²₄+𝛿𝑦_ℎ−𝑦𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)},𝐷₂⩽𝐶(𝛿)^𝑇_0𝜏ℎ³_𝑙𝜕𝜏𝐴∇𝑦_ℎ⋅𝑛−^𝑡₀𝐶(𝑡,𝜏)∇𝑦_ℎ(𝜏)⋅𝑛²𝑑𝑙𝑑𝑡+𝛿^𝑇₀𝜓²_2,Ω𝑑𝑡⩽𝐶𝜉²₅+𝛿𝑦_ℎ−𝑦𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)},𝐷₃⩽𝐶𝜂²₆+𝛿𝑦_ℎ−𝑦𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)}. (5.12)
Letting 𝛿 be small enough, then from (5.11)-(5.12) we have 𝑦_ℎ−𝑦𝑢_ℎ²_{𝐿20,𝑇;𝐿2(Ω)}⩽𝐶𝜉²₄+𝜉²₅+𝜂²₆. (5.13)
Then (5.6) follows from (5.10)–(5.13). The proof of Lemma 5.3 is completed.

5.3. Proof of Theorem 5.1

From Lemmas 4.2 and 5.3, we can easily prove Theorem 5.1.

Proof. By triangle inequality, (4.45), Lemmas 4.2 and 5.3, we can easily prove (5.1) in the same way of getting (4.11). The proof of Theorem 5.1 is completed.

6. Conclusion

In this paper, we first briefly introduce optimal control problem governed by parabolic integrodifferential equations and give the weak formulation for this optimal control problem. For this formulation, we prove the existence and uniqueness of the solution. Then by the theory of optimal control problem, we present the optimality conditions and semidiscrete finite element approximation scheme. The upper a posteriori error estimates for first-order derivative and 𝐿²(0,𝑇;𝐿²(Ω))-norm are derived for both the state and the control approximation for the case of an integral obstacle constraint. We will research on the upper and lower a posteriori error estimates for full discrete finite element approximations of this control problem.

Acknowledgments

This research was supported by Science and Technology Development Planning Project of Shandong Province (no. 2011GGH20118) and Shandong Province Natural Science Foundation (nos. ZR2012AQ024 and ZR2012GM018).