# Adaptive Finite Element Method for Optimal Control Problem Governed by Linear Quasiparabolic Integrodifferential Equations

Authors: Wanfang Shen; Hua Su
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The mathematical formulation for a quadratic optimal control problem governed by a linear quasiparabolic integrodifferential equation is studied. The control constrains are given in an integral sense:Uad={u∈X;∫ΩUu⩾0,t∈[0,T]}. Then the a posteriori error estimates in L∞(0,T;H1(Ω))-norm and L2(0,T;L2(Ω))-norm for both the state and the control approximation are given.


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## Body

## 1. Introduction

Integrodifferential equations of quasiparabolic and their control of this nature appear in applications such as biology mechanics, nuclear reaction dynamics, heat conduction in materials with memory, and viscoelasticity. All these models express a conservation of a certain quantity in any moment for any subdomain and the historical accumulation feature in the physical models. This in many applications is the most desirable feature of the approximation method when it comes to numerical solution of the corresponding initial boundary value problem. The existence and uniqueness of the solution of the quasiparabolic Integrodifferential equations has been studied in [1]. Finite element methods for quasiparabolic Integrodifferential equations problems with a smooth kernel have been discussed in Cui [2]. Although there is so much work for the finite element approximation of this problem, to our knowledge, there has been a lack of a posteriori error estimates for finite element approximation of any quasiparabolic Integrodifferential optimal control problem.The finite element approximation of optimal control problems has been an important topic in engineering design works. There have been extensive theoretical and numerical studies for various optimal control problems, see, for instance, [3–11], although it is impossible to give even a very brief review here. And research on finite element approximation of parabolic optimal control problems can be found in, for example, [12, 13].Among many finite element methods, the adaptive finite element method based on a posteriori error estimates has become a central theme in scientific and engineering computations for its high efficiency. In order to obtain a numerical solution of acceptable accuracy, it is essential for the adaptive finite element method to use a posteriori error estimate indicators to guide the mesh refinement procedure. We only need refine the area where the error indicators are larger, so that a higher density of nodes are distributed over the area where the solution is difficult to approximate. In this sense, adaptive finite element approximation relies very much on the error indicators used, which are often based on a posteriori error estimates of the solutions.The purpose of this paper is to derive the a posteriori error estimates for the semidiscrete finite element approximation of a quadratic optimal control problem governed by a linear quasiparabolic Integrodifferential equation, which paves a way to derive the a posteriori error estimates for the full discrete finite element approximation for this control problem and thus to develop its adaptive finite element schemes. We extend the existing techniques and results in [14–16] to the optimal control problem governed by the Integrodifferential equation of quasiparabolic type.The outline of the paper is as follows. In Section2, we first briefly introduce the optimal control problem and give the optimality conditions, then construct the finite element approximation schemes for the optimal control problem. In Section 3, we give the a posteriori error bounds in L∞(0,T;H1(Ω))-norm for the control problem. And the a posteriori error bounds in L2(0,T;L2(Ω))-norm for the control problem are derived in Section 4.


## 2. Optimal Control Problem and Its Finite Element Approximation

LetΩ and ΩU be bounded convex polygon domains in Rd with Lipschitz boundary ∂Ω and ∂ΩU. In this paper, we adopt the standard notation Wm,q(Ω) for Sobolev spaces on Ω with norm ∥·∥m,q,Ω, and seminorm  |·|m,q,Ω. We set W0m,q(Ω)={w∈Wm,q(Ω):w|∂Ω=0}. We denote Wm,2(Ω)(W0m,2(Ω)) by Hm(Ω)(H0m(Ω)), with norm  ∥·∥m,Ω, and seminorm  |·|m,Ω.We denote byLs(0,T;Wm,q(Ω)) the Banach space of all Ls integrable functions from (0,T) into Wm,q(Ω) with norm ∥v∥Ls(0,T;Wm,q(Ω))=(∫0T‍∥v∥Wm,q(Ω)sdt)1/s for s∈[1,∞) and the standard modification for s=∞. Similarly, one can define the spaces H1(0,T;Wm,q(Ω)) and Ck(0,T;Wm,q(Ω)). The details can be found in [17]. In addition, c or C denotes a general positive constant independent of the mesh size h.In the following, we will give semi-discrete finite element approximation schemes for the optimal control problem governed by a linear quasiparabolic Integrodifferential equation.


### 2.1. Model Problem and Its Weak Formulation

We will take the state spaceW=L2(0,T;V) with V=H01(Ω) and the control space X=L2(0,T;U) with U=L2(ΩU). Let the observation space Y=L2(0,T;H) with H=L2(Ω) and Uad⊆X a convex subset.We are interested in the following optimal control problem:(2.1)min⁡u∈Uad⊂XJ(u,y(u))=12{∫0T‍‖y-zd‖0,Ω2dt+∫0T‍‖u‖0,ΩU2dt},
subject to
(2.2)yt-div⁡(A∇yt+D∇y+∫0t‍C(t,τ)∇y(x,τ)dτ)=f+Bu,inΩ×(0,T],y=0,on⁡∂Ω×[0,T],y|t=0=y0,inΩ,
where u is control, y is state, zd is the observation, Uad is a closed convex subset, f(x,t)∈L2(0,T;L2(Ω)), and zd and y0∈H1(Ω) are some suitable functions to be specified later. B is a linear bounded operator from L2(ΩU) to L2(Ω) independent of t. And
(2.3)A=A(x)=(ai,j(⋅))n×n∈(C∞(Ω¯))n×n,D=D(x)=(di,j(⋅))n×n∈(C∞(Ω¯))n×n,
such that there is a constant c>0 satisfying that for any vector X∈Rn as follows:
(2.4)XtAX≥c‖X‖Rn2,XtDX≥c‖X‖Rn2,C=C(x,t,τ)=(ci,j(x,t,τ))n×n∈(C∞(0,T;L2(Ω¯))n×n).Let(2.5)(f1,f2)=∫Ω‍f1f2,∀(f1,f2)∈H×H,(u,v)U=∫ΩU‍uv,∀(u,v)∈U×U,a(z,ω)=(A∇z,∇ω),d(z,ω)=(D∇z,∇ω),c(t,τ;z,ω)=(C(t,τ)∇z,∇ω),∀z,w∈V×V.
In the case that f1∈V and f2∈V*, the dual pair (f1,f2) is understood as 〈f1,f2〉V×V*.Assume that there are constantscand C, such that for all t and τ in [0,T] as follows:
(2.6)(a)a(z,z)⩾c‖z‖1,Ω2,(b)|a(z,w)|⩽C‖z‖1,Ω‖w‖1,Ω,|d(z,w)|⩽C‖z‖1,Ω‖w‖1,Ω,(c)|c(t,τ;z,w)|⩽C‖z‖1,Ω‖w‖1,Ω.
for any z and w in V.Then the weak form of the state equation reads as(2.7)(yt,w)+a(yt,w)+d(y,w)+∫0t‍c(t,τ;y(τ),w)dτ=(f+Bu,w)∀w∈V,t∈(0,T],y|t=0=y0.
It is well known (see, e.g., [1]) that the above weak formulation has at least one solution in  y∈W(0,T)={w∈L∞(0,T;H1(Ω)),wt′∈L2(0,T;H1(Ω))}.Therefore, the weak form of the control problem (2.1) and (2.2) reads as (OCP)
(2.8)min⁡u∈UadJ(u,y(u)),(yt,w)+a(yt,w)+d(y,w)+∫0t‍c(t,τ;y(τ),w)dτ=(f+Bu,w)∀w∈V,t∈(0,T],y|t=0=y0.
In the following, we first give the existence and uniqueness of the solution of the system (2.8).Theorem 2.1.
Assume that the condition (2.6) (a)–(c) holds. There exists the unique solution (u,y) for the minimization problem (2.8) such that u∈L2(0,T;L2(ΩU)), y∈L∞(0,T;H1(Ω)), and yt∈L2(0,T;H1(Ω)).Proof.
Let{(un,yn)}n=1∞ be a minimization sequence for the system (2.8), then the sequence {un}n=1∞ is bounded in L2(0,T;L2(ΩU)). Thus there is a subsequence of {un}n=1∞ (still denote by {un}n=1∞ ) such that un converges to u* weakly in L2(0,T;L2(ΩU)). For the subsequence {un}n=1∞, we have
(2.9)(ytn,w)+a(ytn,w)+d(yn,w)+∫0t‍c(t,τ;yn(τ),w(t))dτ=(f+Bun,w)∀w∈V,t∈(0,T].
By setting w=yn and integrating from 0 to t in (2.9), we give
(2.10)‖yn(t)‖1,Ω2+∫0t‍‖yn‖1,Ω2dτ⩽C{‖y0‖1,Ω+C∫0t‍(‖f‖-1,Ω2+‖un‖0,ΩU2)dt+∫0t‍∫0τ‍‖y(s)‖1,Ω2dsdτ}.
Applying Gronwall's inequality to (2.10) yields
(2.11)‖yn‖L∞(0,T;H1(Ω))2+‖yn‖L2(0,T;H1(Ω))2⩽C{‖y0‖1,Ω2+∫0T‍(‖f‖-1,Ω2+‖un‖0,ΩU2)}.
So {un}n=1∞ is a bounded set in L2(0,T;L2(ΩU)) and {yn}n=1∞ is a bounded set in L∞(0,T;H1(Ω)). Thus
(2.12)un⟶uweaklyinL2(0,T;L2(ΩU)),yn⟶yweaklyinL∞(0,T;H1(Ω)),yn(T)⟶y(T)weaklyinH1(Ω).
Let W={w;w∈L∞(0,T;H1(Ω)),wt′∈L2(0,T;H1(Ω))}.
By integrating time from0 to T in (2.9) and taking limit as n→∞, we obtain
(2.13)(y(T),w(T))+a(y(T),w(T))-∫0T‍[(y,wt′)+a(y,wt′)+d(y,w)]+∫0T‍∫0t‍c(t,τ;y(τ),w(t))dτdt=(y0,w(0))+a(y0,w(0))+∫0T‍(f+Bu,w),∀w∈W.
Then,
(2.14)(yt,w)+a(yt,w)+d(y,w)+∫0t‍c(t,τ;y(τ),w)dτ=(f+Bu,w),∀w∈V,t∈(0,T].
Furthermore, we have
(2.15)∫0T‍[(yt,yt)+a(yt,yt)+d(y,yt)+∫0t‍c(t,τ;y(τ),yt(t))dτ]dt=∫0T‍(f+Bu,yt).
Then, we get
(2.16)∫0T‍‖yt‖1,Ω2⩽C∫0T‍[‖f‖-1,Ω2+‖u‖0,ΩU2+‖y‖1,Ω2+∫0t‍‖y‖1,Ω2dτ].
This means yt∈L2(0,T;H1(Ω)). So (u,y) is one solution of (2.8).
Since∫0T‍∥y-zd∥0,Ω2 is a convex function on space L2(0,T;L2(Ω)) and (α/2)∫0T‍∥u∥0,ΩU2 is a strictly convex function on U, hence J(u,y(u)) is a strictly convex function on U, so the minimization problem (2.8) has one unique solution.


### 2.2. Optimality Conditions and Their Finite Element Approximation

By the theory of optimal control problem (see [18]), we can similarly deduce the following optimality conditions of the problem (2.8).Theorem 2.2.
A pair(y,u)∈L2(0,T;H01(Ω))×L2(0,T;L2(ΩU)) is the solution of the optimal control problem (2.8), if and only if there exists a costate p∈L2(0,T;H01(Ω)) such that the triple (y,p,u) satisfies the following optimality conditions:
(2.17)(yt,w)+a(yt,w)+d(y,w)+∫0t‍c(t,τ;y(τ),w(t))dτ=(f+Bu,w)∀w∈V,t∈(0,T],y|t=0=y0;(2.18)-(q,pt)-a(q,pt)+d(q,p)+∫tT‍c(τ,t;q(t),p(τ))dτ=(y-zd,q)∀q∈V,t∈[0,T),p|t=T=0;(2.19)∫0T‍(u+B*p,v-u)Udt⩾0,∀v∈Uad,
where B* is the adjoint operator of B.Let us consider the semi-discrete finite element approximation of the control problem (2.8). Here, we only consider triangular and conforming elements.LetΩh be a polygonal approximation to Ω with boundary  ∂Ωh. Let Th be a partitioning of Ωh into disjoint regular n-simplices τ, so that Ω-h=⋃τ∈Th‍τ-. Each element has at most one face on ∂Ωh, and τ- and τ-′ have either only one common vertex or a whole edge or face if τ- and τ-′∈Th. We further require that Pi∈∂Ωh⇒Pi∈∂Ω  where Pi(i=1,…,J) is the vertex set associated with the triangulation Th. As usual, h denotes the diameter of the triangulation Th. For simplicity, we assume that Ω is a convex polygon so that Ω=Ωh.Associated withTh is a finite-dimensional subspace Sh of C(Ω-h), such that χ|τ are polynomials of order m(m≥1) for all χ∈Sh and τ∈Th. Let Vh={vh∈Sh:vh(Pi)=0(i=1,…,J)},Wh=L2(0,T;Vh). Note that we do not impose a continuity requirement. It is easy to see that Vh⊂V,Wh⊂W.LetTUh be a partitioning of ΩUh into disjoint regular n-simplices τU, so that Ω-Uh=⋃τU∈TUh‍τ-U. τ-U and τ-U′ have either only one common vertex or a whole edge or face if τ-U and τ-U′∈TUh. We further require that Pi∈∂ΩUh⇒Pi∈∂ΩU, where Pi(i=1,…,J) is the vertex set associated with the triangulation TUh. For simplicity, we again assume that ΩU is a convex polygon so that ΩU=ΩUh.Associated withTUh is another finite-dimensional subspace Uh of L2(ΩUh), such that χ|τU are polynomials of order m(m⩾0) for all χ∈Uh and τU∈TUh. Here there is no requirement for the continuity. Let Xh=L2(0,T;Uh). It is easy to see that Xh⊂X. Let hτ(hτU) denote the maximum diameter of the element τ(τU) in Th(TUh).Due to the limited regularity of the optimal controlu in general, there will be no advantage in considering higher-order finite element spaces than the piecewise constant space for the control. We therefore only consider the piecewise constant finite element space for the approximation of the control, though higher-order finite element spaces will be used to approximate the state and the co-state. Let P0(Ω) denote all the 0-order polynomial over Ω. Therefore, we always take Xh={u∈X:u(x,t)|x∈τU∈P0(τU),forallt∈[0,T]}. Uadh is a closed convex set in Xh. For ease of exposition, in this paper, we assume that Uadh⊂Uad∩Xh.Then a possible semi-discrete finite element approximation of(OCP) is as follows (OCP)h:(2.20)min⁡uh∈UadhJ(uh,yh)=12{∫0T‍‖yh-zd‖0,Ω2+∫0T‍‖uh‖0,ΩU2},
such that
(2.21)(∂yh∂t,wh)+a(∂yh∂t,wh)+d(yh,wh)+∫0t‍c(t,τ;yh(τ),wh(t))dτ=(f+Buh,wh),∀wh∈Vh,t∈(0,T],yh|t=0=yh0,
where yh∈Wh and yh0∈Vh is the approximation of y0.In the same way of proving Theorem2.1, we can easily prove that the problem (2.20)-(2.21) has a unique solution (yh,uh)∈Wh×Uadh.It is well known (see [18]) that a pair (yh,uh)∈Wh×Uadh is a solution of (2.20)-(2.21), if and only if there exists a co-state ph∈Wh such that the triple (yh,ph,uh) satisfies the following optimality conditions:
(2.22)(∂yh∂t,wh)+a(∂yh∂t,wh)+d(yh,wh)+∫0t‍c(t,τ;yh(τ),wh(t))dτ=(f+Buh,wh),∀wh∈Vh,yh|t=0=yh0,(2.23)-(qh,∂ph∂t)-a(qh,∂ph∂t)+d(qh,ph)+∫tT‍c(τ,t;qh,ph(τ))dτ=(yh-zd,qh),∀qh∈Vh,ph|t=T=0,(2.24)∫0T‍(uh+B*ph,vh-uh)Udt⩾0,∀vh∈Uadh.The optimality conditions in (2.22)–(2.24) are the semi-discrete approximation to the problem (2.17)–(2.19).Introduce the local averaging operatorπh given by
(2.25)(πhw)|τU:=∫τU‍w∫τU‍1,∀τU∈TUh.
Then, we have ∫ΩU‍w=∫ΩU‍πhw for any w∈L2(0,T;L2(ΩU)),t∈[0,T] and (2.24) is equivalent to
(2.26)∫0T‍(uh+πh(B*ph),vh-uh)Udt⩾0,∀vh∈Uadh.
In the following, we derive the a posteriori error estimates for semi-discrete finite element approximation (2.22)–(2.24), allowing different meshes to be used for the state and the control.The following lemmas are important in deriving the a posteriori error estimates of residual type.Lemma 2.3 (see [19]).
Letπ^h be the standard Lagrange interpolation operator. For m=0 or 1,  q>n/2 and v∈W2,q(Ω) as
(2.27)|v-π^hv|m,q,Ω⩽Ch2-m|v|2,q,Ω.Lemma 2.4 (see [20]).
Letπh be the average interpolation operator defined in (2.25). For m=0 or 1, 1⩽q⩽∞ and forallv∈W1,q(Ωh) as
(2.28)|v-πhv|m,q,τ⩽∑τ¯′⋂‍τ¯≠∅‍Chτ1-m|v|1,q,τ′.Lemma 2.5 (see [21]).
For allv∈W1,q(Ω),1⩽q<∞ as
(2.29)‖v‖0,q,∂τ⩽C(hτ-1/q‖v‖0,q,τ+hτ1-1/q|v|1,q,τ).


## 2.1. Model Problem and Its Weak Formulation

We will take the state spaceW=L2(0,T;V) with V=H01(Ω) and the control space X=L2(0,T;U) with U=L2(ΩU). Let the observation space Y=L2(0,T;H) with H=L2(Ω) and Uad⊆X a convex subset.We are interested in the following optimal control problem:(2.1)min⁡u∈Uad⊂XJ(u,y(u))=12{∫0T‍‖y-zd‖0,Ω2dt+∫0T‍‖u‖0,ΩU2dt},
subject to
(2.2)yt-div⁡(A∇yt+D∇y+∫0t‍C(t,τ)∇y(x,τ)dτ)=f+Bu,inΩ×(0,T],y=0,on⁡∂Ω×[0,T],y|t=0=y0,inΩ,
where u is control, y is state, zd is the observation, Uad is a closed convex subset, f(x,t)∈L2(0,T;L2(Ω)), and zd and y0∈H1(Ω) are some suitable functions to be specified later. B is a linear bounded operator from L2(ΩU) to L2(Ω) independent of t. And
(2.3)A=A(x)=(ai,j(⋅))n×n∈(C∞(Ω¯))n×n,D=D(x)=(di,j(⋅))n×n∈(C∞(Ω¯))n×n,
such that there is a constant c>0 satisfying that for any vector X∈Rn as follows:
(2.4)XtAX≥c‖X‖Rn2,XtDX≥c‖X‖Rn2,C=C(x,t,τ)=(ci,j(x,t,τ))n×n∈(C∞(0,T;L2(Ω¯))n×n).Let(2.5)(f1,f2)=∫Ω‍f1f2,∀(f1,f2)∈H×H,(u,v)U=∫ΩU‍uv,∀(u,v)∈U×U,a(z,ω)=(A∇z,∇ω),d(z,ω)=(D∇z,∇ω),c(t,τ;z,ω)=(C(t,τ)∇z,∇ω),∀z,w∈V×V.
In the case that f1∈V and f2∈V*, the dual pair (f1,f2) is understood as 〈f1,f2〉V×V*.Assume that there are constantscand C, such that for all t and τ in [0,T] as follows:
(2.6)(a)a(z,z)⩾c‖z‖1,Ω2,(b)|a(z,w)|⩽C‖z‖1,Ω‖w‖1,Ω,|d(z,w)|⩽C‖z‖1,Ω‖w‖1,Ω,(c)|c(t,τ;z,w)|⩽C‖z‖1,Ω‖w‖1,Ω.
for any z and w in V.Then the weak form of the state equation reads as(2.7)(yt,w)+a(yt,w)+d(y,w)+∫0t‍c(t,τ;y(τ),w)dτ=(f+Bu,w)∀w∈V,t∈(0,T],y|t=0=y0.
It is well known (see, e.g., [1]) that the above weak formulation has at least one solution in  y∈W(0,T)={w∈L∞(0,T;H1(Ω)),wt′∈L2(0,T;H1(Ω))}.Therefore, the weak form of the control problem (2.1) and (2.2) reads as (OCP)
(2.8)min⁡u∈UadJ(u,y(u)),(yt,w)+a(yt,w)+d(y,w)+∫0t‍c(t,τ;y(τ),w)dτ=(f+Bu,w)∀w∈V,t∈(0,T],y|t=0=y0.
In the following, we first give the existence and uniqueness of the solution of the system (2.8).Theorem 2.1.
Assume that the condition (2.6) (a)–(c) holds. There exists the unique solution (u,y) for the minimization problem (2.8) such that u∈L2(0,T;L2(ΩU)), y∈L∞(0,T;H1(Ω)), and yt∈L2(0,T;H1(Ω)).Proof.
Let{(un,yn)}n=1∞ be a minimization sequence for the system (2.8), then the sequence {un}n=1∞ is bounded in L2(0,T;L2(ΩU)). Thus there is a subsequence of {un}n=1∞ (still denote by {un}n=1∞ ) such that un converges to u* weakly in L2(0,T;L2(ΩU)). For the subsequence {un}n=1∞, we have
(2.9)(ytn,w)+a(ytn,w)+d(yn,w)+∫0t‍c(t,τ;yn(τ),w(t))dτ=(f+Bun,w)∀w∈V,t∈(0,T].
By setting w=yn and integrating from 0 to t in (2.9), we give
(2.10)‖yn(t)‖1,Ω2+∫0t‍‖yn‖1,Ω2dτ⩽C{‖y0‖1,Ω+C∫0t‍(‖f‖-1,Ω2+‖un‖0,ΩU2)dt+∫0t‍∫0τ‍‖y(s)‖1,Ω2dsdτ}.
Applying Gronwall's inequality to (2.10) yields
(2.11)‖yn‖L∞(0,T;H1(Ω))2+‖yn‖L2(0,T;H1(Ω))2⩽C{‖y0‖1,Ω2+∫0T‍(‖f‖-1,Ω2+‖un‖0,ΩU2)}.
So {un}n=1∞ is a bounded set in L2(0,T;L2(ΩU)) and {yn}n=1∞ is a bounded set in L∞(0,T;H1(Ω)). Thus
(2.12)un⟶uweaklyinL2(0,T;L2(ΩU)),yn⟶yweaklyinL∞(0,T;H1(Ω)),yn(T)⟶y(T)weaklyinH1(Ω).
Let W={w;w∈L∞(0,T;H1(Ω)),wt′∈L2(0,T;H1(Ω))}.
By integrating time from0 to T in (2.9) and taking limit as n→∞, we obtain
(2.13)(y(T),w(T))+a(y(T),w(T))-∫0T‍[(y,wt′)+a(y,wt′)+d(y,w)]+∫0T‍∫0t‍c(t,τ;y(τ),w(t))dτdt=(y0,w(0))+a(y0,w(0))+∫0T‍(f+Bu,w),∀w∈W.
Then,
(2.14)(yt,w)+a(yt,w)+d(y,w)+∫0t‍c(t,τ;y(τ),w)dτ=(f+Bu,w),∀w∈V,t∈(0,T].
Furthermore, we have
(2.15)∫0T‍[(yt,yt)+a(yt,yt)+d(y,yt)+∫0t‍c(t,τ;y(τ),yt(t))dτ]dt=∫0T‍(f+Bu,yt).
Then, we get
(2.16)∫0T‍‖yt‖1,Ω2⩽C∫0T‍[‖f‖-1,Ω2+‖u‖0,ΩU2+‖y‖1,Ω2+∫0t‍‖y‖1,Ω2dτ].
This means yt∈L2(0,T;H1(Ω)). So (u,y) is one solution of (2.8).
Since∫0T‍∥y-zd∥0,Ω2 is a convex function on space L2(0,T;L2(Ω)) and (α/2)∫0T‍∥u∥0,ΩU2 is a strictly convex function on U, hence J(u,y(u)) is a strictly convex function on U, so the minimization problem (2.8) has one unique solution.


## 2.2. Optimality Conditions and Their Finite Element Approximation

By the theory of optimal control problem (see [18]), we can similarly deduce the following optimality conditions of the problem (2.8).Theorem 2.2.
A pair(y,u)∈L2(0,T;H01(Ω))×L2(0,T;L2(ΩU)) is the solution of the optimal control problem (2.8), if and only if there exists a costate p∈L2(0,T;H01(Ω)) such that the triple (y,p,u) satisfies the following optimality conditions:
(2.17)(yt,w)+a(yt,w)+d(y,w)+∫0t‍c(t,τ;y(τ),w(t))dτ=(f+Bu,w)∀w∈V,t∈(0,T],y|t=0=y0;(2.18)-(q,pt)-a(q,pt)+d(q,p)+∫tT‍c(τ,t;q(t),p(τ))dτ=(y-zd,q)∀q∈V,t∈[0,T),p|t=T=0;(2.19)∫0T‍(u+B*p,v-u)Udt⩾0,∀v∈Uad,
where B* is the adjoint operator of B.Let us consider the semi-discrete finite element approximation of the control problem (2.8). Here, we only consider triangular and conforming elements.LetΩh be a polygonal approximation to Ω with boundary  ∂Ωh. Let Th be a partitioning of Ωh into disjoint regular n-simplices τ, so that Ω-h=⋃τ∈Th‍τ-. Each element has at most one face on ∂Ωh, and τ- and τ-′ have either only one common vertex or a whole edge or face if τ- and τ-′∈Th. We further require that Pi∈∂Ωh⇒Pi∈∂Ω  where Pi(i=1,…,J) is the vertex set associated with the triangulation Th. As usual, h denotes the diameter of the triangulation Th. For simplicity, we assume that Ω is a convex polygon so that Ω=Ωh.Associated withTh is a finite-dimensional subspace Sh of C(Ω-h), such that χ|τ are polynomials of order m(m≥1) for all χ∈Sh and τ∈Th. Let Vh={vh∈Sh:vh(Pi)=0(i=1,…,J)},Wh=L2(0,T;Vh). Note that we do not impose a continuity requirement. It is easy to see that Vh⊂V,Wh⊂W.LetTUh be a partitioning of ΩUh into disjoint regular n-simplices τU, so that Ω-Uh=⋃τU∈TUh‍τ-U. τ-U and τ-U′ have either only one common vertex or a whole edge or face if τ-U and τ-U′∈TUh. We further require that Pi∈∂ΩUh⇒Pi∈∂ΩU, where Pi(i=1,…,J) is the vertex set associated with the triangulation TUh. For simplicity, we again assume that ΩU is a convex polygon so that ΩU=ΩUh.Associated withTUh is another finite-dimensional subspace Uh of L2(ΩUh), such that χ|τU are polynomials of order m(m⩾0) for all χ∈Uh and τU∈TUh. Here there is no requirement for the continuity. Let Xh=L2(0,T;Uh). It is easy to see that Xh⊂X. Let hτ(hτU) denote the maximum diameter of the element τ(τU) in Th(TUh).Due to the limited regularity of the optimal controlu in general, there will be no advantage in considering higher-order finite element spaces than the piecewise constant space for the control. We therefore only consider the piecewise constant finite element space for the approximation of the control, though higher-order finite element spaces will be used to approximate the state and the co-state. Let P0(Ω) denote all the 0-order polynomial over Ω. Therefore, we always take Xh={u∈X:u(x,t)|x∈τU∈P0(τU),forallt∈[0,T]}. Uadh is a closed convex set in Xh. For ease of exposition, in this paper, we assume that Uadh⊂Uad∩Xh.Then a possible semi-discrete finite element approximation of(OCP) is as follows (OCP)h:(2.20)min⁡uh∈UadhJ(uh,yh)=12{∫0T‍‖yh-zd‖0,Ω2+∫0T‍‖uh‖0,ΩU2},
such that
(2.21)(∂yh∂t,wh)+a(∂yh∂t,wh)+d(yh,wh)+∫0t‍c(t,τ;yh(τ),wh(t))dτ=(f+Buh,wh),∀wh∈Vh,t∈(0,T],yh|t=0=yh0,
where yh∈Wh and yh0∈Vh is the approximation of y0.In the same way of proving Theorem2.1, we can easily prove that the problem (2.20)-(2.21) has a unique solution (yh,uh)∈Wh×Uadh.It is well known (see [18]) that a pair (yh,uh)∈Wh×Uadh is a solution of (2.20)-(2.21), if and only if there exists a co-state ph∈Wh such that the triple (yh,ph,uh) satisfies the following optimality conditions:
(2.22)(∂yh∂t,wh)+a(∂yh∂t,wh)+d(yh,wh)+∫0t‍c(t,τ;yh(τ),wh(t))dτ=(f+Buh,wh),∀wh∈Vh,yh|t=0=yh0,(2.23)-(qh,∂ph∂t)-a(qh,∂ph∂t)+d(qh,ph)+∫tT‍c(τ,t;qh,ph(τ))dτ=(yh-zd,qh),∀qh∈Vh,ph|t=T=0,(2.24)∫0T‍(uh+B*ph,vh-uh)Udt⩾0,∀vh∈Uadh.The optimality conditions in (2.22)–(2.24) are the semi-discrete approximation to the problem (2.17)–(2.19).Introduce the local averaging operatorπh given by
(2.25)(πhw)|τU:=∫τU‍w∫τU‍1,∀τU∈TUh.
Then, we have ∫ΩU‍w=∫ΩU‍πhw for any w∈L2(0,T;L2(ΩU)),t∈[0,T] and (2.24) is equivalent to
(2.26)∫0T‍(uh+πh(B*ph),vh-uh)Udt⩾0,∀vh∈Uadh.
In the following, we derive the a posteriori error estimates for semi-discrete finite element approximation (2.22)–(2.24), allowing different meshes to be used for the state and the control.The following lemmas are important in deriving the a posteriori error estimates of residual type.Lemma 2.3 (see [19]).
Letπ^h be the standard Lagrange interpolation operator. For m=0 or 1,  q>n/2 and v∈W2,q(Ω) as
(2.27)|v-π^hv|m,q,Ω⩽Ch2-m|v|2,q,Ω.Lemma 2.4 (see [20]).
Letπh be the average interpolation operator defined in (2.25). For m=0 or 1, 1⩽q⩽∞ and forallv∈W1,q(Ωh) as
(2.28)|v-πhv|m,q,τ⩽∑τ¯′⋂‍τ¯≠∅‍Chτ1-m|v|1,q,τ′.Lemma 2.5 (see [21]).
For allv∈W1,q(Ω),1⩽q<∞ as
(2.29)‖v‖0,q,∂τ⩽C(hτ-1/q‖v‖0,q,τ+hτ1-1/q|v|1,q,τ).


## 3. A Posteriori Error Estimates inL∞(0,T;H1(Ω))-Norm

In this paper, the control constraints are given in an integral sense as follows:(3.1)Uad={v∈X;∫ΩU‍v⩾0,t∈[0,T]}.
The following lemma is the first step to derive the a posteriori error estimates of residual type.Lemma 3.1.
Let(y,p,u) and (yh,ph,uh) be the solutions of (2.17)–(2.19) and (2.22)–(2.24). Then, we have
(3.2)‖u-uh‖L2(0,T;L2(ΩU))2⩽Cη12+C‖ph-p(uh)‖L2(0,T;L2(Ω))2,
where
(3.3)η12=∫0T‍{∑τU‍∫τU‍(B*ph-Ph(B*ph))2}dt,Ph is the L2-projection from L2(Ω) to Uh, and p(uh) is defined by the following system:
(3.4)(∂∂ty(uh),ω)+a(∂∂ty(uh),ω)+d(y(uh),ω)+∫0t‍c(t,τ;y(uh)(τ),ω(t))dτ=(f+Buh,ω),∀ω∈V,y(uh)(0)=y0h(x),x∈Ω,(3.5)-(q,∂∂tp(uh))-a(q,∂∂tp(uh))+d(q,p(uh))+∫tT‍c(τ,t;q(t),p(uh)(τ))dτ=(y(uh)-zd,q),∀q∈V.Proof.
From (2.19), we have
(3.6)(u,u-uh)U⩽-(B*p,u-uh)U.
Then, by (2.24) and (3.6), we have
(3.7)‖u-uh‖L2(0,T;L2(ΩU))2=∫0T‍[(u,u-uh)U-(uh,u-uh)U]dt=∫0T‍-(B*p+uh,u-uh)Udt=-∫0T‍(B*ph+uh,u-vh)Udt-∫0T‍(B*ph+uh,vh-uh)Udt+∫0T‍(B*ph-B*p(uh),u-uh)Udt+∫0T‍(B*p(uh)-B*p,u-uh)Udt⩽inf⁡vh∈Uadh∫0T‍(B*ph+uh,vh-u)Udt+∫0T‍(B*(ph-p(uh)),u-uh)Udt+∫0T‍(B*(p(uh)-p),u-uh)Udt=I1+I2+I3.
Next, we will estimate I1,I2,andI3, respectively.
(1) We first estimateI1. Let Ph be the L2-projection from L2(Ω) to Uh.
We have(3.8)∫ΩU‍(Phv-v)ϕ=0,∀ϕ∈Xh,v∈Uad,t∈(0,T].
Since v∈Uad, so ∫ΩU‍Phv⩾0, then Phv∈Uadh. So that we can take vh=Phu in I1.
For givent∈(0,T], let
(3.9)uh=Ph(-B*ph+max⁡{0,∫ΩU‍B*ph∫ΩU‍1}).
We have uh∈Xh. We will show that uh is the solution of the variational inequality in (2.24) assuming ph is known.
Since∫ΩU‍[Ph(-B*ph+max{0,∫ΩU‍B*ph/∫ΩU‍1})-(-B*ph+max{0,∫ΩU‍B*ph/∫ΩU‍1})]=0, we have
(3.10)∫ΩU‍uh=-∫ΩU‍B*ph+∫ΩU‍max⁡{0,∫ΩU‍B*ph∫ΩU‍1}={-∫ΩU‍B*ph,∫ΩU‍B*ph<0,0,∫ΩU‍B*ph⩾0.
Thus, ∫ΩU‍uh⩾0, we have uh∈Uadh. Note that for all vh∈Uadh,t∈(0,T], we have
(3.11)(uh+B*ph,vh-uh)U=∫ΩU‍[Ph(-B*ph+max⁡{0,∫ΩU‍B*ph∫ΩU‍1})-(-B*ph+max⁡{0,∫ΩU‍B*ph∫ΩU‍1})+max⁡{0,∫ΩU‍B*ph∫ΩU‍1}](vh-uh)=∫ΩU‍max⁡{0,∫ΩU‍B*ph∫ΩU‍1}(vh-uh).
If ∫ΩU‍B*ph<0, then
(3.12)(uh+B*ph,vh-uh)U=∫ΩU‍0⋅(vh-uh)=0.
If ∫ΩU‍B*ph⩾0, since
(3.13)∫ΩU‍uh=∫ΩU‍(-B*ph+max⁡{0,∫ΩU‍B*ph∫ΩU‍1})=0.
we have
(3.14)(uh+B*ph,vh-uh)U=∫ΩU‍B*ph∫ΩU‍1∫ΩU‍(vh-uh)=∫ΩU‍B*ph∫ΩU‍1⋅∫ΩU‍vh⩾0.
From (3.11)–(3.14), we obtain
(3.15)(uh+B*ph,vh-uh)U⩾0,∀vh∈Uadh.
So uh=Ph(-B*ph+max{0,∫ΩU‍B*ph/∫ΩU‍1}) is the solution of the variational inequality in (2.24) assuming ph is known.
Then,(3.16)I1⩽∫0T‍(B*ph+uh,Phu-u)Udt=∫0T‍{∑τU‍∫τU‍[Ph(-B*ph+max⁡{0,∫ΩU‍B*ph∫ΩU‍1})+B*ph](Phu-u)}dt.
Since ∫τU‍(Phu-u)=0, we have
(3.17)I1⩽∫0T‍{∑τU‍∫τU‍(-Ph(B*ph)+B*ph)(Phu-u)}dt=∫0T‍{∑τU‍∫τU‍(-Ph(B*ph)+B*ph)(Ph(u-uh)-(u-uh))}dt⩽C(δ)∫0T‍{∑τU‍∫τU‍(-Ph(B*ph)+B*ph)2}dt+δ‖u-uh‖L2(0,T;L2(ΩU))2=Cη12+δ‖u-uh‖L2(0,T;L2(ΩU))2.(2) Consider
(3.18)I2=∫0T‍(B*(ph-p(uh)),u-uh)Udt⩽C‖ph-p(uh)‖L2(0,T;L2(Ω))2+δ‖u-uh‖L2(0,T;L2(ΩU))2.(3) By (3.4) and (2.17), we have for t∈(0,T](3.19)(∂∂t(y-y(uh)),ω)+a(∂∂t(y-y(uh)),ω)+d(y-y(uh),w)+∫0t‍c(t,τ;(y-y(uh))(τ),ω(t))dτ=(B(u-uh),ω),∀ω∈V,
and from (3.5) and (2.18), we have
(3.20)-(q,∂∂t(p-p(uh)))-a(q,∂∂t(p-p(uh)))+d(q,p-p(uh))+∫tT‍c(τ,t;q(t),(p-p(uh))(τ))dτ=(y-y(uh),q),∀q∈V.
Then, from (3.19), (3.20), and integrating by part we have
(3.21)I3=∫0T‍(B*(p(uh)-p),u-uh)Udt=∫0T‍(p(uh)-p,B(u-uh))Udt=∫0T‍[(∂∂t(y-y(uh)),p(uh)-p)+a(∂∂t(y-y(uh)),p(uh)-p)+d(y-y(uh),p(uh)-p)+∫0t‍c(t,τ;(y-y(uh))(τ),(p(uh)-p)(t))dτ]dt=∫0T‍[-(y-y(uh),∂∂t(p(uh)-p))-a(y-y(uh),∂∂t(p(uh)-p))+d(y-y(uh),p(uh)-p)+∫tT‍c(τ,t;(y-y(uh))(t),(p(uh)-p)(τ))dτ]dt=∫0T‍-(y-y(uh),y-y(uh))dt⩽0.
Following from (3.17)–(3.21), let δ be small enough as
(3.22)‖u-uh‖L2(0,T;L2(ΩU))2⩽Cη12+C‖ph-p(uh)‖L2(0,T;L2(Ω))2.
This completes the proof.Lemma 3.2.
Let(y,p,u) and (yh,ph,uh) be the solutions of (2.17)–(2.19), and (2.22)–(2.24) respectively. Then, there hold the a posteriori error estimates as
(3.23)‖yh-y(uh)‖L∞(0,T;H1(Ω))2+‖∂∂t(yh-y(uh))‖L2(0,T;H1(Ω))2+‖ph-p(uh)‖L∞(0,T;H1(Ω))2+‖∂∂t(ph-p(uh))‖L2(0,T;H1(Ω))2⩽C∑i=26‍ηi2,
where
(3.24)η22=∫0T‍{∑τ‍hτ2∫τ‍(∫tT‍∂ph∂t-div⁡(A*∇∂ph∂t)+div⁡(D*∇ph)+∫tT‍div⁡(C*(τ,t)∇ph(τ))dτ+yh-zd)2dτ}dt,η32=∫0T‍∑τ‍hl∫∂τ‍[-(A*∇∂ph∂t)⋅n+(D*∇ph)⋅n+∫tT‍(C*(τ,t)∇ph(τ))⋅ndτ]2dldt,η42=∫0T‍{-∫0t‍div⁡(C(t,τ)∇yh(τ))dτ-f-Buh)2∑τ‍hτ2∫τ‍(∫0t‍∂∂tyh-div⁡(A∇∂yh∂t)-div⁡(D∇yh)-∫0t‍div⁡(C(t,τ)∇yh(τ))dτ-f-Buh)2}dt,η52=∫0T‍∑τ‍hl∫∂τ‍[(A∇∂yh∂t)⋅n+(A∇yh)⋅n+∫0t‍(C(t,τ)∇yh(τ))⋅ndτ]2dldt,η62=‖y0h-y0‖1,Ω2,
where l is a face of an element τ, hl is the maximum diameter of l,  and  [∇ph·n] and [∇yh·n] are the normal derivative jumps over the interior face l defined by
(3.25)[∇ph⋅n]l=(∇phh|τl1-∇ph|τl2)⋅n,[∇yh⋅n]l=(∇yh|τl1-∇yh|τl2)⋅n,
where n is the unit normal vector on l=τl1∩τl2 outwards τl1. For later convenience, one can define [∇ph·n]l=0 and [∇yh·n]l=0 when l⊂∂Ω.Proof.
Let(3.26)〈R(uh),v〉=-(v,∂∂t(ph-p(uh)))-a(v,∂∂t(ph-p(uh)))+d(v,ph-p(uh))+∫tT‍c(τ,t;v(t),(ph-p(uh))(τ))dτ,
and πh the average interpolation operator defined as in (2.25) and e=ph-p(uh). Then, it follows from (2.23) and (3.5) that
(3.27)-(qh,∂∂t(ph-p(uh)))-a(qh,∂∂t(ph-p(uh)))+d(qh,ph-p(uh))+∫tT‍c(τ,t;qh(t),(ph-p(uh))(τ))dτ=(yh-y(uh),qh),∀qh∈Vh.
We have
(3.28)〈R(uh),v〉=-(v-πhv,∂∂t(ph-p(uh)))-a(v-πhv,∂∂t(ph-p(uh)))+d(v-πhv,ph-p(uh))+∫tT‍c(τ,t;(v-πhv)(t),(ph-p(uh))(τ))dτ-(πhv,∂∂t(ph-p(uh)))-a(πhv,∂∂t(ph-p(uh)))+d(πhv,ph-p(uh))+∫tT‍c(τ,t;πhv(t),(ph-p(uh))(τ))dτ=-(v-πhv,∂ph∂t)-a(v-πhv,∂ph∂t)+d(v-πhv,ph)+∫tT‍c(τ,t;(v-πhv)(t),ph(τ))dτ-(y(uh)-zd,v-πhv)+(yh-y(uh),πhv)=∑τ‍∫τ‍(-∂ph∂t+div⁡(A*∇∂ph∂t)-div⁡(D*∇ph)-∫tT‍div⁡(C*(τ,t)∇ph(τ))dτ-yh+zd)×(v-πhv)+∑τ‍∫∂τ‍[-(A*∇∂ph∂t)⋅n+(D*∇ph)⋅n+∫tT‍(C*(τ,t)∇ph(τ))⋅ndτ]×(v-πhv)+(yh-y(uh),v)⩽{∑τ‍∫τ‍hτ2(∫tT‍-∂ph∂t+div⁡(A*∇∂ph∂t)-div⁡(D*∇ph)-∫tT‍div⁡(C*(τ,t)∇ph(τ))dτ-yh+zd)2+∑τ‍∫∂τ‍hl[-(A*∇∂ph∂t)⋅n+(D*∇ph)⋅n+∫tT‍(C*(τ,t)∇ph(τ))⋅ndτ]2}1/2×‖v‖1,Ω+(yh-y(uh),v).
Taking v=ph-p(uh) in (3.28) and from (2.6), we have
(3.29)-12ddt‖ph-p(uh)‖0,Ω2-12ddta(ph-p(uh),ph-p(uh))+c‖ph-p(uh)‖1,Ω2⩽{+∫tT‍(C*(τ,t)∇ph(τ))⋅ndτ]2∑τ‍∫τ‍hτ2(-∂ph∂t+div⁡(A*∇∂ph∂t)-div⁡(D*∇ph)-∫tT‍div⁡(C*(τ,t)∇ph(τ))dτ-yh+zd∫tT‍)2+∑τ‍∫∂τ‍hl[-(A*∇∂ph∂t)⋅n+(D*∇ph)⋅n]2}1/2×‖ph-p(uh)‖1,Ω+(yh-y(uh),ph-p(uh))-∫tT‍c(τ,t;(ph-p(uh))(t),(ph-p(uh))(τ))dτ.
Integrating time from t to T in (3.29) and by Schwartz inequality, Lemmas 2.4 and 2.5, we have
(3.30)12‖ph-p(uh)‖0,Ω2+c‖ph-p(uh)‖1,Ω2+c∫tT‍‖ph-p(uh)‖1,Ω2dτ⩽∫tT‍∑τ‍hτ2×∫τ‍(∫τT‍∂ph∂t-div⁡(A*∇∂ph∂t)+div⁡(D*∇ph)+∫τT‍div⁡(C*(s,τ)∇ph(s))ds+yh-zd)2dτ+∫tT‍∑τ‍hl∫∂τ‍[-(A*∇∂ph∂t)⋅n+(D*∇ph)⋅n+∫τT‍(C*(s,τ)∇ph(s))⋅nds]2dτ+δ∫tT‍‖ph-p(uh)‖1,Ω2dτ+C∫tT‍‖yh-y(uh)‖0,Ω2dτ+C∫tT‍∫τT‍‖(ph-p(uh))(s)‖1,Ω2dsdτ.
Letting δ be small enough, we have
(3.31)∫tT‍‖ph-p(uh)‖1,Ω2dτ⩽C∫tT‍∑τ‍hτ2∫τ‍(∫τT‍∂ph∂t-div⁡(A*∇∂ph∂t)+div⁡(D*∇ph)+∫τT‍div⁡(C*(s,τ)∇ph(s))ds+yh-zd)2dτ+C∫tT‍∑τ‍hl∫∂τ‍[-(A*∇∂ph∂t)⋅n+(D*∇ph)⋅n+∫τT‍(C*(s,τ)∇ph(s))⋅nds]2dτ+C∫tT‍‖yh-y(uh)‖0,Ω2dτ+C∫tT‍∫τT‍‖(ph-p(uh))(s)‖1,Ω2dsdτ.
Then, from Gronwall inequality and (3.28)–(3.31) we have
(3.32)‖ph-p(uh)‖L2(0,T;H1(Ω))2⩽Cη22+Cη32+C‖yh-y(uh)‖L2(0,T;L2(Ω))2.
Similarly,
(3.33)‖ph-p(uh)‖L∞(0,T;H1(Ω))2⩽C(η22+η32+‖yh-y(uh)‖L2(0,T;L2(Ω))2)+C∫0T‍∫tT‍‖(ph-p(uh))(τ)‖1,Ω2dτdt⩽Cη22+Cη32+C‖yh-y(uh)‖L2(0,T;L2(Ω))2.
In the same way of getting (3.32),by setting v=(∂/∂t)(ph-p(uh)) in (3.28), we have
(3.34)‖∂∂t(ph-p(uh))‖L2(0,T;H1(Ω))2⩽Cη22+Cη32+C‖yh-y(uh)‖L2(0,T;L2(Ω))2.
Similarly analysis for ∥yh-y(uh)∥L∞(0,T;H1(Ω)), we let
(3.35)〈Q(uh),v〉=(∂∂t(yh-y(uh)),v)+a(∂∂t(yh-y(uh)),v)+d(yh-y(uh),v)+∫0t‍c(t,τ;(yh-y(uh))(τ),v(t))dτ.
From (2.22) and (3.4), we obtain
(3.36)(ωh,∂∂t(yh-y(uh)))+a(∂∂t(yh-y(uh)),ωh)+d(yh-y(uh),ωh)+∫0t‍c(t,τ;(yh-y(uh))(τ),ωh(t))dτ=0,∀ωh∈Vh.
We have
(3.37)〈Q(uh),v〉=(∂∂t(yh-y(uh)),v-πhv)+a(∂∂t(yh-y(uh)),v-πhv)+d(yh-y(uh),v-πhv)+∫0t‍c(t,τ;(yh-y(uh))(τ),(v-πhv)(t))dτ=(∂yh∂t,v-πhv)+a(∂yh∂t,v-πhv)+d(yh,v-πhv)+∫0t‍c(t,τ;yh(τ),(v-πhv)(t))dτ-(f+Buh,v-πhv)=∑τ‍∫τ‍(∂yh∂t-div⁡(A∇∂yh∂t)-div⁡(D∇yh)-∫0t‍div⁡(C(t,τ)∇yh)dτ-f-Buh)(v-πhv)+∑τ‍∫∂τ‍[(A∇∂yh∂t)⋅n+(D∇yh)⋅n+∫0t‍(C(t,τ)∇yh)⋅ndτ](v-πhv)⩽{∑τ‍∫τ‍hτ2(∂∂tyh-div⁡(A∇∂yh∂t)-div⁡(D∇yh)-∫0t‍div⁡(C(t,τ)∇yh)dτ-f-Buh)2+∑τ‍∫∂τ‍hl[(A∇∂yh∂t)⋅n+(D∇yh)⋅n+∫0t‍(C(t,τ)∇yh)⋅ndτ]2}1/2‖v‖1,Ω.
By setting v=yh-y(uh) and Swartz inequality, we have
(3.38)12ddt‖yh-y(uh)‖0,Ω2+12ddta(yh-y(uh),yh-y(uh))+c‖yh-y(uh)‖1,Ω2⩽∑τ‍∫τ‍hτ2(∂yh∂t-div⁡(A∇∂yh∂t)-div⁡(D∇yh)-∫0t‍div⁡(C(t,τ)∇yh)dτ-f-Buh)2+∑τ‍∫∂τ‍hl[(A∇∂yh∂t)⋅n+(D∇yh)⋅n+∫0t‍(C(t,τ)∇yh)⋅ndτ]2+δ‖yh-y(uh)‖1,Ω2-∫0t‍c(t,τ;(yh-y(uh))(τ),(yh-y(uh))(t))dτ.
Integrating time from 0 to t in (3.38), we obtain
(3.39)‖yh-y(uh)‖1,Ω2+c∫0t‍‖yh-y(uh)‖1,Ω2dτ⩽C{∫0t‍∑τ‍hτ2∫τ‍(∂yh∂t-div⁡(A∇∂yh∂t)-div⁡(D∇yh)-∫0τ‍div⁡(C(τ,s)∇yh)ds-f-Buh)2dτ+∫0t‍∑τ‍hl∫∂τ‍[(A∇∂yh∂t)⋅n+(D∇yh)⋅n+∫0τ‍(C(τ,s)∇yh)⋅nds]2dt}+δ∫0t‍‖yh-y(uh)‖1,Ω2dτ+C∫0t‍∫0τ‍‖yh-y(uh)‖1,Ω2dsdτ+C‖y0-y0h‖1,Ω2.
Since δ is small enough, then from (3.39) and Gronwall inequality, we have
(3.40)∫0t‍‖yh-y(uh)‖1,Ω2dt⩽C∫0t‍∑τ‍hτ2∫τ‍(∫0τ‍∂yh∂t-div⁡(A∇∂yh∂t)-div⁡(D∇yh)-∫0τ‍div⁡(C(τ,s)∇yh)ds-f-Buh)2dτ+C∫0t‍∑τ‍hl∫∂τ‍[(A∇∂yh∂t)⋅n+(D∇yh)⋅n+∫0τ‍(C(τ,s)∇yh)⋅nds]2dt+C‖y0-y0h‖1,Ω2.
Then,
(3.41)‖yh-y(uh)‖L2(0,T;H1(Ω))2⩽C(η42+η52+η62),‖yh-y(uh)‖L∞(0,T;H1(Ω))2⩽C(η42+η52+η62).
In the same way of getting (3.34), we can similarly obtain
(3.42)‖∂∂t(yh-y(uh))‖L2(0,T;H1(Ω))2⩽C(η42+η52+η62).
Then the desired results (3.23) follow from (3.32)–(3.34) and (3.41)-(3.42).From Lemmas3.1 and 3.2, we have the following results.Theorem 3.3.
Let(y,p,u) and (yh,ph,uh) be the solutions of(2.17)–(2.19) and (2.22)–(2.24) respectively. Then, there hold the a posteriori error estimates as
(3.43)‖u-uh‖L2(0,T;L2(ΩU))2+‖y-yh‖L∞(0,T;H1(Ω))2+‖∂∂t(y-yh)‖L2(0,T;H1(Ω))2+‖p-ph‖L∞(0,T;H1(Ω))2+‖∂∂t(p-ph)‖L2(0,T;H1(Ω))2⩽C∑i=16‍ηi2,
where η12 is defined in Lemma 3.1.Proof.
First,  from (3.27) and (3.36), and [2], we have the following stability results:
(3.44)‖y-y(uh)‖L∞(0,T;H1(Ω))2+‖∂∂t(y-y(uh))‖L2(0,T;H1(Ω))2⩽C‖u-uh‖L2(0,T;L2(ΩU))2,‖p-p(uh)‖L∞(0,T;H1(Ω))2+‖∂∂t(p-p(uh))‖L2(0,T;H1(Ω))2⩽C‖y-y(uh)‖L2(0,T;L2(Ω))2⩽C‖u-uh‖L2(0,T;L2(ΩU))2.
Then, the desired results (3.43) follows from triangle inequality, (3.44)  and Lemmas 3.1 and 3.2.
This completes the proof.


## 4. A Posteriori Error Estimates inL2(0,T;L2(Ω))-Norm

In the following, we will derive the a posteriori error estimates inL2(0,T;L2(Ω))-norm.We will introduce the following dual equations.For givenF∈L2(0,T;L2(Ω)), we have
(4.1)∂ϕ∂t-div⁡(A∇∂ϕ∂t)-div⁡(D∇ϕ)-∫0t‍div⁡(C(t,τ)∇ϕ(τ))dτ=F,(x,t)∈Ω×(0,T],ϕ∂Ω=0,t∈(0,T],ϕ(x,0)=0,x∈Ω,
and its dual equation
(4.2)-∂ψ∂t+div⁡(A*∇∂ψ∂t)-div⁡(D*∇ψ)-∫tT‍div⁡(C*(τ,t)∇ψ(τ))dτ=F,(x,t)∈Ω×(0,T],ψ∂Ω=0,t∈(0,T],ψ(x,T)=0,x∈Ω.
From [1, 2], we have the following stability results.Lemma 4.1.
Assume thatΩ is a convex domain. Let ϕ and ψ be the solution of (4.1) and (4.2), respectively. Then,
(4.3)‖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)),‖∂∂tϕ‖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)),‖∂∂tψ‖L2(0,T;L2(Ω))⩽C‖F‖L2(0,T;L2(Ω)),
where D2ϕ=∂2ϕ/∂xi∂xj,1⩽i,j⩽n, and D2ψ is defined similarly.Using Lemmas3.1 and 4.1, we have the following upperbounds.Lemma 4.2.
Let(y,p,u) and (yh,ph,uh) be the solutions of (2.17)–(2.19) and (2.22)–(2.24), respectively. Then, there hold the a posteriori error estimates as
(4.4)‖yh-y(uh)‖L2(0,T;L2(Ω))2+‖ph-p(uh)‖L2(0,T;L2(Ω))2⩽C(∑i=25‍ξi2+η62),
where η62 is defined in Lemma 3.2, and
(4.5)ξ22=∫0T‍{∑τ‍hτ4∫τ‍(∫tT‍∂ph∂t-div⁡(A*∇∂ph∂t)+div⁡(D*∇ph)+∫tT‍div⁡(C*(τ,t)∇ph(τ))dτ+yh-zd)2}dt,ξ32=∫0T‍∑τ‍hl3∫∂τ‍[-(A*∇∂ph∂t)⋅n+(D*∇ph)⋅n+∫tT‍(C*(τ,t)∇ph(τ))⋅ndτ]2dldt,ξ42=∫0T‍{∑τ‍hτ4∫τ‍(∫0t‍∂yh∂t-div⁡(A∇∂yh∂t)-div⁡(D∇yh)-∫0t‍div⁡(C(t,τ)∇yh(τ))dτ-f-Buh)2}dt,ξ52=∫0T‍∑τ‍hl3∫∂τ‍[(A∇∂yh∂t)⋅n+(D∇yh)⋅n+∫0t‍(C(t,τ)∇yh(τ))⋅ndτ]2dldt.Proof.
We first estimate∥ph-p(uh)∥L2(0,T;L2(Ω))2.
Letϕ be the solution of (4.1) with F=ph-p(uh),  and  ϕI=π^hϕ the interpolation of ϕ in Lemma 2.3.
From (4.1), (3.27), and by integrating by parts we obtain
(4.6)‖ph-p(uh)‖L2(0,T;L2(Ω))2=∫0T‍(F(t),(ph-p(uh))(t))dt=∫0T‍[∫0t‍-(∂∂t(ph-p(uh)),ϕ)-a(ϕ,∂∂t(ph-p(uh)))+d(ϕ,ph-p(uh))+∫0t‍c(t,τ;ϕ(τ),(ph-p(uh))(t))dτ(ϕ,∂∂t(ph-p(uh)))]dt=∫0T‍[-(∂∂t(ph-p(uh)),ϕ-ϕI)-a(ϕ-ϕI,∂∂t(ph-p(uh)))+d(ϕ-ϕI,ph-p(uh))+∫0t‍c(t,τ;(ϕ-ϕI)(τ),(ph-p(uh))(t))dτ-(∂∂t(ph-p(uh)),ϕI)-a(ϕI,∂∂t(ph-p(uh)))+d(ϕI,ph-p(uh))+∫0t‍c(t,τ;ϕI(τ),(ph-p(uh))(t))dτ]dt=∫0T‍[∫0t‍-(∂ph∂t,ϕ-ϕI)-a(ϕ-ϕI,∂ph∂t)+d(ϕ-ϕI,ph)+∫0t‍c(t,τ;(ϕ-ϕI)(τ),ph(t))dτ+(∂p(uh)∂t,ϕ-ϕI)+a(ϕ-ϕI,∂p(uh)∂t)-d(ϕ-ϕI,p(uh))-∫0t‍c(t,τ;(ϕ-ϕI)(τ),p(uh)(t))dτ-(∂∂t(ph-p(uh)),ϕI)-a(ϕI,∂∂t(ph-p(uh)))+d(ϕI,ph-p(uh))+∫0t‍c(t,τ;ϕI(τ),(ph-p(uh))(t))dτ]dt=∫0T‍[∫tT‍-(∂ph∂t,ϕ-ϕI)-a(ϕ-ϕI,∂ph∂t)+d(ϕ-ϕI,ph)+∫tT‍c(τ,t;(ϕ-ϕI)(t),ph(τ))dτ+(∂p(uh)∂t,ϕ-ϕI)+a(ϕ-ϕI,∂p(uh)∂t)-d(ϕ-ϕI,p(uh))-∫tT‍c(τ,t;(ϕ-ϕI)(t),p(uh)(τ))dτ-(∂∂t(ph-p(uh)),ϕI)-a(ϕI,∂∂t(ph-p(uh)))+d(ϕI,ph-p(uh))+∫tT‍c(τ,t;ϕI(t),(ph-p(uh))(τ))dτ]dt=∫0T‍[-(∂ph∂t,ϕ-ϕI)-a(ϕ-ϕI,∂ph∂t)+d(ϕ-ϕI,ph)+∫tT‍c(τ,t;(ϕ-ϕI)(t),ph(τ))dτ]dt-∫0T‍(y(uh)-zd,ϕ-ϕI)dt+∫0T‍(yh-y(uh),ϕI)dt=∫0T‍{∑τ‍∫τ‍(∫tT‍-∂ph∂t+div⁡(A*∇∂ph∂t)-div⁡(D*∇ph)+∫tT‍div⁡(C*(τ,t)∇ph(τ))dτ+yh-zd)(ϕ-ϕI)}dt+∫0T‍∑τ‍∫∂τ‍[-(A*∇∂ph∂t)⋅n+(D*∇ph)⋅n+∫tT‍(C*(τ,t)∇ph(τ))⋅ndτ]×(ϕ-ϕI)dldt+∫0T‍(yh-y(uh),ϕ)dt=J1+J2+J3.
It follows from Lemmas 2.3, 2.5, and 4.1 that
(4.7)J1⩽C(δ)∫0T‍{∑τ‍hτ4∫τ‍(∫tT‍∂ph∂t-div⁡(A*∇∂ph∂t)+div⁡(D*∇ph)+∫tT‍div⁡(C*(τ,t)∇ph(τ))dτ+yh-zd)2}dt+δ∫0T‍|ϕ|2,Ω2dt⩽C(δ)ξ22+δ‖ph-p(uh)‖L2(0,T;L2(Ω))2,J2⩽C(δ)∫0T‍∑τ‍hl3∫∂τ‍[-(A*∇∂ph∂t)⋅n+(D*∇ph)⋅n+∫tT‍(C*(τ,t)∇ph(τ))⋅ndτ]2dldt+δ∫0T‍|ϕ|0,Ω2dt⩽C(δ)ξ32+δ‖ph-p(uh)‖L2(0,T;L2(Ω))2.
By Schwartz inequality, we have
(4.8)J3⩽C(δ)‖yh-y(uh)‖L2(0,T;L2(Ω))2+δ‖ph-p(uh)‖L2(0,T;L2(Ω))2.
Letting δ be small enough, it follows from (4.6)–(4.8) that
(4.9)‖ph-p(uh)‖L2(0,T;L2(Ω))2⩽C∑i=23‍ξi2+C‖yh-y(uh)‖L2(0,T;L2(Ω))2.
Next, we estimate ∥yh-y(uh)∥L2(0,T;L2(Ω))2. Similarly let ψ be the solution of (4.2) with F=yh-y(uh),  and  ψI=π^hψ the interpolation of ψ in Lemma 2.3.
Then, it follows from (3.36) and integrating by parts that
(4.10)‖yh-y(uh)‖L2(0,T;L2(Ω))2=∫0T‍(F(t),(yh-y(uh))(t))dt=∫0T‍[∫tT‍(∂∂t(yh-y(uh)),ψ)+a(∂∂t(yh-y(uh)),ψ)+d(yh-y(uh),ψ)+∫tT‍c(τ,t;(yh-y(uh))(t),ψ(τ))dτ]dt+a(y0h-y0,ψ(0))+(y0h-y0,ψ(0))=∫0T‍[∫tT‍(∂∂t(yh-y(uh)),ψ-ψI)+a(∂∂t(yh-y(uh)),ψ-ψI)+d(yh-y(uh),ψ-ψI)+∫tT‍c(τ,t;(yh-y(uh))(t),(ψ-ψI)(τ))dτ]dt+∫0T‍[∫tT‍(∂∂t(yh-y(uh)),ψI)+a(∂∂t(yh-y(uh)),ψI)+d(yh-y(uh),ψI)+∫tT‍c(τ,t;(yh-y(uh))(t),ψI(τ))dτ]dt+a(y0h-y0,ψ(0))+(y0h-y0,ψ(0))=∫0T‍[∫tT‍(∂yh∂t,ψ-ψI)+a(∂yh∂t,ψ-ψI)+d(yh,ψ-ψI)+∫0t‍c(t,τ;yh(τ),(ψ-ψI)(t))dτ]dt-∫0T‍(f+Buh,ψ-ψI)dt+a(y0h-y0,ψ(0))+(y0h-y0,ψ(0))=∫0T‍{∑τ‍∫τ‍(∂yh∂t-div⁡(A∇∂yh∂t)-div⁡(D∇yh)-∫0t‍div⁡(C(τ,t)∇yh(τ))dτ-f-Buh)(ψ-ψI)}dt+∫0T‍∑τ‍∫∂τ‍[(A∇∂yh∂t)⋅n+(D∇yh)⋅n-∫0t‍(C(t,τ)∇yh(τ))⋅n]dτ(ψ-ψI)dldt+a(y0h-y0,ψ(0))+(y0h-y0,ψ(0))=D1+D2+D3+D4.
Similarly, it follows from Lemmas 2.3,  2.5 and 4.1 that
(4.11)D1⩽C(δ)∫0T‍{∑τ‍hτ4∫τ‍(∫0t‍∂yh∂t-div⁡(A∇∂yh∂t)-div⁡(D∇yh)-∫0t‍div⁡(C(τ,t)∇yh(τ))dτ-f-Buh)2}dt+δ∫0T‍|ψ|2,Ω2dt⩽Cξ42+δ‖yh-y(uh)‖L2(0,T;L2(Ω))2,D2⩽C(δ)∫0T‍∑τ‍hl3∫∂τ‍[(A∇∂yh∂t)⋅n+(D∇yh)⋅n-∫0t‍(C(t,τ)∇yh(τ))⋅n]2dldt+δ∫0T‍|ψ|2,Ω2dt⩽Cξ52+δ‖yh-y(uh)‖L2(0,T;L2(Ω))2,D3+D4⩽Cη62+δ‖yh-y(uh)‖L2(0,T;L2(Ω))2.
Letting δ be small enough, then from (4.10)–(4.11), we have
(4.12)‖yh-y(uh)‖L2(0,T;L2(Ω))2⩽C(ξ42+ξ52+η62).
The desired results (4.4) follows from (4.9)–(4.12).This completes the proof.Using Lemmas3.1 and 4.2, we have the following upper bounds.Theorem 4.3.
Let(y,p,u) and (yh,ph,uh) be the solutions of (2.17)–(2.19) and (2.22)–(2.24), respectively. Then, there hold the a posteriori error estimates as
(4.13)‖u-uh‖L2(0,T;L2(ΩU))2+‖y-yh‖L2(0,T;L2(Ω))2+‖p-ph‖L2(0,T;L2(Ω))2⩽C(η12+∑i=25‍ξi2+η62).Proof.
By triangle inequality, (3.44) and (3.38), Lemmas 3.1 and 4.2, we can easily prove (4.13) in the same way of getting (3.43).
This completes the proof.


## 5. Conclusion

In this paper, we study the semi-discrete adaptive finite element method for optimal control problem governed by a linear quasiparabolic Integrodifferential equation. We extend the existing methods in studying adaptive finite element approximation of optimal control governed by a parabolic Integrodifferential equation to the control governed by a quasiparabolic Integrodifferential equation. After presenting the weak form and the existence and uniqueness of the solution for the optimal control problem, the a posteriori error estimates for semi-discrete finite element approximations inL∞(0,T;H1(Ω))-norm and L2(0,T;L2(Ω))-norm are derived. The work will pave a way to derive the a posteriori error estimates of full discrete finite element approximations of this optimal control problem



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