New a posteriori error estimates for hp version of finite element methods of nonlinear parabolic optimal control problems

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R E S E A R C H

Open Access

New

a posteriori

error estimates for

hp

version of ﬁnite element methods of

nonlinear parabolic optimal control problems

Zuliang Lu

1,2,3*

_{, Hongyan Liu}

4

_{, Chunjuan Hou}

5

_{and Longzhou Cao}

1

*_{Correspondence:}

[email protected]

1_{Key Laboratory of Signal and}

Information Processing, Chongqing Three Gorges University, Chongqing, 404000, P.R. China

2_{Key Laboratory for Nonlinear}

Science and System Structure, Chongqing Three Gorges University, Chongqing, 404000, P.R. China

3_{Research Center for Mathematics}

and Economics, Tianjin University of Finance and Economics, Tianjin, 300222, P.R. China

Full list of author information is available at the end of the article

Abstract

In this paper, we investigate residual-baseda posteriorierror estimates for thehp

version of the finite element approximation of nonlinear parabolic optimal control problems. By using thehpfinite element approximation for both the state and the co-state variables and thehpdiscontinuous Galerkin finite element approximation for the control variable, we derivehpresidual-baseda posteriorierror estimates for both the state and the control approximation. Such estimates, which are apparently not available in the literature, can be used to construct a reliablehpadaptive finite element approximation for the nonlinear parabolic optimal control problems.

MSC: 49J20; 65N30

Keywords: residual-baseda posteriorierror estimates; nonlinear parabolic optimal control problems;hpversion of ﬁnite element method

1 Introduction

Recently, optimal control problems have attracted substantial interest due to their appli-cations in atmospheric pollution problems, exploration and extraction of oil and gas re-sources, and engineering. They must be solved with efficient numerical methods. The hpversion of the finite element method is an important numerical method for the opti-mal control problems governed by partial differential equations. There have been exten-sive studies of the convergence of the finite element approximation for optimal control problems. Let us mention two early papers devoted to linear optimal control problems by Falk [] and Geveci []. A systematic introduction of the finite element method for opti-mal control problems can be found in [–], but there are much less published results on this topic forhp-finite element methods for optimal control problems. The adaptive finite element method has been investigated extensively and became one of the most popular methods in the scientific computation and numerical modeling. In [], the authors studied a posteriorierror estimates for adaptive finite element discretizations of boundary control problems.A posteriorierror estimates and adaptive finite element approximation for pa-rameter estimation problems have been obtained in [, ]. The adaptive finite element approximation is among the most important means to boost the accuracy and efficiency of the finite element discretization. There are three main versions in adaptive finite ele-ment approximation,i.e., thep-version,h-version, andhp-version. Thep-version of finite

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element methods uses a fixed mesh and improves the approximation of the solution by increasing the degrees of piecewise polynomials. Theh-version is based on mesh refine-ment and piecewise polynomials of low and fixed degrees. In thehp-version adaptation, one has the option to split an element (h-refinement) or to increase its approximation or-der (p-refinement). Generally speaking, a localp-refinement is the more efficient method on regions where the solution is smooth, while a localh-refinement is the strategy suitable on elements where the solution is not smooth in []. There have been many theoretical studies as regards thehpfinite element method in [–].

To the best of our knowledge, there are manyh-versions of adaptive finite element meth-ods for optimal control problems in [, ]. In fact, comparable literature for high order elements such as thehp-version of the finite element method for optimal control prob-lems is rather limited. For the constrained optimal control problem governed by elliptic equations, the authors have deriveda posteriorierror estimates for thehpfinite element approximation in []. The purpose of this work is to derivehp a posteriorierror estimates for optimal control problems governed by nonlinear parabolic equations.

In this paper, we adopt the standard notationWm,p_{() for Sobolev spaces on}_{with a}

norm · m,p given byvpm,p=

|α|≤mDαvpLp₍₎, a semi-norm| · |m,p given by|v|pm,p=

|α|=mDαvpLp₍₎. We set W_m,p() = {v∈Wm,p() :v|∂ = }. For p= , we denote

Hm_{() =}_Wm,_(),_Hm

() =W m,

 (), and · m= · m,, · = · ,. We denote by

Ls_(,_T_;_Wm,p_{()) the Banach space of all}_Ls_{integrable functions from}_J_into_Wm,p_{() with}

normvLs_(J;Wm,p₍₎₎= (T

 v s

Wm,p₍₎dt)



s for_s∈[,∞), and the standard modiﬁcation for

s=∞. Similarly, one can deﬁne the spacesH(J;Wm,p()) andCk₍_J_;_Wm,p_{()). The details}

can be found in [].

The paper is organized as follows. In Section , we shall construct thehpﬁnite element approximation for nonlinear parabolic optimal control problems. In Section , we derive hp a posteriorierror estimates for the optimal control problems. In the last section, we brieﬂy give conclusions and some possible future work.

2 Thehpﬁnite element of nonlinear parabolic optimal control

In this section, we study thehpﬁnite element method and the backward Euler discretiza-tion approximadiscretiza-tion of convex optimal control problems governed by nonlinear parabolic equations. We shall take the state spaceW=L_(,_T_;_Y_{) with}_Y₌_H

(), the control space

X=L_(,_T_;_U_{) with}_U₌_L₍

U) andH=L() to ﬁx the idea. LetBbe a linear

continu-ous operator fromXtoL_(,_T_;_Y_{). We are interested in the following nonlinear parabolic}

optimal control problems:

min

u∈K

 

T



y–yd_L₍₎+u_L₍

U)

dt

, (.)

yt–div(A∇y) +φ(y) =f+Bu, x∈,t∈(,T], (.)

y(x,t) = , x∈∂,t∈(,T], (.)

y(x, ) =y(x), x∈, (.)

whereandU are bounded open sets inRwith a Lipschitz boundary∂and∂U,

K is a set deﬁned byK={v∈X:_T

Uv dx dt≥}, andf,yd∈L

_(,_T_;_H_),_y

(x)∈V=

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ξt_A_ξ_≥_c_ξ_,_ξ _∈_R_{. The function}_φ(_·₎_∈_W,∞_(–_R_,_R_{) for any}_R_{> ,}_φ₍_y₎_∈_L_{() for}

anyy∈H_{(), and}_φ₍_y₎_≥_.

Let a(v,w) = (A∇v)· ∇w dx, ∀v,w∈ V, (f,f) =

ffdx, ∀f,f ∈ H, (v,w)U =

Uvw dx, ∀v,w∈U. It follows from the assumptions onAthat there are constants c

andC>  such that

a(v,v)≥cv_H₍₎, a(v,w) ≤C|v|_H₍₎|w|_H₍₎, ∀v,w∈Y.

Then a weak formula of the convex nonlinear parabolic optimal control problems (.)-(.) reads

min

u(t)∈K

 

T



y–yd_L₍₎+u_L₍

U)

dt

, (.)

wherey∈W,u∈X,u(t)∈K, subject to

(yt,w) +a(y,w) +

φ(y),w= (f+Bu,w), ∀w∈Y,t∈(,T], (.)

y(x, ) =y(x), x∈. (.)

It is well known (see,e.g., []) that the optimal control problem (.)-(.) has at least a solution (y,u), and if that a pair (y,u) is the solution of (.)-(.), then there is a co-state p∈W such that the triplet (y,p,u) satisﬁes the following optimality conditions:

(yt,w) +a(y,w) +

φ(y),w= (f+Bu,w), ∀w∈Y,y() =y(x), (.)

–(pt,w) +a(q,p) +

φ(y)p,q= (y–yd,q), ∀q∈Y,p(T) = , (.) T



u+B∗p,v–u_Udt≥, ∀v∈K, (.)

whereB∗is the adjoint operators ofB, and (·,·)Uis the inner product ofU, which will be

simply written as (·,·) in the rest of the paper when no confusion is caused.

Due to the special structure of the control constraint setK, we can derive a relation-ship between the control variable and the co-state variable of (.)-(.) in the following lemma.

Lemma . Let(y,p,u)be the solution of(.)-(.).Then we have

u=max,B∗p–B∗p,

where B∗p= T



_UB∗p dx dt

T



_Udx dt

denotes the integral average onU×[,T]of the function B∗p.

Now, we consider thehpﬁnite element approximation for the nonlinear parabolic opti-mal control problems. We assume thatandUare polygonal. We consider the

triangu-lationT of the set⊂R_{, which is a collection of elements}_τ_∈_T _{associated with each}

elementτ, and an aﬃne element mapFτ :τ →τ, where the reference elementτ is the

reference triangle

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We consider the triangulationT which satisﬁes the standard conditions deﬁned in [] and writehτ =diamτ. Additionally we assume that triangulationT isγ-shape regular,

i.e.,

h–τ Fτ_L∞₍τ)+hτF τ

–

L∞(τ)≤γ. (.)

This implies that there exists a constantC>  that depends solely onγ such that

C–hτ≤hτ≤Chτ, τ,τ∈T withτ∩τ=∅, (.)

and there exists a constantM∈Nthat depends solely onγ such that no more thanM elements share a common vertex. We assume that the triangulationTUofUwhich is a

collection of elementsτU∈TU, isγ-shape regular which satisﬁes the standard conditions

asT. Associated with each elementτUis an aﬃne element mapFτU:τ→τU. We further

assume the triangulationT satisﬁes the relation between the patch and the reference patch in [].

For each elementτ∈T, we denoteE(τ) the set of edges ofτandN(τ) the set of vertices ofτ, and choose a polynomial degreepτ∈Nand collect these numbers in the polynomial

degree vector p= (pτ)τ∈T. Similarly, for each elementτU∈TU, we choose a polynomial

degree vector p= (pτU)τU∈TU(pτU∈N).N(T) denotes the set of all vertices ofT,E(T)

denotes the set of all edges. Additionally, we introduce the following notation (V∈N(T), e∈E(T)):

N(e) =V∈N(T) :V∈e,

wV=

x∈:x∈τ andτ∩ {V} =∅,

w_e=

V∈N(e)

wV, wτ =

V∈N(τ)

wV,

hτU=diamτU, pe=max

pτ:e∈E(τ)

,

where χ denotes the interior of the setχ. Finally, we denote byhe the length of the

edgee.

Next, we define thehp-finite element spaceSp₍T₎⊂_H_{() and the}_hp_{-discontinuous} Galerkin finite element spaceUp₍T

U)⊂L(U) by

Sp₍T_{) =}_v∈_C_{() :}_v|

τ◦Fτ∈pτ(τ)

,

Up₍T

U) =

v∈L(U) :v|τU◦FτU∈pτU(τ)

,

where we set

k(τ) = ⎧ ⎨ ⎩

Pk=span{xiyj: ≤i+j≤k}, ifτ =T,

Qk=span{xiyj: ≤i,j≤k}, ifτ =S.

We also assume that the polynomial degree vector psatisﬁes

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Then we can introduce the ﬁnite dimensional spaces Khp=K∩Up(TU), Vhp=V ∩

Sp₍T_).

The semidiscretehpﬁnite element approximation of (.)-(.) is as follows:

min

uhp∈Khp

 

T



yhp–yd_L₍₎+uhp_L₍

U)

dt

, (.)

∂yhp

∂t ,whp

+a(yhp,whp) +

φ(yhp),w

= (f+Buhp,whp), ∀whp∈Vhp, (.)

yhp(x, ) =yhp(x), x∈, (.)

whereyhp∈H(,T;Vhp) andyhp ∈Vhpis an ﬁnite element approximation ofy.

It follows that the optimal control problems (.)-(.) has at least a solution (yhp,uhp)

and if that a pair (yhp,uhp) is the solution of (.)-(.), then there is a co-statephp∈Vhp

such that the triplet (yhp,php,uhp) satisﬁes the following optimality conditions:

∂yhp

∂t ,whp

+a(yhp,whp) +

φ(yhp),w

= (f+Buhp,whp), ∀whp∈Vhp, (.)

yhp(x, ) =yhp(x), x∈,

–

∂php

∂t ,qhp

+a(qhp,php) +

φ(yhp)php,qhp

= (yhp–yd,qhp), ∀whp∈Vhp, (.)

php(x,T) = , x∈,

uhp+B∗php,vhp–uhp

U≥, ∀vhp∈Khp. (.)

Furthermore, we consider the fully discrete ﬁnite element approximation for the above semidiscrete problems by using the backward Euler scheme. Let  =t<t<· · ·<tM–<

tM=T,ki=ti–ti–,i= , , . . . ,M,k=_≤i≤Mmax{ki}.

Fori= , , . . . ,M, construct thehpﬁnite element approximation spacesV_hpi ⊂H_() (similar to Vhp) on theith time step. Similarly, we construct thehpﬁnite element

ap-proximation spaces K_hpi ⊂L₍

U) (similar toKhp) on theith time step. The fully

dis-cretehpﬁnite element approximation scheme (.)-(.) is to ﬁnd (yi

hp,uihp)∈Vhpi ×Khpi ,

i= , , . . . ,M, such that

min

ui_hp∈K_hpi

_M

i=

 y

i

hp–yd(x,ti)  L₍₎+

 u

i hp

 L₍

U)

, (.)

_yi hp–yi–hp

ki

,whp

+ayi_hp,whp

+φyi_hp,whp

=f(x,ti) +Buihp,whp

, ∀whp∈Vhpi , (.)

y_hp(x) =yhp_ (x), x∈. (.)

It follows that the optimal control problem (.)-(.) has at least a solution (Y_hpi ,U_hpi ), and if a pair (Yi

hp,Uhpi )∈Vhpi ×Khpi is the solution of (.)-(.), then there is a co-state

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optimality conditions:

_Yi hp–Yhpi–

ki

,whp

+aY_hpi ,whp

+φY_hpi ,whp

=f(x,ti) +BUhpi ,whp

, (.)

∀whp∈Vhpi ⊂V=H(), i= , , . . . ,M, Yhp(x) =y hp

 (x), x∈, _Pi–

hp –Pihp

ki

,qhp

+aqhp,Pi–hp

+φY_hpi–Pi–_hp,qhp

=Y_hpi –yd(x,ti),qhp

, (.)

∀qhp∈Vhpi ⊂V=H(), i=M, . . . , , , PMhp(x) = , x∈,

U_hpi +B∗Pi–_hp,vhp–Uhpi

U≥, ∀vhp∈K i

hp,i= , , . . . ,M. (.)

Fori= , , . . . ,M, let

Yhp|(ti–,ti]=

(ti–t)Yhpi–+ (t–ti–)Yhpi

/ki,

Php|(ti–,ti]=

(ti–t)Phpi–+ (t–ti–)Pihp

/ki,

Uhp|(ti–,ti]=U

i hp.

For any functionw∈C(,T;L()), letwˆ(x,t)|t∈(ti–,ti]=w(x,ti),w˜(x,t)|t∈(ti–,ti]=w(x,ti–).

Then the optimality conditions (.)-(.) can be restated as

∂Yhp

∂t ,whp

+a(Yˆhp,whp) +

φ(Yˆhp),whp

= (fˆ+BUhp,whp), (.)

∀whp∈Vhpi ⊂V=H(), t∈(ti–,ti],i= , , . . . ,M,

Yhp(x, ) =yhp(x), x∈,

–

∂Php

∂t ,qhp

+a(qhp,P˜hp) +

φ(Y˜hp)P˜hp,qhp

= (Yˆhp–yˆd,qhp), (.)

∀qhp∈Vhpi ⊂V=H(), t∈(ti–,ti],i=M, . . . , , ,

Php(x,T) = , x∈,

Uhp+B∗P˜hp,vhp–Uhp

U≥, (.)

∀vhp∈Khpi , t∈(ti–,ti],i= , , . . . ,M.

The following lemmas [, , ] are important in derivinga posteriorierror estimates of residual type.

Lemma . There exist a constant C> independent of v,hτU,and pτU and a mapping

πhτU

pτ_U :H(τU)→Ppτ_U(τU)such that∀v∈H(τU),τU∈TUthe following inequality is valid: v–πhτU

pτ_U_L_(τ

U)≤C

hτU

pτU

v_H_(τ

U),

where we will write v∈Ppτ_U(τU)if the following satisﬁed:v|τU◦FτU ∈PpτU(τ)ifτU is a

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Lemma . Letp be an arbitrary polynomial degree distribution satisﬁes(.).Then

there exists a linear operator E:H()→Sp(T)∩H(),and there exists a constant

C> depending solely onγ such that for every v∈H

()and all elementsτ∈T and all

edges e∈E(T):

v–EvL_(τ₎+

hτ

pτ

_∇₍_v_–_E__v₎

L_(τ₎≤C

hτ

pτ∇

v_L_(w_τ),

v–EvL_(e)≤C

he

pe 



∇v_L_(w

e).

Lemma . Letp be an arbitrary polynomial degree distribution satisfying(.)and

pτ ≥, ∀τ ∈T. Then there exists a bounded linear operator E : H()∩H()→

Sp₍T₎∩_H

(),and there exists a constant C> that depends solely onγ such that for

every v∈H_()∩H()and all elementsτ ∈T and all edges e∈E(T):

v–EvL_(τ₎+

hτ

pτ

_∇(v–Ev)_L_(τ)≤C

hτ

pτ 

|v|_H_(w_τ),

v–EvL_(e)≤C

he

pe 



|v|_H_(w

e).

Forϕ∈Wh, we shall write

φ(ϕ) –φ(ρ) = –φ˜(ϕ)(ρ–ϕ) = –φ(ρ)(ρ–ϕ) +φ˜(ϕ)(ρ–ϕ), (.)

where

˜

φ(ϕ) =





φϕ+s(ρ–ϕ)ds, φ˜(ϕ) =





( –s)φρ+s(ϕ–ρ)ds

are bounded functions in¯ [].

3 A posteriori error estimates

In this section, we shall derive somea posteriorierror estimates for thehpﬁnite element approximation of the optimal control problems governed by nonlinear parabolic equa-tions.

Let

S(u) =  

y–yd_L₍₎+u_L₍

U)

, (.)

Shp(Uhp) =

 

Yhp–yd_L₍₎+Uhp_L₍

U)

. (.)

It can be shown [] that

S(u),v=u+B∗p,v, (.)

S(Uhp),v

=Uhp+B∗p(Uhp),v

, (.)

S_hp (Uhp),v

=Uhp+B∗P˜hp,v

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It is clear thatSandShpare well deﬁned and continuous onKandKhp. Also the

func-tionalShpcan be naturally extended onK. Then (.) and (.) can be represented as

min

u∈K

T



S(u)dt

(.)

and

min

Uhp∈Khp

T



Shp(Uhp)dt

. (.)

In many application,S(·) is uniform convex near the solutionu. The convexity ofS(·) is closely related to the second order suﬃcient conditions of the optimal control problems, which are assumed in many studies on numerical methods of the problem. For instance, in many applications, there is ac> , independent ofh, such that

T



S(u) –S(Uhp),u–Uhp

Udt≥cu–Uhp 

L_(J;L₍₎₎. (.)

The following theorem is the ﬁrst step to derivea posteriorierror estimates.

Theorem . Let(y,u,p)and(Yhp,Php,Uhp)be the solutions of(.)-(.)and(.)

-(.),respectively.Assume that(S_hp(Uhp))|τ ∈Hs(τ),∀τ∈Th(s= , ),and there is a vhp∈

Khpsuch that

S_hp (Uhp),vhp–u ≤C

τ∈Th

hτS_hp(Uhp)_Hs_(τ₎u–Uhps_L_(τ). (.)

Then we have

u–Uhp_L_(J;L₍₎₎≤Cη+Cp(Uhp) –P˜hp 

L_(J_;L₍₎₎, (.)

where

η_=

T



τ∈Th

h+sτ Uhp+B∗P˜hp +s H_(τ₎dt,

and p(Uhp)is deﬁned by the following equations:

∂

∂ty(Uhp),w

+ay(Uhp),w

+φy(Uhp)

,w= (f+BUhp,w), ∀w∈V, (.)

y(Uhp)(x, ) =y(x), x∈,

–

∂

∂tp(Uhp),q

+aq,p(Uhp)

+φy(Uhp)

p(Uhp),q

=y(Uhp) –yd,q

, ∀q∈V, (.)

(9)

Proof It follows from (.) and (.) that_T(S(u),u–v)dt≤,∀v∈K, and_T(S_hp(Uhp),

Uhp–vhp)dt≤,∀vhp∈Khp⊂K. Then it follows from the assumptions (.), (.), and

the Schwartz inequality that

cu–Uhp_L_(J;L₍₎₎≤

T



S(u) –S(Uhp),u–Uhp

dt

≤ T



S_hp (Uhp),vhp–u

+S_hp (Uhp) –S(Uhp),u–Uhp

dt

≤C

T



τ∈Th

h+s_τ S_hp(Uhp) +s

Hs_(τ)+Shp(Uhp) –S(Uhp)  L₍₎

dt

+c

u–Uhp



L_(J;L₍₎₎. (.)

It is not diﬃcult to show

S_hp(Uhp) =Uhp+B∗P˜hp, S(Uhp) =Uhp+B∗p(Uhp), (.)

wherep(Uhp) is deﬁned in (.)-(.). Thanks to (.), it is easy to derive

S_hp (Uhp) –S(Uhp)_L₍₎=p(Uhp) –P˜hp_L₍₎. (.)

Then by using the estimates (.) and (.) we can prove the requested result (.).

Next, we are in the position to estimate the errors Yhp – y(Uhp)_L_(,T;H₍₎₎ and

Php–p(Uhp)_L_(,T;H₍₎₎.

Theorem . Let(Yhp,Php,Uhp)be the solutions of(.)-(.),let (y(Uhp),p(Uhp))be

deﬁned by(.)-(.).Then we have

Yhp–y(Uhp) 

L_(,T;H₍₎₎+Php–p(Uhp) 

L_(,T;H₍₎₎≤C



i=

η_i, (.)

where

η_=

T



τ

h τ

p τ

τ

ˆ

f +BUhp+div(A∇ ˆYhp) –φ(Yˆhp) –

∂Yhp

∂t



dx dt,

η_=

T



τ

h τ

p τ

τ

ˆ

Yhp–yˆd+div

A∗∇ ˜Php

–φ(Y˜hp)P˜hp+

∂Php

∂t



dx dt,

η_=

T



e

he

pe

(A∇ ˆYhp)·n 

de dt+

T



e

he

pe

A∗∇ ˜Php

·nde dt,

η_=

T



A∇(Yˆhp–Yhp) dx dt+ T



A∗∇(P˜hp–Php) dx dt,

η_=Yhp–Y˜hp_L_(,T;L₍₎₎+Yhp–Yˆhp_L_(,T;L₍₎₎+yd–yˆd_L_(,T;L₍₎₎

(10)

where e=τ¯

e ∩ ¯τe,τe,τeare two neighboring elements inT, [A∇ ˆYhp·n]eand[A∗∇ ˜Php·n]e

are the A-normal and A∗-normal derivative jumps over the interior edge e,respectively, deﬁned by

(A∇ ˆYhp)·n

e= (A∇ ˆYhp|τe–A∇ ˆYhp|τe)·n,

A∗∇ ˜Php

·n_e=A∗∇ ˜Php|τe–A

∗_{∇ ˜}_P hp|τe

·n,

where n is the unit normal vector on e=τ¯_e∩ ¯τ_e outwardsτ_e.For later convenience,we deﬁne[A∇ ˆYhp·n]e= and[A∗∇ ˜Php·n]e= when e⊂∂.

Proof Letrhp=p(Uhp) –PhpandEbe the linear operator deﬁned in Lemma .. Note that

(p(Uhp) –Php)(x,T) = , hence

T



–

∂(p(Uhp) –Php)

∂t ,rhp

dt≥. (.)

By using the assumption ofφ, (.), and (.), we obtain

crhp_L_(,T;H₍₎₎

≤ T



arhp,p(Uhp) –Php

dt+

T



φy(Uhp)

p(Uhp) –Php

,p(Uhp) –Php dt ≤ T 

∇rhp,A∗∇

p(Uhp) –Php

dt–

T



∂(p(Uhp) –Php)

∂t ,rhp

dt + T 

φy(Uhp)

p(Uhp) –P˜hp

,p(Uhp) –Php dt + T 

φy(Uhp)

(P˜hp–Php),p(Uhp) –Php dt = T 

∇rhp,A∗∇

p(Uhp) –P˜hp

dt–

T



∂(p(Uhp) –Php)

∂t ,rhp

dt + T 

φy(Uhp)

p(Uhp) –φ(Y˜hp)P˜hp,p(Uhp) –Php dt + T  _˜

φ(Y˜hp) _˜

Yhp–y(Uhp) _˜

Php,p(Uhp) –Php dt + T 

φy(Uhp)

(P˜hp–Php),p(Uhp) –Php dt + T 

∇rhp,A∗∇(P˜hp–Php)

dt. (.)

Connecting (.), (.), and (.), we have

crhp_L_(,T;H₍₎₎

≤ T



∇(rhp–Erhp),A∗∇

p(Uhp) –P˜hp dt – T 

∂(p(Uhp) –Php)

∂t ,rhp–Erhp

(11)

+

T



φy(Uhp)

p(Uhp) –φ(Y˜hp)P˜hp,rhp–Erhp dt – T 

∂(p(Uhp) –Php)

∂t ,Erhp

dt+

T



∇Erhp,A∗∇

p(Uhp) –P˜hp dt + T 

φy(Uhp)

p(Uhp) –φ(Y˜hp)P˜hp,Erhp dt + T  _˜

φ(Y˜hp) _˜

φy(Uhp)

(P˜hp–Php),p(Uhp) –Php

dt+

T



∇rhp,A∗∇(P˜hp–Php) dt = T 

y(Uhp) –yd+div

A∗∇ ˜Php

–φ(Y˜hp)P˜hp+

∂Php

dt + T  e e

A∗∇ ˜Php

·n(rhp–Erhp)de dt

+

T

 _˜

φ(Y˜hp) _˜

y(Uhp) –Yˆhp,Erhp

dt+

T



(yˆd–yd,Erhp)dt

+

T



φy(Uhp)

(P˜hp–Php),p(Uhp) –Php dt + T 

dt. (.)

Then we have

cp(Uhp) –Php_L_(,T;H₍₎₎

≤ T



ˆ

Yhp–yˆd+div

A∗∇ ˜Php

–φ(Y˜hp)P˜hp+

∂Php

dt + T  e e

A∗∇ ˜Php

+

T



y(Uhp) –Yˆhp,rhp

dt+

T



(yˆd–yd,rhp)dt

+

T



∇rhp,A∗∇(P˜hp–Php) dt + T  _˜

φ(Y˜hp) _˜

φy(Uhp)

(P˜hp–Php),p(Uhp) –Php dt ≡  i=

(12)

It follows from Lemma . that

I= T



ˆ

Yhp–yˆd+div

A∗∇ ˜Php

–φ(Y˜hp)P˜hp+

∂Php

dt

≤C(δ)

T



τ

h τ

p τ

τ

ˆ

Yhp–yˆd+div

A∗∇ ˜Php

–φ(Y˜hp)P˜hp+

∂Php

∂t



dx dt

+δ

T



rhp_H₍₎dt

=C(δ)η_+δp(Uhp) –Php 

L_(,T;H₍₎₎. (.)

Similarly,

I= T



e

A∗∇ ˜Php

≤C(δ)

T



e

he

pe

A∗∇ ˜Php

·nde dt+δ

T



rhp_H₍₎dt

≤C(δ)η_+δp(Uhp) –Php_L_(,T;H₍₎₎. (.)

ForI-I, we have

I= T



y(Uhp) –Yˆhp,rhp

dt

≤C(δ)

T



y(Uhp) –Yˆhp 

dx dt+δp(Uhp) –Php 

L_(,T;H₍₎₎

≤C(δ)y(Uhp) –Yhp 

L_(,T;H₍₎₎+C(δ)Yhp–Yˆhp_L_(,T;L₍₎₎

+δp(Uhp) –Php 

L_(,T;H₍₎₎

≤C(δ)η_+C(δ)y(Uhp) –Yhp 

L_(,T;H₍₎₎+δp(Uhp) –Php 

L_(,T;H₍₎₎, (.)

I= T



(ˆyd–yd,rhp)dt

≤C(δ)yd–yˆd_L_(,T;L₍₎₎+δp(Uhp) –Php_L_(,T;H₍₎₎

≤C(δ)η_+δp(Uhp) –Php_L_(,T;H₍₎₎, (.)

and

I= T



dt

≤C(δ)

T



_A∗_∇₍_P_˜

hp–Php) 

dx dt+δ

T



|∇rhp|dx dt

≤C(δ)η_+δp(Uhp) –Php 

(13)

Similarly, we can obtain

I= T

 _˜

φ(Y˜hp) _˜

Php,p(Uhp) –Php

dt

≤C(δ)

T



y(Uhp) –Y˜hp 

L_(,T;H₍₎₎

≤C(δ)y(Uhp) –Yhp 

L_(,T;H₍₎₎+C(δ)Yhp–Y˜hp_L_(,T;L₍₎₎

+δp(Uhp) –Php 

L_(,T;H₍₎₎

≤C(δ)η_+C(δ)y(Uhp) –Yhp 

L_(,T;H₍₎₎+δp(Uhp) –Php 

L_(,T;H₍₎₎ (.)

and

I= T



φy(Uhp)

(P˜hp–Php),p(Uhp) –Php

dt

≤C(δ)˜Php–Php_L_(,T;L₍₎₎+δp(Uhp) –Php 

L_(,T;H₍₎₎

≤C(δ)

T



A∗∇(P˜hp–Php) 

L_(,T;H₍₎₎

≤C(δ)η_+δp(Uhp) –Php 

L_(,T;H₍₎₎. (.)

Then letδbe small enough, from (.)-(.), we derive

p(Uhp) –Php 

L_(,T;H₍₎₎≤C(δ)



i=

η_i +C(δ)y(Uhp) –Yhp 

L_(,T;L₍₎₎. (.)

Furthermore, we estimate the errory(Uhp) –Yhp_L_(,T;L₍₎₎. Letehp=y(Uhp) –Yhpand

Ebe the linear operator deﬁned in Lemma .. Note that

T



∂(y(Uhp) –Yhp)

∂t ,ehp

dt

=

T



∂(y(Uhp) –Yhp)

∂t ehpdx dt

=

y(Uhp) –Yhp

(x,T)dx–

y(Uhp) –Yhp

(x, )dx

–

T



∂(y(Uhp) –Yhp)

∂t ,ehp

dt.

Thus

T



∂(y(Uhp) –Yhp)

∂t ,ehp

dt+

y(x) –Yhp(x, )



L₍₎≥. (.)

From (.), it is easy to see that

φy(Uhp)

–φ(Yhp),ehp

=φ˜y(Uhp)

y(Uhp) –Yhp

,ehp

(14)

By using (.) and (.), we have

cehp_L_(,T;H₍₎₎

≤ T 

ay(Uhp) –Yhp,ehp

dt+

T



φy(Uhp)

–φ(Yhp),ehp

dt

≤ T



ay(Uhp) –Yhp,ehp

dt+

T



φy(Uhp)

–φ(Yˆhp),ehp dt + T 

∂(y(Uhp) –Yhp)

∂t ,ehp

dt+

T



φ(Yˆhp) –φ(Yhp),ehp

dt

+

y(x) –Yhp(x, )

 L₍₎

=

T



A∇y(Uhp) –Yˆhp

,∇ehp

dt+

T



φy(Uhp)

–φ(Yˆhp),ehp dt + T 

∂(y(Uhp) –Yhp)

∂t ,ehp

dt+

T



A∇(Yˆhp–Yhp),∇ehp dt + T 

φ(Yˆhp) –φ(Yhp),ehp

dt

+

y(x) –Yhp(x, )



L₍₎. (.)

Connecting (.), (.), (.), and (.), we obtain

cehp_L_(,T;H₍₎₎

≤ T



ˆ

f+BUhp+div(A∇ ˆYhp) –φ(Yˆhp) –

∂Yhp

∂t ,ehp–Eehp

dt + T  e e

(A∇ ˆYhp)·n

(ehp–Eehp)de dt+ T



(f–fˆ,ehp)dt

+

T



A∇(Yˆhp–Yhp),∇ehp

dt+

y(x) –Yhp(x, )

 L₍₎

+

T

 _˜

φ(Yˆhp)(Yˆhp–Yhp),ehp

dt

≤C(δ)

T  τ h τ p τ τ ˆ

f +BUhp+div(A∇ ˆYhp) –φ(Yˆhp) –

∂Yhp

∂t



dx dt

+C(δ)

T  e e he pe

(A∇ ˆYhp)·n 

de dt+C(δ)f–fˆ_L_(,T;L₍₎₎

+C(δ)

T



A∇(Yˆhp–Yhp) 

dx dt+C(δ) ˆYhp–Yhp_L_(,T;H₍₎₎

+

y(x) –Yhp(x, )



L₍₎+δy(Uhp) –Yhp 

L_(,T;H₍₎₎

≤C(δ)



i=

η_i+δy(Uhp) –Yhp 

(15)

Hence, letδbe small enough, we have

y(Uhp) –Yhp_L_(,T;H₍₎₎≤C(δ)



i=

η_i. (.)

Then (.) follows from (.) and (.).

Finally, collecting Theorems .-., we derive the following residual-baseda posteriori error estimates.

Theorem . Let(y,p,u)and(Yhp,Php,Uhp)are the solutions of(.)-(.)and(.)

-(.)respectively.Assume that all the conditions in Theorem.are valid.Then

y–Yhp_L_(,T;H₍₎₎+p–Php_L_(,T;H₍₎₎+u–Uhp_L_(,T;L₍

U))

≤C



i=

η_i, (.)

whereη_i,i= , . . . , are deﬁned in Theorem.and Theorem..

Proof It follows from Theorem . and Theorem . that

u–Uhp_L_(,T;L₍

U))≤Cη



+CP˜hp–p(Uhp) 

L_(,T;L₍₎₎

≤Cη_+C˜Php–Php_L_(,T;L₍₎₎

+CPhp–p(Uhp) 

L_(,T;L₍₎₎

≤C



i=

η_i +C˜Php–Php_L_(,T;L₍₎₎. (.)

Note thatAis positive deﬁnite, it follows from the Poincaré inequality that

˜Php–Php_L_(,T;L₍₎₎≤

T



A∗∇(P˜hp–Php) dx dt

≤Cη_. (.)

Then it follows from (.) and (.) that

u–Uhp_L_(,T;L₍

U))≤C



i=

η_i. (.)

Note that

y–Yhp_L_(,T;H₍₎₎≤y–y(Uhp) 

L_(,T;H₍₎₎+y(Uhp) –Yhp 

L_(,T;H₍₎₎, (.)

p–Php_L_(,T;H₍₎₎≤p–p(Uhp) 

L_(,T;H₍₎₎+p(Uhp) –Php 

(16)

and

_y_–_y₍_U_hp₎

L_(,T;H₍₎₎≤Cu–Uhp_L_(,T;L₍

U)), (.)

p–p(Uhp) 

L_(,T;H₍₎₎≤y–y(Uhp) 

L_(,T;L₍₎₎≤Cu–Uhp_L_(,T;L₍

U)). (.)

Therefore, we obtain (.) from (.) and (.)-(.).

4 Conclusion and future work

In this paper, we present thehpversion of the finite element approximation for the opti-mal control problems governed by nonlinear parabolic equations. By using thehpfinite element approximation for both the state and the co-state variables and thehp discontinu-ous Galerkin finite element approximation for the control variable, we derivehp residual-baseda posteriorierror estimates for the nonlinear parabolic optimal control problems. To the best of our knowledge in the context of optimal control problems, these residual-baseda posteriorierror estimates for the nonlinear parabolic optimal control problems are new.

In future, we shall consider thehpversion of the ﬁnite element method for hyperbolic optimal control problems. Furthermore, we shall considera posteriorierror estimates and the superconvergence of the hpﬁnite element solutions for hyperbolic optimal control problems.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

ZL and LC have participated in the sequence alignment and drafted the manuscript. HL and CH have made substantial contributions to the conception and design. All authors read and approved the ﬁnal manuscript.

Author details

1_{Key Laboratory of Signal and Information Processing, Chongqing Three Gorges University, Chongqing, 404000, P.R.}

China.2_{Key Laboratory for Nonlinear Science and System Structure, Chongqing Three Gorges University, Chongqing,}

404000, P.R. China.3_{Research Center for Mathematics and Economics, Tianjin University of Finance and Economics,}

Tianjin, 300222, P.R. China.4_{Chongqing Wanzhou Long Bao Middle School, Chongqing, 404001, P.R. China.}5_Huashang

College, Guangdong University of Finance, Guangzhou, 511300, P.R. China.

Acknowledgements

The authors express their thanks to the referees for their helpful suggestions, which led to improvements of the presentation. This work is supported by National Science Foundation of China (11201510), China Postdoctoral Science Foundation (2015M580197), Chongqing Research Program of Basic Research and Frontier Technology

(cstc2015jcyjA20001), Ministry of education Chunhui projects (Z2015139) and Science and Technology Project of Wanzhou District of Chongqing (2013030050).

Received: 17 June 2015 Accepted: 1 February 2016 References

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