Skip to content

Advertisement

  • Research
  • Open Access

A posteriori error estimates of fully discrete finite-element schemes for nonlinear parabolic integro-differential optimal control problems

Advances in Difference Equations20142014:15

https://doi.org/10.1186/1687-1847-2014-15

Received: 12 August 2013

Accepted: 11 December 2013

Published: 15 January 2014

Abstract

The aim of this work is to study the optimality conditions and the adaptive multi-mesh fully discrete finite-element schemes for quadratic nonlinear parabolic integro-differential optimal control problems. We derive a posteriori error estimates in L 2 ( J ; H 1 ( Ω ) ) -norm and L 2 ( J ; L 2 ( Ω ) ) -norm for both the coupled state and control approximation. Such estimates can be used to construct reliable adaptive finite-element approximation for nonlinear parabolic integro-differential optimal control problems.

MSC:49J20, 65N30.

Keywords

nonlinear parabolic integro-differential optimal control problemsadaptive multi-mesh finite-element methodsa posteriori error estimatesfully discrete

1 Introduction

Parabolic integro-differential optimal control problems are very important for modeling in science. They have various physical backgrounds in many practical applications such as population dynamics, visco-elasticity, and heat conduction in materials with memory. The finite-element approximation of parabolic integro-differential optimal control problems plays a very important role in the numerical methods for these problems. The finite-element approximation of an optimal control problem by piecewise constant functions has been investigated by Falk [1] and Geveci [2]. The discretization for semilinear elliptic optimal control problems is discussed by Arada et al. in [3]. In [4], Brunner and Yan analyzed the finite-element Galerkin discretization for a class of optimal control problems governed by integral equations and integro-differential equations. Systematic introductions of the finite-element method for optimal control problems can be found in [510].

The adaptive finite-element approximation is the most important method to boost the accuracy of the finite-element discretization. It ensures a higher density of nodes in a certain area of the given domain, where the solution is discontinuous or more difficult to approximate, using an a posteriori error indicator. A posteriori error estimates are computable quantities in terms of the discrete solution and measure the actual discrete errors without the knowledge of exact solutions. They are essential in designing algorithms for a mesh which equidistribute the computational effort and optimize the computation. The literature for this is huge. Some techniques directly relevant to our work can be found in [11, 12]. Recently, in [1316], we derived a priori error estimates and superconvergence for linear quadratic optimal control problems using mixed finite-element methods. A posteriori error estimates of mixed finite-element methods for general semilinear optimal control problems were addressed in [17].

In this paper, we adopt the standard notation W m , p ( Ω ) for Sobolev spaces on Ω with a norm m , p given by v m , p p = | α | m D α v L p ( Ω ) p , a semi-norm | | m , p given by | v | m , p p = | α | = m D α v L p ( Ω ) p . We set W 0 m , p ( Ω ) = { v W m , p ( Ω ) : v | Ω = 0 } . For p = 2 , we denote H m ( Ω ) = W m , 2 ( Ω ) , H 0 m ( Ω ) = W 0 m , 2 ( Ω ) , and m = m , 2 , = 0 , 2 . We denote by L s ( 0 , T ; W m , p ( Ω ) ) the Banach space of all L s integrable functions from J into W m , p ( Ω ) with norm v L s ( J ; W m , p ( Ω ) ) = ( 0 T v W m , p ( Ω ) s d t ) 1 s for s [ 1 , ) , and the standard modification for s = . The details can be found in [18].

The problems that we are interested in are the following nonlinear parabolic integro-differential optimal control problems:
min u ( t ) K { 0 T ( 1 2 y y 0 2 + α 2 u 2 ) d t }
(1.1)
subject to the state equations
y t div ( A y ( x , t ) ) 0 t div ( ψ ( t , τ ) y ( x , τ ) ) d τ + ϕ ( y ) = f + B u , x Ω , t J ,
(1.2)
y ( x , t ) = 0 , x Ω , t J ,
(1.3)
y ( x , 0 ) = y 0 ( x ) , x Ω ,
(1.4)
where the bounded open set Ω R 2 is a 2 regular convex polygon with boundary Ω, J = ( 0 , T ] , f L 2 ( Ω ) , ψ = ψ ( x , t , τ ) = ψ i , j ( x , t , τ ) 2 × 2 C ( 0 , T ; L 2 ( Ω ¯ ) ) 2 × 2 , y 0 H 1 ( Ω ) , α is a positive constant, and B is a continuous linear operator from K to L 2 ( Ω ) . For any R > 0 the function ϕ ( ) W 2 , ( R , R ) , ϕ ( y ) L 2 ( Ω ) for any y L 2 ( J ; H 0 1 ( Ω ) ) , and ϕ ( y ) 0 . We assume the coefficient matrix A ( x ) = ( a i , j ( x ) ) 2 × 2 ( W 1 , ( Ω ) ) 2 × 2 is a symmetric positive definite matrix and there is a constant c > 0 satisfying for any vector X R 2 , X t A X c X R 2 2 . Here, K denotes the admissible set of the control variable, defined by
K = { u ( x , t ) L 2 ( J ; L 2 ( Ω ) ) : Ω u ( x , t ) d x 0 , t J } .
(1.5)

The plan of this paper is as follows. In the next section, we construct the optimality conditions and present the finite-element discretization for nonlinear parabolic integro-differential optimal control problems. A posteriori error estimates of finite-element solutions for those problems are established in Section 3. Finally, we analyze the conclusion and future work in Section 4.

2 Finite elements for integro-differential optimal control

We shall now construct the optimality conditions and the finite element discretization of the nonlinear parabolic integro-differential optimal control problem (1.1)-(1.4). Let V = H 0 1 ( Ω ) , W = L 2 ( Ω ) . Let
a ( y , w ) = Ω ( A y ) w , y , w V ,
(2.1)
ψ ( t , τ ; z , w ) = ( ψ ( t , τ ) z , w ) , z , w V ,
(2.2)
( u , v ) = Ω u v , u , v W ,
(2.3)
( f 1 , f 2 ) = Ω f 1 f 2 , f 1 , f 2 W .
(2.4)
Then the nonlinear parabolic integro-differential optimal control problem (1.1)-(1.4) can be restated as
min u ( t ) K { 0 T ( 1 2 y y 0 2 + α 2 u 2 ) d t }
(2.5)
subject to
( y t , w ) + a ( y , w ) + 0 t ψ ( t , τ ; y ( τ ) , w ) d τ + ( ϕ ( y ) , w ) = ( f + B u , w ) , w V , t J ,
(2.6)
y ( x , 0 ) = y 0 ( x ) , x Ω ,
(2.7)

where the inner product in L 2 ( Ω ) or L 2 ( Ω ) 2 is indicated by ( , ) .

It is well known (see, e.g., [4]) that the optimal control problem has a solution ( y , u ) , and that if a pair ( y , u ) is the solution of equations (2.5)-(2.7), then there is a co-state p V such that the triplet ( y , p , u ) satisfies the following optimality conditions:
( y t , w ) + a ( y , w ) + 0 t ψ ( t , τ ; y ( τ ) , w ) d τ + ( ϕ ( y ) , w ) = ( f + B u , w ) , w V ,
(2.8)
y ( x , 0 ) = y 0 ( x ) , x Ω ,
(2.9)
( p t , w ) + a ( q , p ) + t T ψ ( τ , t ; q , p ( τ ) ) d τ + ( ϕ ( y ) p , q ) = ( y y 0 , q ) , q V ,
(2.10)
p ( x , T ) = 0 , x Ω ,
(2.11)
0 T ( α u + B p , v u ) d t 0 , v K ,
(2.12)

where B is the adjoint operator of B.

Let us consider the finite-element approximation of the optimal control problem (2.5)-(2.7). Again here we consider only n-simplex elements and conforming finite elements.

For ease of exposition we will assume that Ω is a polygon. Let T h be regular partition of Ω. Associated with T h is a finite-dimensional subspace V h of C ( Ω ¯ ) , such that χ | τ are polynomials of order m ( m 1 ) χ V h and τ T h . It is easy to see that V h V . Let h τ denote the maximum diameter of the element τ in T h , h = max τ T h { h τ } . In addition C or c denotes a general positive constant independent of h.

Due to the limited regularity of the optimal control u in general, there will be no advantage in considering higher-order finite element spaces rather than the piecewise constant space for the control. So, we only consider piecewise constant finite elements for the approximation of the control, though higher-order finite elements will be used to approximate the state and the co-state.

Let P 0 ( τ ) denote the piecewise constant space over τ. Then we take K h = { u K : u ( x , t ) | τ P 0 ( τ ) } . By the definition of the finite-element subspace, the finite-element discretization of equations (2.5)-(2.7) is as follows: compute ( y h , u h ) V h × K h such that
min u h K h { 0 T ( 1 2 y h y 0 2 + α 2 u h 2 ) d t } ,
(2.13)
( y h t , w h ) + a ( y h , w h ) + 0 t ψ ( t , τ ; y h ( τ ) , w h ) d τ + ( ϕ ( y h ) , w h ) = ( f + B u h , w h ) ,
(2.14)
y h ( x , 0 ) = y 0 h ( x ) , x Ω ,
(2.15)

where w h V h , y 0 h V h is an approximation of y 0 .

Again, it follows that the optimal control problem (2.13)-(2.15) has a solution ( y h , u h ) , and that if a pair ( y h , u h ) is the solution of equations (2.13)-(2.15), then there is a co-state p h V h such that triplet ( y h , p h , u h ) satisfies the following optimality conditions:
( y h t , w h ) + a ( y h , w h ) + 0 t ψ ( t , τ ; y h ( τ ) , w h ) d τ + ( ϕ ( y h ) , w h ) = ( f + B u h , w h ) ,
(2.16)
y h ( x , 0 ) = y 0 h ( x ) , x Ω ,
(2.17)
( p h t , w h ) + a ( q h , p h ) + t T ψ ( τ , t ; q h , p h ( τ ) ) d τ + ( ϕ ( y h ) p h , q h ) = ( y h y 0 , q h ) ,
(2.18)
p h ( x , T ) = 0 , x Ω ,
(2.19)
( α u h + B p h , v h u h ) U 0 ,
(2.20)

where w h , q h V h , v h K h .

We now consider the fully discrete finite-element approximation for the semidiscrete problem. Let t > 0 , N = T / t Z , and t i = i t , i R . Also, let
ξ i = ξ i ( x ) = ξ ( x , t i ) , d t ξ i = ξ i ξ i 1 t .
For i = 1 , 2 , , N , construct the finite-element spaces V i h V with the mesh T h i (similar to  V h ). Similarly, construct the finite-element spaces K i h L 2 ( Ω ) with the mesh T h i (similar as I h ). Let h τ i denote the maximum diameter of the element τ i in T h i . Define mesh functions τ ( ) and mesh size functions h τ ( ) such that τ ( t ) | t ( t i 1 , t i ] = τ i , h τ ( t ) | t ( t i 1 , t i ] = h τ i . For ease of exposition, we shall denote τ ( t ) and h τ ( t ) by τ and h τ , respectively. Then the fully discrete finite-element approximation of equations (2.13)-(2.15) is as follows: compute ( y h i , u h i ) V i h × K i h , i = 1 , 2 , , N , such that
min u h i K i h { i = 1 N Δ t ( 1 2 y h i y 0 2 + α 2 u h i 2 ) } ,
(2.21)
( d t y h i , w h ) + a ( y h i , w h ) + 0 t ψ ( t , τ ; y h i ( τ ) , w h ) d τ + ( ϕ ( y h i ) , w h ) = ( f ( x , t i ) + B u h i , w h ) ,
(2.22)
w h V i h , i = 1 , 2 , , N , y h 0 ( x ) = y 0 h ( x ) , x Ω ,
(2.23)

where y 0 h V h is an approximation of y 0 .

Now, it follows that the optimal control problem (2.21)-(2.23) has a solution ( Y h i , U h i ) , i = 1 , 2 , , N , and that if a pair ( Y h i , U h i ) , i = 1 , 2 , , N , is the solution of (2.21)-(2.23), then there is a co-state P h i 1 V i h , i = N , , 2 , 1 , such that triplet ( Y h i , P h i 1 , U h i ) satisfies the following optimality conditions:
( d t Y h i , w h ) + a ( Y h i , w h ) + 0 t ψ ( t , τ ; Y h i ( τ ) , w h ) d τ + ( ϕ ( Y h i ) , w h ) = ( f + B U h i , w h ) ,
(2.24)
w h V i h , i = 1 , 2 , , N , Y h 0 ( x ) = y 0 h ( x ) , x Ω ,
(2.25)
( d t P h i , q h ) + a ( q h , P h i 1 ) + t T ψ ( τ , t ; q h , P h i 1 ( τ ) ) d τ + ( ϕ ( Y h i 1 ) P h i 1 , q h )
(2.26)
= ( Y h i y 0 , q h ) , q h V i h , i = N , , 2 , 1 , P h N ( x ) = 0 , x Ω ,
(2.27)
( α U h i + B P h i , v h U h i ) 0 , v h K i h , i = 1 , 2 , , N .
(2.28)
For i = 1 , 2 , , N , let
Y h | ( t i 1 , t i ] = ( ( t i t ) Y h i 1 + ( t t i 1 ) Y h i ) / Δ t ,
(2.29)
P h | ( t i 1 , t i ] = ( ( t i t ) P h i 1 + ( t t i 1 ) P h i ) / Δ t ,
(2.30)
U h | ( t i 1 , t i ] = U h i .
(2.31)
For any function w C ( 0 , T ; L 2 ( Ω ) ) , let w ˆ ( x , t ) | t ( t i 1 , t i ] = w ( x , t i ) , w ˜ ( x , t ) | t ( t i 1 , t i ] = w ( x , t i 1 ) . Then the optimality conditions (2.24)-(2.28) can be restated as
( Y h t , w h ) + a ( Y ˆ h , w h ) + 0 t ψ ( t , τ ; Y ˆ h ( τ ) , w h ) d τ + ( ϕ ( Y ˆ h ) , w h ) = ( f ˆ + B U h , w h ) ,
(2.32)
w h V i h , i = 1 , 2 , , N , Y h 0 ( x ) = y 0 h ( x ) , x Ω ,
(2.33)
( P h t , q h ) + a ( q h , P ˜ h ) + t T ψ ( τ , t ; q h , P ˜ h ( τ ) ) d τ + ( ϕ ( Y ˜ h ) P ˜ h , q h ) = ( Y ˆ h y 0 , q h ) ,
(2.34)
q h V i h , i = N , , 2 , 1 , P h ( x , T ) = 0 , x Ω ,
(2.35)
( α U h + B P ˜ h , v h U h ) 0 , v h K i h , i = 1 , 2 , , N .
(2.36)
In the rest of the paper, we shall use some intermediate variables. For any control function U h K , we first define the state solution ( y ( U h ) , p ( U h ) ) which satisfies
( y t ( U h ) , w ) + a ( y ( U h ) , w ) + 0 t ψ ( t , τ ; y ( U h ) ( τ ) , w ) d τ + ( ϕ ( y ( U h ) ) , w ) = ( f + B U h , w ) ,
(2.37)
w V , y ( U h ) ( x , 0 ) = y 0 ( x ) , x Ω ,
(2.38)
( p t ( U h ) , q ) + a ( q , p ( U h ) ) + t T ψ ( τ , t ; q , p ( U h ) ( τ ) ) d τ + ( ϕ ( y ( U h ) ) p ( U h ) , q )
(2.39)
= ( y ( U h ) y 0 , q ) , q V , p ( U h ) ( x , T ) = 0 , x Ω .
(2.40)

Now we restate the following well-known estimates in [18].

Lemma 2.1 Let π ˆ h be the Clément type interpolation operator defined in [18]. Then for any v H 1 ( Ω ) and all elements τ,
v π ˆ h v L 2 ( τ ) + h τ ( v π ˆ h v ) L 2 ( τ ) C h τ τ ¯ τ ¯ | v | L 2 ( τ ) ,
(2.41)
v π ˆ h v L 2 ( l ) C h l 1 / 2 l τ ¯ | v | L 2 ( τ ) ,
(2.42)

where l is the edge of the element.

3 A posteriori error estimates

In this section we will obtain a posteriori error estimates in L 2 ( J ; H 1 ( Ω ) ) and L 2 ( J ; L 2 ( Ω ) ) for the coupled state and control approximation. Firstly, we estimate the error y ( U h ) Y ˆ h L 2 ( J ; H 1 ( Ω ) ) .

Theorem 3.1 Let ( y ( U h ) , p ( U h ) ) and ( Y h , P h ) be the solutions of equations (2.37)-(2.39) and equations (2.32)-(2.34), respectively. Then
y ( U h ) Y ˆ h L 2 ( J ; H 1 ( Ω ) ) 2 C i = 1 6 η i 2 ,
(3.1)
where
η 1 2 = 0 T τ T h h τ 2 τ ( f ˆ + B U h Y h t + div ( A Y ˆ h ) η 1 2 = + 0 t div ( ψ ( t , τ ) Y ˆ h ( τ ) ) d τ ϕ ( Y ˆ h ) ) 2 d t , η 2 2 = 0 T l Ω = h l l [ A Y ˆ h n + 0 t ( ( ψ ( t , τ ) Y ˆ h ( τ ) ) n ) d τ ] 2 d t , η 3 2 = 0 T l Ω h l l [ A Y ˆ h n + 0 t ( ( ψ ( t , τ ) Y ˆ h ( τ ) ) n ) d τ ] 2 d t , η 4 2 = f f ˆ L 2 ( J ; L 2 ( Ω ) ) 2 , η 5 2 = Y h Y ˆ h L 2 ( J ; H 1 ( Ω ) ) 2 , η 6 2 = Y h ( x , 0 ) y 0 ( x ) L 2 ( Ω ) 2 ,
where l is a face of an element τ, h l is the size of face l, [ A y h n ] is the A-normal derivative jump over the interior face l, defined by
[ A Y h n ] l = ( A Y h | τ l 1 A Y h | τ l 2 ) n ,

where n is the unit normal vector on l = τ ¯ l 1 τ ¯ l 2 outwards of τ l 1 .

Proof Let e y = y ( U h ) Y h , and let e I y be the Clément type interpolator of e y defined in Lemma 2.1. Note that
0 T ( y t ( U h ) Y h t , e y ) d t = 0 T Ω ( y t ( U h ) Y h t ) e y d x d t = 1 2 Ω ( ( y ( U h ) Y h ) ( x , T ) ) 2 d x 1 2 Y h ( x , 0 ) y 0 ( x ) L 2 ( Ω ) 2 .
(3.2)
From equation (3.2), we have
0 T ( y t ( U h ) Y h t , e y ) d t + 1 2 Y h ( x , 0 ) y 0 ( x ) L 2 ( Ω ) 2 0 .
(3.3)
By using the assumptions of A and ϕ, thus we can obtain the following result:
c e y L 2 ( J ; H 1 ( Ω ) ) 2 0 T ( A ( y ( U h ) Y h ) , e y ) d t + 0 T ( ϕ ( y ( U h ) ) ϕ ( Y h ) , e y ) d t = 0 T ( A ( y ( U h ) Y h ) , ( e y e I y ) ) d t + 0 T ( ϕ ( y ( U h ) ) ϕ ( Y h ) , e y e I y ) d t + 0 T ( A ( y ( U h ) Y h ) , e I y ) d t + 0 T ( ϕ ( y ( U h ) ) ϕ ( Y h ) , e I y ) d t 0 T ( A ( y ( U h ) Y h ) , ( e y e I y ) ) d t + 0 T ( ϕ ( y ( U h ) ) ϕ ( Y h ) , e y e I y ) d t + 0 T ( y t ( U h ) Y h t , e y e I y ) d t + 1 2 Y h ( x , 0 ) y 0 ( x ) L 2 ( Ω ) 2 + 0 T ( A ( y ( U h ) Y h ) , e I y ) d t + 0 T ( ϕ ( y ( U h ) ) ϕ ( Y h ) , e I y ) d t + 0 T ( y t ( U h ) Y h t , e I y ) d t .
(3.4)
By using equations (2.32), (2.37), and (3.4), we infer that
c e y L 2 ( J ; H 1 ( Ω ) ) 2 0 T τ T h τ ( f ˆ Y h t + div ( A Y ˆ h ) + 0 t div ( ψ ( t , τ ) Y ˆ h ( τ ) ) d τ ϕ ( Y ˆ h ) ) × ( e y e I y ) d t + 0 T l Ω = l ( A Y ˆ h n + 0 t ( ( ψ ( t , τ ) Y ˆ h ( τ ) ) n ) d τ ) ( e y e I y ) d t + 0 T l Ω l ( A Y ˆ h n + 0 t ( ( ψ ( t , τ ) Y ˆ h ( τ ) ) n ) d τ ) ( e y e I y ) d t + 0 T ( A ( Y h Y ˆ h ) , e y ) d t + 0 T ( ϕ ( Y h ) ϕ ( Y ˆ h ) , e y ) d t + 0 T ( f f ˆ , e y ) d t + 1 2 Y h ( x , 0 ) y 0 ( x ) L 2 ( Ω ) 2 I 1 + I 2 + I 3 + I 4 + I 5 + I 6 + 1 2 Y h ( x , 0 ) y 0 ( x ) L 2 ( Ω ) 2 .
(3.5)
Let us bound each of the terms on the right-hand side of equation (3.5). By Lemma 2.1 we have
I 1 = 0 T τ T h τ ( f ˆ Y h t + div ( A Y ˆ h ) + 0 t div ( ψ ( t , τ ) Y ˆ h ( τ ) ) d τ ϕ ( Y ˆ h ) ) × ( e y e I y ) d t C 0 T τ T h h τ 2 τ ( f ˆ Y h t + div ( A Y ˆ h ) + 0 t div ( ψ ( t , τ ) Y ˆ h ( τ ) ) d τ ϕ ( Y ˆ h ) ) 2 d t + C δ 0 T τ T h h τ 2 τ | e y e I y | 2 d t C 0 T τ T h h τ 2 τ ( f ˆ Y h t + div ( A Y ˆ h ) + 0 t div ( ψ ( t , τ ) Y ˆ h ( τ ) ) d τ ϕ ( Y ˆ h ) ) 2 d t + C δ e y L 2 ( J ; H 1 ( Ω ) ) 2 .
(3.6)
Next, using Lemma 2.1, we get
I 2 = 0 T l Ω = l ( A Y ˆ h n + 0 t ( ( ψ ( t , τ ) Y ˆ h ( τ ) ) n ) d τ ) ( e y e I y ) d t C 0 T l Ω = h l l [ A Y ˆ h n + 0 t ( ( ψ ( t , τ ) Y ˆ h ( τ ) ) n ) d τ ] 2 d t + C δ 0 T τ T h h τ 2 τ | e y e I y | 2 d t + C δ 0 T τ T h τ | ( e y e I y ) | 2 d t C 0 T l Ω = h l l [ A Y ˆ h n + 0 t ( ( ψ ( t , τ ) Y ˆ h ( τ ) ) n ) d τ ] 2 d t + C δ e y L 2 ( J ; H 1 ( Ω ) ) 2 ,
(3.7)
and
I 3 = 0 T l Ω l ( A Y ˆ h n + 0 t ( ( ψ ( t , τ ) Y ˆ h ( τ ) ) n ) d τ ) ( e y e I y ) d t C 0 T l Ω h l l ( A Y ˆ h n + 0 t ( ( ψ ( t , τ ) Y ˆ h ( τ ) ) n ) d τ ) 2 d t + C δ 0 T τ T h h τ 2 τ | e y e I y | 2 d t + C δ 0 T τ T h τ | ( e y e I y ) | 2 d t C 0 T l Ω h l l ( A Y ˆ h n + 0 t ( ( ψ ( t , τ ) Y ˆ h ( τ ) ) n ) d τ ) 2 d t + C δ e y L 2 ( J ; H 1 ( Ω ) ) 2 .
(3.8)
For the right-hand terms I 4 - I 6 of equation (3.5), the Schwarz inequality implies
I 4 = 0 T ( A ( Y h Y ˆ h ) , e y ) d t I 4 C Y h Y ˆ h L 2 ( J ; H 1 ( Ω ) ) 2 + C δ e y L 2 ( J ; H 1 ( Ω ) ) 2 ,
(3.9)
I 5 = 0 T ( ϕ ( Y h ) ϕ ( Y ˆ h ) , e y ) d t I 5 = 0 T ( ϕ ˜ ( Y h ) ( Y h Y ˆ h ) , e y ) d t I 5 C Y h Y ˆ h L 2 ( J ; H 1 ( Ω ) ) 2 + C δ e y L 2 ( J ; H 1 ( Ω ) ) 2 ,
(3.10)
and
I 6 = 0 T ( f f ˆ , e y ) d t C f f ˆ L 2 ( J ; L 2 ( Ω ) ) 2 + C δ e y L 2 ( J ; H 1 ( Ω ) ) 2 .
(3.11)
Let δ be small enough, and add inequalities (3.5)-(3.11) to obtain
y ( U h ) Y ˆ h L 2 ( J ; H 1 ( Ω ) ) 2 C i = 1 6 η i 2 .
(3.12)

This completes the proof. □

Analogously to the proof of Theorem 3.1, we can obtain the following estimates.

Theorem 3.2 Let ( y ( U h ) , p ( U h ) ) and ( Y h , P h ) be the solutions of equations (2.37)-(2.39) and equations (2.32)-(2.34), respectively. Then
p ( U h ) P ˜ h L 2 ( J ; H 1 ( Ω ) ) 2 C i = 1 11 η i 2 ,
(3.13)
where
η 7 2 = 0 T τ T h h τ 2 η 7 2 = × τ ( P h t + div ( A P ˜ h ) + t T div ( ψ ( τ , t ) P ˜ h ( τ ) ) d τ + Y ˆ h y 0 ϕ ( Y ˜ h ) P ˜ h ) 2 d t , η 8 2 = 0 T l Ω = h l l [ A P ˜ h n + t T ( ψ ( τ , t ) P ˜ h ( τ ) ) n d τ ] 2 d t , η 9 2 = 0 T l Ω h l l [ A P ˜ h n + t T ( ψ ( τ , t ) P ˜ h ( τ ) ) n d τ ] 2 d t , η 11 2 = P h P ˜ h L 2 ( J ; H 1 ( Ω ) ) 2 , η 10 2 = Y h Y ˜ h L 2 ( J ; H 1 ( Ω ) ) 2 ,
where η 1 - η 6 are defined in Theorem  3.1, l is a face of an element τ, [ A P ˜ h n ] is the A-normal derivative jump over the interior face l, defined by
[ A P ˜ h n ] l = ( A P ˜ h | τ l 1 A P ˜ h | τ l 2 ) n ,

where n is the unit normal vector on l = τ ¯ l 1 τ ¯ l 2 outwards of τ l 1 .

For given u K , let M be the inverse operator of the state equation (2.8), such that y ( u ) = M B u is the solution of the state equation (2.8). Similarly, for given U h K h , Y h ( U h ) = M h B U h is the solution of the discrete state equation (2.32). Let
S ( u ) = 1 2 M B u y 0 2 + α 2 u 2 , S h ( U h ) = 1 2 M h B U h y 0 2 + α 2 U h 2 .
It is clear that S and S h are well defined and continuous on K and K h . Also the functional S h can be naturally extended on K. Then equations (2.5) and (2.21) can be represented as
min u K { S ( u ) } ,
(3.14)
min U h K h { S h ( U h ) } .
(3.15)
It can be shown that
( S ( u ) , v ) = ( α u + B p , v ) , ( S ( U h ) , v ) = ( α U h + B p ( U h ) , v ) , ( S h ( U h ) , v ) = ( α U h + B P ˜ h , v ) ,

where p ( U h ) is the solution of equations (2.37)-(2.39).

In many applications, S ( ) is uniform convex near the solution u (see, e.g., [19]). The convexity of S ( ) is closely related to the second-order sufficient conditions of the control problems, which are assumed in many studies on numerical methods of the problems. If S ( ) is uniformly convex, then there is a c > 0 , such that
0 T ( S ( u ) S ( U h ) , u U h ) c u U h L 2 ( J ; L 2 ( Ω ) ) 2 ,
(3.16)

where u and U h are the solutions of equations (3.14) and (3.15), respectively. We will assume the above inequality throughout this paper.

In order to have sharp a posteriori error estimates, we divide Ω into some subsets:
Ω i = { x Ω : ( B P ˜ h ) ( x , t i ) α U h i } , Ω i = { x Ω : ( B P ˜ h ) ( x , t i ) > α U h i , U h i = 0 } , Ω i + = { x Ω : ( B P ˜ h ) ( x , t i ) > α U h i , U h i > 0 } .

Then, it is clear that the three subsets do not intersect, and Ω = Ω i Ω i Ω i + , i = 1 , 2 , , N .

Let p ( U h ) be the solution of equations (2.37)-(2.39); we establish the following error estimate, which can be proved similarly to the proofs given in [10].

Theorem 3.3 Let u and U h be the solutions of equations (2.5) and (2.36), respectively. Then
u U h L 2 ( J ; L 2 ( Ω ) ) 2 C ( η 12 2 + P ˜ h p ( U h ) L 2 ( J ; H 1 ( Ω ) ) 2 ) ,
(3.17)
where
η 12 2 = i = 1 N t i 1 t i Ω i | B P ˜ h + α U h | 2 d x d t .
Proof It follows from the inequality (3.16) that
c u U h L 2 ( J ; L 2 ( Ω ) ) 2 0 T ( S ( u ) , u U h ) ( S ( U h ) , u U h ) d t 0 T ( S ( U h ) , u U h ) d t = 0 T ( S h ( U h ) , U h u ) d t + 0 T ( S h ( U h ) S ( U h ) , u U h ) d t .
(3.18)
Note that
0 T ( S h ( U h ) , U h u ) d t = i = 1 N t i 1 t i Ω i ( B P ˜ h + α U h ) ( U h u ) d x d t + i = 1 N t i 1 t i Ω i ( B P ˜ h + α U h ) ( U h u ) d x d t + i = 1 N t i 1 t i Ω i + ( B P ˜ h + α U h ) ( u ) d x d t .
(3.19)
It is easy to see that
Ω i ( B P ˜ h + α U h ) ( U h u ) d x Ω i | B P ˜ h + α U h | 2 d x + δ u U h L 2 ( J ; L 2 ( Ω ) ) 2 = C η 12 2 + δ u U h L 2 ( J ; L 2 ( Ω ) ) 2 .
(3.20)
Since U h is piecewise constant, U h | s > 0 if s Ω i + is not empty. If u h | s > 0 , there exist ε > 0 and β U h , such that β 0 , β L ( s ) = 1 and ( u h ε β ) | s 0 . For example, one can always find such a required β from one of the shape functions on s. Hence, u ˆ h K h , where u ˆ h = U h ε β as x s and otherwise u ˆ = U h . Then, it follows from equation (2.36) that
s ( B P ˜ h + α U h ) β d x = ε 1 s ( B P ˜ h + α U h ) ( U h ( U h ε β ) ) d x ε 1 Ω ( B P ˜ h + α U h ) ( U h ( U h ε β ) ) d x 0 .
(3.21)
Note that on Ω i + , B P ˜ h + α U h B P ˜ h > 0 and from equation (3.20) we have
s Ω i + | B P ˜ h + α U h | β d x = s Ω i + ( B P ˜ h + α U h ) β d x s Ω i ( B P ˜ h + α U h ) β d x s Ω i | B P ˜ h + α U h | d x .
(3.22)
Let s ˆ be the reference element of s, s 0 = s Ω i + , and s ˆ 0 s ˆ be a part mapped from s ˆ 0 . Note that ( s | | 2 ) 1 / 2 , s | | β are both norms on L 2 ( s ) . In such a case for the function β fixed above, it follows from the equivalence of the norm in the finite-dimensional space that
s Ω i + | B P ˜ h + α U h | 2 d x = s 0 | B P ˜ h + α U h | 2 d x C h s 2 s ˆ 0 | B P ˜ h + α U h | 2 d x C h s 2 ( s ˆ 0 | B P ˜ h + α U h | β d x ) 2 C h s 2 ( s Ω i + | B P ˜ h + α U h | β d x ) 2 C h s 2 ( s Ω i | B P ˜ h + α U h | d x ) 2 C s Ω i | B P ˜ h + α U h | 2 d x ,
(3.23)
where the constant C can be made independent of β since it is always possible to find the required β from the shape functions on s. Thus
Ω i + ( B P ˜ h + α U h ) ( U h u ) d x C Ω i + | B P ˜ h + α U h | 2 d x + δ u U h L 2 ( J ; L 2 ( Ω ) ) 2 C Ω i | B P ˜ h + α U h | 2 d x + δ u U h L 2 ( J ; L 2 ( Ω ) ) 2 C η 12 2 + δ u U h L 2 ( J ; L 2 ( Ω ) ) 2 .
(3.24)
It follows from the definition of Ω i that B P ˜ h + α U h > 0 on Ω i . Note that u 0 , we have
Ω i ( B P ˜ h + α U h ) ( u ) d x 0 .
(3.25)
It is easy to show that
( S h ( U h ) S ( U h ) , u U h ) = ( B P ˜ h + α U h , u U h ) ( B p ( U h ) + α U h , u U h ) = ( B ( P ˜ h p ( U h ) ) , u U h ) C P ˜ h p ( U h ) L 2 ( J ; L 2 ( Ω ) ) 2 + δ u U h L 2 ( J ; L 2 ( Ω ) ) 2 C P ˜ h p ( U h ) L 2 ( J ; H 1 ( Ω ) ) 2 + δ u U h L 2 ( J ; L 2 ( Ω ) ) 2 .
(3.26)

Therefore, equation (3.17) follows from equations (3.18)-(3.20) and (3.24)-(3.26). □

Hence, we combine Theorems 3.1-3.3 to conclude that

Theorem 3.4 Let ( y , p , u ) and ( Y h , P h , U h ) be the solutions of equations (2.8)-(2.12) and equations (2.32)-(2.36), respectively. Then
u U h L 2 ( J ; L 2 ( Ω ) ) 2 + y Y h L 2 ( J ; H 1 ( Ω ) ) 2 + p P h L 2 ( J ; H 1 ( Ω ) ) 2 C i = 1 12 η i 2 ,
(3.27)

where η 1 - η 12 are defined in Theorems  3.1-3.3, respectively.

Proof From equations (2.8)-(2.11) and (2.37)-(2.40), we obtain the error equations
( y t y t ( U h ) , w ) + a ( y y ( U h ) , w ) + 0 t ψ ( t , τ ; ( y y ( U h ) ) ( τ ) , w ) d τ + ( ϕ ( y ) ϕ ( y ( U h ) ) , w ) = ( B ( u U h ) , w ) ,
(3.28)
( p t p t ( U h ) , q ) + a ( q , p p ( U h ) ) + t T ψ ( τ , t ; q , ( p p ( U h ) ) ( τ ) ) d τ + ( ϕ ( y ) p ϕ ( y ( U h ) ) p ( U h ) , q ) = ( y y ( U h ) , q ) ,
(3.29)
for all w V and q V . Thus it follows from equations (3.28)-(3.29) that
( y t y t ( U h ) , w ) + a ( y y ( U h ) , w ) + 0 t ψ ( t , τ ; ( y y ( U h ) ) ( τ ) , w ) d τ + ( ϕ ˜ ( y ) ( y y ( U h ) ) , w ) = ( B ( u U h ) , w ) ,
(3.30)
( p t p t ( U h ) , q ) + a ( q , p p ( U h ) ) + t T ψ ( τ , t ; q , ( p p ( U h ) ) ( τ ) ) d τ + ( ϕ ( y ( U h ) ) ( p p ( U h ) ) , q ) = ( ϕ ˜ ( y ( U h ) ) ( y ( U h ) y ) p , q ) .
(3.31)
By using the stability results in [20, 21], then we obtain
y y ( U h ) L 2 ( J ; H 1 ( Ω ) ) 2 C u U h L 2 ( J ; L 2 ( Ω ) ) 2 ,
(3.32)
and
p p ( U h ) L 2 ( J ; H 1 ( Ω ) ) 2 y y ( U h ) L 2 ( J ; H 1 ( Ω ) ) 2 C u U h L 2 ( J ; L 2 ( Ω ) ) 2 .
(3.33)

Finally, combining Theorems 3.1-3.3 and equations (3.32)-(3.33) leads to equation (3.27). □

4 Conclusion and future work

In this paper we discuss the finite-element methods of the nonlinear parabolic integro-differential optimal control problems (1.1)-(1.4). We have established a posteriori error estimates for both the state, the co-state, and the control variables. The posteriori error estimates for those problems by finite-element methods seem to be new.

In our future work, we shall use the mixed finite-element method to deal with nonlinear parabolic integro-differential optimal control problems. Furthermore, we shall consider a posteriori error estimates and superconvergence of mixed finite-element solution for nonlinear parabolic integro-differential optimal control problems.

Declarations

Acknowledgements

The author expresses his gratitude 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), Chongqing Research Program of Basic Research and Frontier Technology (cstc2012jjA00003), Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJ121113), and Science and Technology Project of Wanzhou District of Chongqing (2013030050).

Authors’ Affiliations

(1)
Key Laboratory for Nonlinear Science and System Structure, School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing, China

References

  1. Falk FS: Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 1973, 44: 28–47. 10.1016/0022-247X(73)90022-XMATHMathSciNetView ArticleGoogle Scholar
  2. Geveci T: On the approximation of the solution of an optimal control problem governed by an elliptic equation. RAIRO. Anal. Numér. 1979, 13: 313–328.MATHMathSciNetGoogle Scholar
  3. Arada N, Casas E, Tröltzsch F: Error estimates for the numerical approximation of a boundary semilinear elliptic control problem. Comput. Optim. Appl. 2005, 31: 193–219. 10.1007/s10589-005-2180-2MathSciNetView ArticleGoogle Scholar
  4. Brunner H, Yan NN: Finite element methods for optimal control problems governed by integral equations and integro-differential equations. Appl. Numer. Math. 2003, 47: 173–187. 10.1016/S0168-9274(03)00054-0MathSciNetView ArticleGoogle Scholar
  5. French DA, King JT: Approximation of an elliptic control problem by the finite element method. Numer. Funct. Anal. Optim. 1991, 12: 299–315. 10.1080/01630569108816430MATHMathSciNetView ArticleGoogle Scholar
  6. Gunzburger MD, Hou SL: Finite dimensional approximation of a class of constrained nonlinear control problems. SIAM J. Control Optim. 1996, 34: 1001–1043. 10.1137/S0363012994262361MATHMathSciNetView ArticleGoogle Scholar
  7. Mossino J: An application of duality to distributed optimal control problems with constraints on the control and the state. J. Math. Anal. Appl. 1975, 50: 223–242. 10.1016/0022-247X(75)90019-0MATHMathSciNetView ArticleGoogle Scholar
  8. Neittaanmaki P, Tiba D: Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications. Dekker, New York; 1994.Google Scholar
  9. Lu Z: A posteriori error estimates of finite element methods for nonlinear quadratic boundary optimal control problem. Numer. Anal. Appl. 2011, 4: 210–222. 10.1134/S1995423911030037View ArticleGoogle Scholar
  10. Chen Y, Liu WB: A posteriori error estimates for mixed finite element solutions of convex optimal control problems. J. Comput. Appl. Math. 2008, 211: 76–89. 10.1016/j.cam.2006.11.015MATHMathSciNetView ArticleGoogle Scholar
  11. Liu WB, Yan NN: A posteriori error estimates for distributed convex optimal control problems. Numer. Math. 2005, 101: 1–27. 10.1007/s00211-005-0608-3MathSciNetView ArticleGoogle Scholar
  12. Verfurth R: A posteriori error estimates for nonlinear problems. Math. Comput. 1994, 62: 445–475.MathSciNetView ArticleGoogle Scholar
  13. Chen Y, Lu Z: Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problems. Comput. Methods Appl. Mech. Eng. 2010, 199: 1415–1423. 10.1016/j.cma.2009.11.009MATHView ArticleGoogle Scholar
  14. Chen Y, Lu Z: Error estimates for parabolic optimal control problem by fully discrete mixed finite element methods. Finite Elem. Anal. Des. 2010, 46: 957–965. 10.1016/j.finel.2010.06.011MathSciNetView ArticleGoogle Scholar
  15. Lu Z, Chen Y: L -error estimates of triangular mixed finite element methods for optimal control problem govern by semilinear elliptic equation. Numer. Anal. Appl. 2009, 2: 74–86. 10.1134/S1995423909010078View ArticleGoogle Scholar
  16. Chen Y, Lu Z, Huang Y: Superconvergence of triangular Raviart-Thomas mixed finite element methods for bilinear constrained optimal control problem. Comput. Math. Appl. 2013, 66: 1498–1513. 10.1016/j.camwa.2013.08.019MathSciNetView ArticleGoogle Scholar
  17. Lu Z, Chen Y: A posteriori error estimates of triangular mixed finite element methods for semilinear optimal control problems. Adv. Appl. Math. Mech. 2009, 1: 242–256.MATHMathSciNetGoogle Scholar
  18. Lions JL: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin; 1971.MATHView ArticleGoogle Scholar
  19. Liu WB, Yan NN: A posteriori error estimates for control problems governed by nonlinear elliptic equation. Adv. Comput. Math. 2001, 15: 285–309. 10.1023/A:1014239012739MATHMathSciNetView ArticleGoogle Scholar
  20. Sloan IH, Thome V: Time discretization of an integro-differential equation of parabolic type. SIAM J. Numer. Anal. 1986, 23: 1052–1061. 10.1137/0723073MATHMathSciNetView ArticleGoogle Scholar
  21. Yanik EG, Fairweather G: Finite element methods for parabolic and hyperbolic partial integro-differential equations. Nonlinear Anal. 1988, 12: 785–809. 10.1016/0362-546X(88)90039-9MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Lu; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Advertisement