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Adaptive fully-discrete finite element methods for nonlinear quadratic parabolic boundary optimal control

Zuliang Lu

Author Affiliations

School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing, 404000, P.R. China

College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan, 411105, P.R. China

Boundary Value Problems 2013, 2013:72  doi:10.1186/1687-2770-2013-72


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/72


Received:18 January 2013
Accepted:14 March 2013
Published:4 April 2013

© 2013 Lu; licensee Springer

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.

Abstract

The aim of this work is to study adaptive fully-discrete finite element methods for quadratic boundary optimal control problems governed by nonlinear parabolic equations. We derive a posteriori error estimates for the state and control approximation. Such estimates can be used to construct reliable adaptive finite element approximation for nonlinear quadratic parabolic boundary optimal control problems. Finally, we present a numerical example to show the theoretical results.

1 Introduction

In this paper, we study the fully-discrete finite element approximation for quadratic boundary optimal control problems governed by nonlinear parabolic equations. Optimal control problems are very important models in engineering numerical simulation. They have various physical backgrounds in many practical applications. Finite element approximation of optimal control problems plays a very important role in the numerical methods for these problems. The finite element approximation of a linear elliptic optimal control problem is well investigated by Falk [1] and Geveci [2]. The discretization for semilinear elliptic optimal control problems is discussed by Arada, Casas, and Tröltzsch in [3]. Systematic introductions of the finite element method for optimal control problems can be found in [4-6].

As one of important kinds of optimal control problems, the boundary optimal control is widely used in scientific and engineering computing. The literature in this aspect is huge; see, e.g., [7-10]. For some quadratic boundary optimal control problems, Liu and Yan [11,12] investigated a posteriori error estimates and adaptive finite element methods. Alt and Mackenroth [13] were concerned with error estimates of finite element approximations to state constrained convex parabolic boundary optimal control problems. Arada et al. discussed the numerical approximation of boundary optimal control problems governed by semilinear elliptic equations with pointwise constraints on the control in [14]. Although a priori error estimates and a posteriori error estimates of finite element approximation are widely used in numerical simulations, they have not yet been utilized in nonlinear parabolic boundary optimal control problems.

Adaptive finite element approximation is the most important method to boost 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 a posteriori error indicator. A posteriori error estimates are computable quantities in terms of the discrete solution that measure the actual discrete errors without the knowledge of exact solutions. They are essential in designing algorithms for mesh which equidistribute the computational effort and optimize the computation. Recently, in [15-18], we derived a priori error estimates, a posteriori error estimates and superconvergence for optimal control problems using mixed finite element methods.

In this paper, we adopt the standard notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M1">View MathML</a> for Sobolev spaces on Ω with a norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M2">View MathML</a> given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M3">View MathML</a> and a semi-norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M4">View MathML</a> given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M5">View MathML</a>. We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M6">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M7">View MathML</a>, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M9">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M11">View MathML</a>. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M12">View MathML</a> the Banach space of all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M13">View MathML</a> integrable functions from J into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M1">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M15">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M16">View MathML</a>, and the standard modification for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M17">View MathML</a>. The details can be found in [19].

In this paper, we derive a posteriori error estimates for a class of boundary optimal control problems governed by a nonlinear parabolic equation. To our best knowledge, in the context of nonlinear parabolic boundary optimal control problems, these estimates are new. The problem that we are interested in is the following nonlinear quadratic parabolic boundary optimal control problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M18">View MathML</a>

(1)

subject to the state equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M19">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M20">View MathML</a>

(3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M21">View MathML</a>

(4)

where the bounded open set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M22">View MathML</a> is 2 regular convex polygon with boundary Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M26">View MathML</a>, and α is a positive constant. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M27">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M29">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M30">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M31">View MathML</a>. We assume the coefficient matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M32">View MathML</a> is a symmetric positive definite matrix, and there is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M33">View MathML</a> satisfying for any vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M35">View MathML</a>. Here, K denotes the admissible set of the control variable defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M36">View MathML</a>

(5)

The plan of this paper is as follows. In the next section, we present a finite element discretization for nonlinear quadratic parabolic boundary optimal control problems. A posteriori error estimates are established for the finite element approximation solutions in Section 3. In Section 4, we give a numerical example to prove the theoretical results.

2 Finite element methods for parabolic boundary optimal control

We shall now describe a finite element discretization of nonlinear quadratic parabolic boundary optimal control problem (1)-(4). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M39">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M40">View MathML</a>

(6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M41">View MathML</a>

(7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M42">View MathML</a>

(8)

Then quadratic parabolic boundary optimal control problem (1)-(4) can be restated as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M43">View MathML</a>

(9)

subject to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M44">View MathML</a>

(10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M45">View MathML</a>

(11)

where the inner product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M46">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M47">View MathML</a> is indicated by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M48">View MathML</a>, and B is a continuous linear operator from U to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M46">View MathML</a>.

It is well known (see, e.g., [12]) that the optimal control problems have at least a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M50">View MathML</a>, and that if a pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M50">View MathML</a> is the solution of (9)-(11), then there is a co-state <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M52">View MathML</a> such that the triplet <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M53">View MathML</a> satisfies the following optimality conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M54">View MathML</a>

(12)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M55">View MathML</a>

(13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M56">View MathML</a>

(14)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M57">View MathML</a>

(15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M58">View MathML</a>

(16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M59">View MathML</a> is the adjoint operator of B. In the rest of the paper, we shall simply write the product as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M48">View MathML</a> whenever no confusion should be caused.

Let us consider the finite element approximation of control problem (9)-(11). Again, here we consider only n-simplex elements and conforming finite elements.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M61">View MathML</a> be a regular partition of Ω. Associated with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M61">View MathML</a> is a finite dimensional subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M63">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M64">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M65">View MathML</a> are polynomials of m-order (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M66">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M67">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M68">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M69">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M70">View MathML</a> be a partition of Ω into disjoint regular <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M71">View MathML</a>-simplices s, so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M72">View MathML</a>. Associated with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M70">View MathML</a> is another finite dimensional subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M74">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M75">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M65">View MathML</a> are polynomials of m-order (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M77">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M79">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M80">View MathML</a> denote the maximum diameter of the element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M81">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M83">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M84">View MathML</a>. In addition C or c denotes a general positive constant independent of h.

By the definition of a finite element subspace, the finite element discretization of (9)-(11) is as follows: compute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M85">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M86">View MathML</a>

(17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M87">View MathML</a>

(18)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M88">View MathML</a>

(19)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M90">View MathML</a> is an approximation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M91">View MathML</a>.

Again, it follows that optimal control problem (17)-(19) has at least a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M92">View MathML</a>, and that if a pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M92">View MathML</a> is the solution of (17)-(19), then there is a co-state <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M94">View MathML</a> such that the triplet <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M95">View MathML</a> satisfies the following optimality conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M96">View MathML</a>

(20)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M97">View MathML</a>

(21)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M98">View MathML</a>

(22)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M99">View MathML</a>

(23)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M100">View MathML</a>

(24)

We now consider the fully discrete approximation for the semidiscrete problem. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M101">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M102">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M104">View MathML</a>. Also, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M105">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M106">View MathML</a>, we construct the finite element spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M107">View MathML</a> with the mesh <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M108">View MathML</a> (similar to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M109">View MathML</a>). Similarly, we construct the finite element spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M110">View MathML</a> with the mesh <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M108">View MathML</a> (similar to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M112">View MathML</a>). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M113">View MathML</a> denote the maximum diameter of the element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M114">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M115">View MathML</a>. Define mesh functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M117">View MathML</a> and mesh size functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M119">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M123">View MathML</a>. For ease of exposition, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M125">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M126">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M127">View MathML</a> by τ, s, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M128">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M129">View MathML</a>, respectively.

Then the fully discrete finite element approximation of (17)-(19) is as follows. Compute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M131">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M132">View MathML</a>

(25)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M133">View MathML</a>

(26)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M134">View MathML</a>

(27)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M90">View MathML</a> is an approximation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M91">View MathML</a>.

Now, it follows that optimal control problem (25)-(27) has at least a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M139">View MathML</a>, and that if a pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M139">View MathML</a>, is the solution of (25)-(27), then there is a co-state <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M139">View MathML</a>, such that the triplet <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M144">View MathML</a> satisfies the following optimality conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M145">View MathML</a>

(28)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M146">View MathML</a>

(29)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M147">View MathML</a>

(30)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M148">View MathML</a>

(31)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M149">View MathML</a>

(32)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M139">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M151">View MathML</a>

(33)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M152">View MathML</a>

(34)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M153">View MathML</a>

(35)

For any function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M154">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M156">View MathML</a>. Then the optimality conditions (28)-(32) can be restated as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M157">View MathML</a>

(36)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M158">View MathML</a>

(37)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M159">View MathML</a>

(38)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M160">View MathML</a>

(39)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M161">View MathML</a>

(40)

In the rest of the paper, we shall use some intermediate variables. For any control function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M162">View MathML</a>, we define that the state solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M163">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M164">View MathML</a>

(41)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M165">View MathML</a>

(42)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M166">View MathML</a>

(43)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M167">View MathML</a>

(44)

Now we restate the following well-known estimates in [19].

Lemma 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M168">View MathML</a>be the Clément-type interpolation operator defined in[19]. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M169">View MathML</a>and all elementτ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M170">View MathML</a>

(45)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M171">View MathML</a>

(46)

wherelis the edge of the element.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M172">View MathML</a>, we write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M173">View MathML</a>

(47)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M174">View MathML</a>

are bounded functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M175">View MathML</a>[20].

3 A posteriori error estimates

In this section we obtain a posteriori error estimates for nonlinear quadratic parabolic boundary optimal control problems. Firstly, we estimate the error <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M176">View MathML</a>.

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M163">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M178">View MathML</a>be the solutions of (41)-(44) and (36)-(40), respectively. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M179">View MathML</a>

(48)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M180">View MathML</a>

wherelis a face of an elementτ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M181">View MathML</a>is the size of the facel, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M182">View MathML</a>is theA-normal derivative jump over the interior faceldefined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M183">View MathML</a>

wherenis the unit normal vector on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M184">View MathML</a>outwards<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M185">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M186">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M187">View MathML</a> be the Clément-type interpolator of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M188">View MathML</a> defined in Lemma 2.1. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M189">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M190">View MathML</a>

Using equations (36) and (41), we infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M191">View MathML</a>

(49)

Let us bound each of the terms on the right-hand side of (49). By Lemma 2.1 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M192">View MathML</a>

(50)

Next, using Lemma 2.1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M193">View MathML</a>

(51)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M194">View MathML</a>

(52)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M195">View MathML</a>-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M196">View MathML</a>, the Schwarz inequality implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M197">View MathML</a>

(53)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M198">View MathML</a>

(54)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M199">View MathML</a>

(55)

Finally, add inequalities (49)-(55) to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M200">View MathML</a>

(56)

This completes the proof. □

Analogously to Theorem 3.1, we show the following estimates.

Theorem 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M163">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M178">View MathML</a>be the solutions of (41)-(44) and (36)-(40), respectively. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M203">View MathML</a>

(57)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M204">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M205">View MathML</a>-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M206">View MathML</a>are defined in Theorem 3.1, lis a face of an elementτ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M207">View MathML</a>is theA-normal derivative jump over the interior faceldefined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M208">View MathML</a>

wherenis the unit normal vector on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M184">View MathML</a>outwards<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M185">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M211">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M212">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M168">View MathML</a> is the Clément-type interpolator defined in Lemma 2.1. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M214">View MathML</a>, then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M215">View MathML</a>

Using equations (38) and (43), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M216">View MathML</a>

(58)

Now let us bound each of the terms on the right-hand side of (58). By Lemma 2.1 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M217">View MathML</a>

(59)

Next, using Lemma 2.1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M218">View MathML</a>

(60)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M219">View MathML</a>

(61)

The Schwarz inequality implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M220">View MathML</a>

(62)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M221">View MathML</a>

(63)

Next, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M222">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M223">View MathML</a>

(64)

Finally, add inequalities (58)-(64) and combine Theorem 3.1 to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M224">View MathML</a>

(65)

This completes the proof. □

For given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M225">View MathML</a>, let M be the inverse operator of the state equation (12) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M226">View MathML</a> is the solution of the state equation (12). Similarly, for given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M227">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M228">View MathML</a> is the solution of the discrete state equation (36). Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M229">View MathML</a>

It is clear that S and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M230">View MathML</a> are well defined and continuous on K and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M231">View MathML</a>. Also, the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M230">View MathML</a> can be naturally extended on K. Then (9) and (25) can be represented as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M233">View MathML</a>

(66)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M234">View MathML</a>

(67)

It can be shown that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M235">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M236">View MathML</a> is the solution of equations (41)-(43).

In many applications, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M237">View MathML</a> is uniform convex near the solution u (see, e.g., [21]). The convexity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M237">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M237">View MathML</a> is uniformly convex, then there is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M33">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M241">View MathML</a>

(68)

where u and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M112">View MathML</a> are the solutions of (66) and (67), respectively. We assume the above inequality throughout this paper.

In order to have sharp a posteriori error estimates, we divide Ω into some subsets:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M243">View MathML</a>

Then it is clear that three subsets do not intersect each other, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M244">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M131">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M236">View MathML</a> be the solution of (41)-(44). We establish the following error estimate, which can be proved similarly to the proofs given in [22].

Theorem 3.3Letuand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M112">View MathML</a>be the solutions of (66) and (67), respectively. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M248">View MathML</a>

(69)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M249">View MathML</a>

Proof It follows from the inequality (68) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M250">View MathML</a>

(70)

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M251">View MathML</a>

(71)

It is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M252">View MathML</a>

(72)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M112">View MathML</a> is piecewise constant, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M254">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M255">View MathML</a> is not empty. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M256">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M257">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M258">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M259">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M260">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M261">View MathML</a>. For example, one can always find such a required β from one of the shape functions on s. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M262">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M263">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M264">View MathML</a> and otherwise <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M265">View MathML</a>. Then it follows from (40) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M266">View MathML</a>

(73)

Note that on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M267">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M268">View MathML</a>, and from (72) we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M269">View MathML</a>

(74)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M270">View MathML</a> be the reference element of s, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M271">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M272">View MathML</a> be a part mapped from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M273">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M274">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M275">View MathML</a> are both norms on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M276">View MathML</a>. In such a case, for the function β fixed above, it follows from the equivalence of the norm in the finite-dimensional space that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M277">View MathML</a>

(75)

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 so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M278">View MathML</a>

(76)

It follows from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M279">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M280">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M279">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M282">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M283">View MathML</a>

(77)

It is easy to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M284">View MathML</a>

(78)

Therefore, (69) follows from (70)-(72) and (76)-(78). □

Hence, we combine Theorems 3.1-3.3 to conclude the following.

Theorem 3.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M53">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M286">View MathML</a>be the solutions of (12)-(16) and (36)-(40), respectively. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M287">View MathML</a>

(79)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M288">View MathML</a> , and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M289">View MathML</a>are defined in Theorems 3.1-3.3, respectively.

Proof From (12)-(15) and (41)-(44), we obtain the error equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M290">View MathML</a>

(80)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M291">View MathML</a>

(81)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M292">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M293">View MathML</a>. Thus it follows from (80)-(81) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M294">View MathML</a>

(82)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M295">View MathML</a>

(83)

By using the stability results in [23], we can prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M296">View MathML</a>

(84)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M297">View MathML</a>

(85)

Finally, combining Theorems 3.1-3.3 and (84)-(85) leads to (79). □

4 Numerical example

In the section, we use a posteriori error estimates presented in our paper as an indicator for the adaptive finite element approximation. The optimization problem is solved numerically by a preconditioned projection algorithm, with codes developed based on AFEPACK. The optimal control problem is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M298">View MathML</a>

In the example, we choose the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M299">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M300">View MathML</a>. Let Ω be partitioned into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M301">View MathML</a> as described in Section 2. We use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M289">View MathML</a> as the control mesh refinement indicator and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M205">View MathML</a>-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M304">View MathML</a> as the states and co-states.

For the constrained optimization problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M305">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M306">View MathML</a> is a convex functional on U, the iterative scheme reads (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M307">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M308">View MathML</a>

(86)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M309">View MathML</a> is a symmetric and positive definite bilinear form such that there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M310">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M311">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M312">View MathML</a>

(87)

and the projection operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M313">View MathML</a> is defined as follows. For given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M314">View MathML</a>, find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M315">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M316">View MathML</a>

(88)

The bilinear form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M309">View MathML</a> provides suitable preconditioning for the projection algorithm. An application of (86) to the discretized nonlinear parabolic boundary optimal control problem yields the following algorithm:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M318">View MathML</a>

(89)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M319">View MathML</a>

(90)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M320">View MathML</a>

(91)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M321">View MathML</a>

(92)

The main computational effort is to solve the state and co-state equations and to compute the projection <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M322">View MathML</a>. In this paper we use a fast algebraic multigrid solver to solve the state and co-state equations. Then it is clear that the key to saving computing time is finding how to compute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M322">View MathML</a> efficiently. For the piecewise constant elements, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M324">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M325">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M326">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M327">View MathML</a> is the average of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M328">View MathML</a> over T. In solving our discretized optimal control problem, we use the preconditioned projection gradient method (89)-(92) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M329">View MathML</a> and a fixed step size <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/72/mathml/M330">View MathML</a>. In the numerical simulation, we use a piecewise linear finite element space for the approximation of y and p, and a piecewise constant for u.

It can be clearly seen from Table 1 that on the adaptive meshes one may use less degree of freedom to produce a given control error reduction. Then it is clear that these a posteriori error estimates are very good for the parabolic boundary optimal control, and the adaptive finite element method is more efficient.

Table 1. Comparison of uniform mesh and adaptive mesh

Competing interests

The author declares that he has no competing interests.

Author’s contributions

ZL participated in the design of all the study and drafted the manuscript.

Acknowledgements

This work is supported by National Science Foundation of China (11201510), Mathematics TianYuan Special Funds of the National Natural Science Foundation of China (11126329), China Postdoctoral Science Foundation funded project (2011M500968), Natural Science Foundation Project of CQ CSTC (cstc2012jjA00003), and Natural Science Foundation of Chongqing Municipal Education Commission (KJ121113). The author expresses his thanks to the referees for their helpful suggestions, which led to improvements of the presentation.

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