Finite element error estimates for an optimal control problem governed by the Burgers equation
We derive a-priori error estimates for the finite-element approximation of a distributed optimal control problem governed by the steady one-dimensional Burgers equation with pointwise box constraints on the control. Here the approximation of the state and the control is done by using piecewise linear functions. With this choice, an superlinear order of convergence for the control is obtained; moreover, under a further assumption on the regularity structure of the optimal control this error estimate can be improved to . The theoretical findings are tested experimentally by means of numerical examples.
Key words and phrases:error estimates, finite element method, optimal control, Burgers equation
2000 Mathematics Subject Classification:49J20, 80M10, 49N05, 49K20, 35Q53,41A25
We consider the finite element approximation of the following optimal control problem of the steady one-dimensional Burgers equation with pointwise control constraints:
The Burgers equation is a well-known one dimensional model for turbulence and its control has been studied by several authors c.f. , ,. Our aim in this paper consists in deriving a-priori error estimates for the optimal control problem in the -norm.
Finite element approximations for control constrained control problems in fluid mechanics have been previously considered in  and  for piecewise constant controls. In particular, in the last, the authors report an error order of if the control space is not discretized, whereas an order of is obtained for the piecewise constant discretization. It is natural to expect that these error estimates also holds in the case of the Burgers equation using the theory developed in . However, if the control space is discretized by piecewise linear functions, results were only obtained for the semilinear case in  and in  for the linear–quadratic case. Since the optimal control is Lipschitz continuous, its approximations by piecewise linear functions seems to be a natural choice, which in addition piecewise linear functions have less degrees of freedom than piecewise constant functions. Here we aim to perform this task by combining the arguments in  and  to obtain a superlinear error of convergence for the –norm estimate of the control. In addition, by considering a stronger assumption on the structure of the optimal control and relying on the one–dimensional setting of our problem, we are able to improve the order of the error to .
The paper is organized as follows: first we briefly comment the properties the optimal control problem and its conditions for optimality, next we refer to the finite element method approximation of the Burgers equation and the corresponding error estimates . Next, we discuss the approximation of the optimal control problem by piecewise linear functions by establishing a superlinear order of convergence for the optimal control. We finish the theory by showing that the superlinear error of convergence can be improved under certain assumptions on the regularity of the optimal control. Finally, we discuss some numerical experiments to confront our theoretical findings.
2. The control problem
We consider the discretization analysis for the following optimal control problem, governed by Burgers equation:
Here, is the set of admissible controls defined by with constants and satisfying . is the usual Tychonoff parameter. We shall denote by and by the norm and the scalar product in , respectively. is the indicator function defined in an open subinterval , whereas denotes the viscosity parameter which is assumed that satisfies (5). For different spaces, the open ball centered in with radius will be denoted by if there is no risk of confusion.
2.1. The state equation equation
The steady Burgers equation is given by
The weak formulation of the homogeneous Dirichlet problem for the Burgers equation is as follows: given , find such that
where: is the continuous, bilinear and symetric form defined by
and stands for the continuous trilinear form defined by
The trilinear enjoys the following important properties c.f. 
It is well known, cf. [13, Theorem 2.10] that if the condition
holds, the Burgers equation (LABEL:eq:burg1) has a unique solution depending on the right hand side. Indeed, for every , there exists which satisfies (LABEL:eq:burg1) and fulfill the relation . In addition, by taking the nonlinearity to the right hand side and relying on elliptic regularity results, it can be shown that belongs to for every integer , provided that . In the following, we link to its associated state as the solution of (LABEL:eq:burg1) and we will indicate this explicitly by writing to emphasize that the state is generated by the right-hand side . The following property will be useful in the forthcoming sections.
Let be a sequence of functions which converges weakly to in satisfying , then the sequence of the corresponding associated states converges strongly to in .
The result is a straightforward consequence of the properties of the solutions of the Burgers equation and the compactness of the usual embeddings. ∎
2.2. Existence of solution for the optimal control problem
The arguments for proving existence of an optimal control are standard since is a nonempty, closed and convex set in .
In the following, will denote the set feasible pairs for , that is, those pairs such that (86b) is satisfied with . Note that is nonempty.
If the inequality (5) holds then the problem has a solution.
Despite the strict convexity of the objective functional and uniqueness of the solution of the state equation, uniqueness of the optimal control can not be guaranteed since is not necessarily convex.
3. Optimality conditions
In this section we shall derive first-order necessary and second-order sufficient conditions for local solutions of , both play an important role in the derivation of error estimates. Therefore, we make precise the notion of local minimum.
A pair will be referred as local optimal pair for if there exist positive reals and such that
For convenience, we introduce the following operator.
We define the operator by the relation
where denotes the dual pairing between and .
Note that indicates that is the weak solution of the state equation (86b) associated to the control .
In the next lemmas we study the differentiability of the operator .
Let . The operator given in Definition 2 has first and second derivatives given by:
respectively for any , and in accordingly, and for all .
The result follows from the linear properties of , and the scalar product in . ∎
3.1. First-order necessary conditions
The following first-order necessary conditions are derived in the spirit of .
Let be a local optimal pair for , then is a regular point for in the sense of .
The regular point condition of for the problem is achieved by noting that for every the linear equation
has a unique solution of the form , with and . ∎
Let be a solution of such that , then there exists an adjoint state such that the following optimality system is fullfilled:
Moreover, (7c) can be equivalently expressed in terms of the projection operator:
It is worth to point out that extra regularity of the optimal quantities can be deduced using standard elliptic regularity results from . Indeed, by taking and since and we have that . From the characterization of given by (8) and properties of the projection operator , it follows that . By bootstrapping arguments on the state equation, we have that solves Poisson’s equation with a right hand side in therefore, by elliptic regularity results, we conclude that . Furthermore, if is assumed in to be in then is also in . This high regularity of , however, can not be transferred to the optimal control because the projection operator.
For our forthcoming analysis, we introduce the Lagrangian defined by:
whose corresponding first and second derivatives (with respect to the first and second variable) at , are given by:
3.2. Second-order sufficient optimality conditions
The forthcoming analysis of Section 5.1 concerning the approximation of the optimal control problem by the finite element method, requires the formulation of second-order sufficient optimality conditions. By the nature of the nonlinearity of the Burgers equation, it shall be notice that the two norm-discrepancy does not occur in our formulation. In order to establish second-order sufficient conditions we introduce the critical cone. For , we define the set
The critical cone consists of those directions , such that
The next theorem states second-order sufficient conditions for . For a better presentation we will use the notation .
Let be a feasible pair for satisfying first-order necessary conditions formulated in Theorem 2, with adjoint state . In addition, suppose there are and , such that the coercivity property
is satisfied for all and all such that , then there exist constants and such that
holds for every and obeying .
We argue by contradiction by adapting the the proof of Theorem 4.1 in . Therefore, we assume the existence of a sequence in converging to , such that the sequence of their associated states converge to . Therefore, is an admissible pair which fulfills the relation
Let us define the sequence of directions , and the sequence , with , for every . Clearly is bounded, with and so is also bounded in ; this implies the existence of subsequences denoted again by and respectively, such that in and in . Moreover, since belongs to the closed and convex set (and therefore weakly closed) , it follows that also belongs to . Let us check that satisfies . From the definition of and Lemma 2, the pair satisfies:
Taking the limit in (15), by the convergence properties of and we see that the pair satisfies the linearized equation:
By applying the mean value theorem to the Lagrangian in (14), we obtain
On the other hand, we find that holds for every in view of the first-order necessary conditions expressed in (10). After passing to the limit and using the same convergence arguments it follows that
thus we have that .
Now, we show that . We recall that if then because is the unique solution of the linearized equation (16). By using the second-order Taylor expansion of the Lagrangian and having in mind (6b) and (9b) we get
From the definition of and and (10) the first term in the last inequality is nonnegative, therefore we have
Once again, since in and in then
4. Finite element approximation of the Burgers equation
This section is devoted to the approximation of Burgers equation by using the finite element method and the derivation of the corresponding error of convergence. Let be a positive integer, we define and a uniform mesh on the interval denoted by , which consists of subintervals: of for , such that and . We also introduce the finite dimensional space defined by
where is the space of polynomials of degree less or equal than one. Therefore, we define the discrete Burgers equation in as follows: given find such that
If is such that then, equation (24) has a unique solution such that
The proof is completely analogous to the proof in [13, Theorem 2.10]. ∎
Let us denote by the usual Lagrange interpolation operator such that for every , the element is the unique element in which satisfies for .
For convenience, we recall a well known result which establishes an estimate for the interpolation error cf. .
Let nonnegative integers and and . If the embeddings
hold, then there exists a constant independent of such that the following interpolation error is satisfied
where is the restriction of to an element of the discretization of the domain with dimension .
Moreover, Lemma 4 implies that
The proof for this result can be found in [4, Lemma 7]. We are interested in the error estimate for the approximation of the solution of the Burgers equation using linear finite elements, to this purpose we convent that denotes a generic constant which is positive and independent of .
Let be such that and such that . If denotes the corresponding solution of the discrete equation (24) with right-hand side ; then, the estimate
Since and satisfy equations (LABEL:eq:burg1) and (24) respectively, after subtracting both equations we get
In particular, if is an arbitrary element in , we choose in (29), resulting in
from which, we conclude that
Observe that the coefficient on the left-hand side is a positive number. Taking in (32) and using the fact that is a continuous operator, it follows that
Finally, the last inequality implies the desired estimate as follows
Let be such that and let and be the solutions of the equations (LABEL:eq:burg1) and (24) respectively. Then the estimate
In the process of deriving error estimates for the finite element approximation of the optimal control problem , we will need the following estimate in the –norm.
Let be such that and let and the solutions of the state equations (LABEL:eq:burg1) and (24) respectively. Then, the estimate