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Error estimates for a pointwise tracking optimal control problem of a semilinear elliptic equation

We consider a pointwise tracking optimal control problem for a semilinear elliptic partial differential equation. We derive the existence of optimal solutions and analyze first and, necessary and sufficient, second order optimality conditions. We devise two strategies of discretization to approximate a solution of the optimal control problem: a semidiscrete scheme where the control variable is not discretized -- the so-called variational discretization approach -- and a fully discrete scheme where the control variable is discretized with piecewise constant functions. For both solution techniques, we analyze convergence properties of discretizations and derive error estimates.

1. Introduction. In this work we are interested in the analysis and discretization of a pointwise tracking optimal control problem for a semilinear elliptic partial differential equation (PDE). This PDE-constrained optimization problem entails the minimization of a cost functional that involves point evaluations of the state; control constrains are also considered. To make matters precise, we let Ω ⊂ R d , with d ∈ {2, 3}, be an open, bounded, and convex polytope with boundary ∂Ω and D be a finite ordered subset of Ω with cardinality #D < ∞. Given a set of desired states {y t } t∈D ⊂ R and a regularization parameter α > 0, we define the cost functional The control bounds a, b ∈ R are such that a < b. Assumptions on the nonlinear function a will be deferred until section 2.2.
The study of a priori error estimates for finite element approximations of distributed semilinear optimal control problems has previously been considered in a number of works. To the best of our knowledge, the work [5] appears to be the first to provide error estimates for such a class of problems; control constraints are considered. Within a general setting, the authors consider the cost functional J(y, u) :=´Ω L(x, y(x), u(x))dx, where L satisfies the conditions stated in [5, assumption A2], and devise finite element techniques to solve the underlying optimal control problem. To be precise, the authors propose a fully discrete scheme on quasiuniform meshes that discretize the control variable with piecewise constant functions and the state and adjoint variables with piecewise linear functions. Assuming that Ω ⊂ R d , with d ∈ {2, 3}, is a convex domain with a boundary ∂Ω of class C 1,1 and that the mesh-size is sufficiently small, the authors derive a priori estimates for the error approximation of an optimal control variable in the L 2 (Ω)-norm [5, Theorem 5.1] and the L ∞ (Ω)-norm [5,Theorem 5.2]. Since the publication of [5], several additional studies have enriched the understanding of error estimates for semilinear PDE-contrained optimization problems. We refer the reader to [14] for references and also for an up-to-date discussion including linear approximation of the optimal control, the so-called variational discretization approach, superconvergence, post-processing schemes, and time dependent problems.
For the particular case a ≡ 0, there are several works available in the literature that provide a priori error estimates for finite element discretizations of (1.1)-(1.3). In two and three dimensions and utilizing that the associated adjoint variable belongs to W 1,r 0 (Ω), for every r < d/(d − 1), the authors of [19] obtain a priori and a posteriori error estimates for the so-called variational discretization approach; the state and adjoint equations are discretized with continuous piecewise linear functions. The a priori estimate for the error approximation of the control variable behaves as O(h 2−d/2 ) [19,Theorem 3.2]. Later, the authors of [9] analyze a fully discrete scheme that approximates the optimal state, adjoint, and control variables with piecewise linear functions and obtain a O(h) rate of convergence for the error approximation of the control variable in two dimensions [9,Theorem 5.1]. The authors also analyze the variational discretization scheme and derive error estimates [9,Theorem 5.2]. In [4], the authors invoke the theory of Muckenhoupt weights and weighted Sobolev spaces and analyze a numerical scheme that discretizes the control variable with piecewise constant functions; the state and adjoint equations are discretized as above. In two and three dimensions, the authors derive estimates for the error approximation of the optimal control variable; the one in two dimensions being nearly-optimal in terms of approximation [4,Theorem 4.3]. In three dimensions the estimate is suboptimal: O(h 1/2 | log h|). This has been recently improved to O(h| log h|) in [7,Theorem 6.6]. In addition, the authors of [7] provide, in two dimensions, a O(h 2 | log h| 2 ) error estimate for the variational discretization approach [7,Theorem 7.5] and a post-processing scheme [7,Theorem 7.12]. We finally mention the works [2,23,6] for extensions of the aforementioned results to the Stokes system and the work [31] for the analysis and discretization of an optimal control problem for a semilinear PDE with a control variable that corresponds to the amplitude of forces modeled as point sources.
In contrast to the aforementioned advances and to the best of our knowledge, this exposition is the first one that studies approximation techniques for a pointwise tracking optimal control problem involving a semilinear elliptic PDE. In what follows we list what, we believe, are the main contributions of our work: • Existence of an optimal control: Assuming that a = a(x, y) is a Carathéodory function that is monotone increasing and locally Lipschitz in y with a(·, 0) ∈ L 2 (Ω), we show that our control problem admits a solution (Theorem 3.1). • Optimality conditions: Under additional assumptions on a, we obtain first order optimality conditions in Theorem 3.3 and second order necessary and sufficient optimality conditions with a minimal gap in section 3.3. Since the cost functional of our problem involves point evaluations of the state, we have thatp ∈ W 1,r 0 (Ω) \ (H 1 0 (Ω) ∩ C(Ω)), where r < d/(d − 1). This requires a suitable adaption of the arguments available in the literature [14, section 6], [36]: the arguments in [14, section 6] utilize thatp ∈ W 2,p (Ω) with p > d.
• Convergence of discretization: We prove the existence of subsequences of global solutions of suitable discrete problems that converge to a global solution of the continuous control problem. We also prove that continuous strict local solutions can be approximated by local minima of discrete problems. • Error estimates: We obtain a nearly-optimal local error estimate in L ∞ for semilinear PDEs (Theorem 4.1). This estimate is instrumental to derive error estimates for the solution schemes that we propose (Theorems 6.1, 6.5, 6.8, and 6.10). The analysis involves estimates in L ∞ -norm and W 1,p -spaces, combined with having to deal with first and second order optimality conditions. This subtle intertwining of ideas is one of the highlights of this contribution. The outline of this manuscript is as follows. We set notation and introduce the functional framework we shall work with in section 2. In section 3, we analyze a weak version of the optimal control problem (1.1)-(1.3). In particular, we show existence of solutions and obtain first and second order optimality conditions. Section 4 presents two finite element discretizations for (1.1)-(1.3) and review some results related to the discretization of the state and adjoint equations. The convergence of discretizations is provided in section 5. In section 6, we derive estimates for the error approximation of an optimal control variable. We conclude in section 7 by providing a numerical example that illustrates the theory.
2. Notation and assumptions. Let us set notation and describe the setting we shall operate with.
2.1. Notation. Throughout this work d ∈ {2, 3} and Ω ⊂ R d is an open, bounded, and convex polytopal domain. If X and Y are Banach function spaces, we write X → Y to denote that X is continuously embedded in Y . We denote by · X the norm of X . Given r ∈ (1, ∞), we denote by r its Hölder conjugate, i.e., the real number such that 1/r + 1/r = 1. The relation a b indicates that a ≤ Cb, with a positive constant that depends neither on a, b nor on the discretization parameters. The value of C might change at each occurrence.

2.2.
Assumptions. We will operate under the following assumptions on the nonlinear function a. We notice, however, that some of the results that we present in this work are valid under less restrictive requirements. When possible we explicitly mention the assumptions on a that are needed to obtain a particular result.

State equation.
Here, we collect some facts on problem (1.2) that are well-known and will be used repeatedly. For f ∈ L q (Ω), with q > d/2, we introduce the following weak formulation of problem (1.2): Find y ∈ H 1 0 (Ω) such that We begin with the following result that states the well-posedness of problem (2.1) and further regularity properties for its solution y.
Theorem 2.1 (well-posedness and regularity). Let f ∈ L q (Ω) with q > d/2. Let a = a(x, y) : Ω × R → R be a Carathéodory function that is monotone increasing and locally Lipschitz in y a.e. in Ω. If Ω denotes an open and bounded domain with Lipschitz boundary and a(·, 0) ∈ L q (Ω), with q > d/2, then problem (2.1) has a unique solution y ∈ H 1 0 (Ω) ∩ L ∞ (Ω). If, in addition, Ω is convex and f, a(·, 0) ∈ L 2 (Ω), then , with a hidden constant that is independent of a and f .

Theorem 2.2 (Lipschitz property). Let Ω be an open and bounded domain with
Lipschitz boundary and let f 1 , f 2 ∈ L q (Ω) with q > d/2. Let a = a(x, y) : Ω×R → R be a Carathéodory function of class C 1 with respect to y such that (A.2) holds. Assume that, for every m > 0, |∂a/∂y(x, y)| ≤ C m for a.e. x ∈ Ω and y ∈ [−m, m]. If a(·, 0) ∈ L q (Ω), with q > d/2, then 3. The pointwise tracking optimal control problem. In this section, we analyze the following weak version of problem (1.1)-(1.3): Find subject to the monotone, semilinear and elliptic state equation Let a = a(x, y) : Ω × R → R be a Carathéodory function that is monotone increasing and locally Lipschitz in y with a(·, 0) ∈ L 2 (Ω). Since Ω is convex, Theorem 2.1 yields the existence of a unique solution y ∈ H 2 (Ω) ∩ H 1 0 (Ω) of problem (3.2). We immediately notice that, in view of the continuous embedding H 2 (Ω) → C(Ω), which holds because d ∈ {2, 3}, point evaluations of y in (1.1) are well-defined.
3.1. Existence of optimal controls. Let us introduce the control to state map S : L 2 (Ω) → H 1 0 (Ω) ∩ H 2 (Ω), which given a control u associates to it the state y that solves (3.2), and define j : L 2 (Ω) → R by j(u) := J(Su, u). With these ingredients at hand, the existence of an optimal state-control pair (ȳ,ū) is as follows.
3.2. First order necessary optimality conditions. Since the optimal control problem (3.1)-(3.2) is not convex, we discuss optimality conditions in the context of local solutions in L 2 (Ω) [36, section 4.4.2]. In this section, we formulate first order necessary optimality conditions. To accomplish this task, we begin by analyzing differentiability properties of the control to state operator S.
We are now in position to present first order necessary optimality conditions.
The next result further illustrates regularity properties ofp; see [12,Theorem 3.4]. To present it, we follow [7] and introduce Observe that, if t ∈ D \ E, then the coefficientȳ(t) − y t in (3.6) vanishes.
, be the solution to (3.6) with y =ȳ = Sū. Then, Here, B t ⊂ Ω denotes a suitable open ball centered at t ∈ E of strictly positive radius.
We now present a regularity results for a locally optimal controlū. Proof. The proof of the fact thatū ∈ H 1 (Ω) follows directly from the projection formula (3.12) and [16,Lemma 3.3]. The C 0,1 (Ω)-regularity ofū can be found in [12,Theorem 3.4] and [18,Theorem 4.2]. For the sake of completeness, we briefly present a proof. If E = ∅, the desired regularity follows. Let us thus assume that E = ∅ and let t ∈ E. The following asymptotic behavior holds as Ω x → t: [33, eq. (4.9)] and [27,Theorem 3.3]. Here, C 2 ∈ R. As a result, for every M > 0, there exists a ball B t of strictly positive radius centered at t such that |p(x)| > M for x ∈ B t . Since this argument holds for each t ∈ E, we conclude that, for every Set M sufficiently large and invoke the projection formula (3.12) to deduce that, for x ∈ ∪ t∈E B t , we have either u(x) = a orū(x) = b. The C 0,1 (Ω)-regularity ofū thus follow from (3.14).
3.3. Second order optimality conditions. In this section, we formulate necessary and sufficient second order optimality conditions.
We begin our studies with the following result.
where p solves (3.6) and Proof. The fact that j is of class C 2 is an immediate consequence of the differentiability properties of the control to state map S given in Theorem 3.2. It thus suffices to derive (3.16) and (3.17). To accomplish this task, we begin with a basic computation, which reveals that, for every u, v 1 , v 2 ∈ L 2 (Ω), we have where z, z v1 , z v2 ∈ H 2 (Ω) ∩ H 1 0 (Ω) are as in the statement of Theorem 3.2. Set w = z in (3.6) and invoke a similar approximation argument to that used in the proof of Theorem 3.3, that essentially allows us to set w = p in (3.4), to obtain .
A collection of the previous bounds yield We now estimate II t in (3.19). Combining the estimate (3.22) with a stability estimate for (3.3), it immediately follows that . We conclude (3.17) by replacing the obtained estimates for I and II t into (3.19).
We are now in position to present second order necessary and sufficient optimality conditions. While it is fair to say that for distributed semilinear optimal control problems such a theory is well-understood, our main source of difficulty here is that, under the most likely scenario where Theorem 3.7 (second order necessary optimality condition). Ifū ∈ U ad denotes a locally optimal control for problem Proof. The proof essentially follows the same arguments as those elaborated in the proof of [14,Theorem 23]. For brevity, we skip details.
We now derive, in the spirit of [14, Theorem 23], a sufficient optimality condition with a minimal gap with respect to the necessary one stated in Theorem 3.7.
Proof. We follow [14,Theorem 23] and proceed by contradiction. Assume that (3.26) does not hold. Hence, for any k ∈ N, there is an element u k ∈ U ad such that Define ρ k := u k −ū L 2 (Ω) and v k := (u k −ū)/ρ k . We assume that (up to a subsequence if necessary) v k v in L 2 (Ω). In what follows we prove that v ∈ Cū and that v = 0. Since the set of elements satisfying condition (3.25) is closed and convex in L 2 (Ω), it is weakly closed. Consequently, v satisfies (3.25) as well. To verify the remaining condition in (3.24), we invoke the mean value theorem and (3.27) to arrive at whereũ k =ū + θ k (u k −ū) and θ k ∈ (0, 1). Defineỹ k := Sũ k andp k as the unique solution to (3.6) with y =ỹ k . Sinceũ k →ū in L 2 (Ω) as k ↑ ∞, an application of On the other hand, in view of (3.7) we obtain´Ωp( Proceeding as in the proof of [14,Theorem 23], we can thus conclude that v ∈ Cū. We now prove that v = 0. To accomplish this task, we invoke Taylor's theorem, the inequality in (3.27), and j (ū)(u k −ū) ≥ 0, for every k ∈ N, to arrive at Letẑ v k and z v be the solutions to (3.3) with forcing terms v k and v, respectively. Invoke (3.16) and write Since In addition, we have lim inf k j (û k )v 2 k ≤ 0 yields α ≤ 0. This is a contradiction and concludes the proof. Let us introduce C τ u := {v ∈ L 2 (Ω) satisfying (3.25) and v(x) = 0 if |p(x)| > τ } and present the following result.
Proof. The proof essentially follows the same arguments as those elaborated in the proof of [14,Theorem 25]. For brevity, we skip details.

Finite element approximation.
We now introduce the discrete setting in which we will operate [8,20,21]. We denote by T h = {T } a conforming partition, or mesh, ofΩ into closed simplices T with size h T = diam(T ). Define h := max T ∈T h h T . We denote by T = {T h } h>0 a collection of conforming and quasi-uniform meshes T h .
Given a mesh T h ∈ T, we define the finite element space In the following sections we present finite element approximations of the state and adjoint equations and the optimal control problem (3.1)-(3.2).

Discrete state equation.
Let f ∈ L 2 (Ω). We define the Galerkin approximation of the solution y to problem (2.1) by Let a = a(x, y) : Ω × R → R be a Carathéodory function that is monotone increasing and locally Lipschitz in y, a.e. in Ω, with a(·, 0) ∈ L 2 (Ω). An application of Brouwer's fixed point theorem yields the existence of a unique solution to (4.2). In addition, we have ∇y h L 2 (Ω) f − a(·, 0) L 2 (Ω) ; see [30,Theorem 3.2] and [14, section 7]. We now provide a local regularity result for the solution y of problem (2.1) that will be of importance to derive error estimates. Let Since y can be seen as the solution to we can invoke [7, Lemma 4.2] to deduce that where C t behaves as Ct, with C > 0, as t ↑ ∞. Notice that we have further assumed that a satisfies a(·, 0) ∈ L t (Ω 0 ), which, since a = a(x, y) is locally Lipschitz in y, implies that a(·, y) L t (Ω0) f −a(·, 0) L 2 (Ω) + a(·, 0) L t (Ω0) .  = a(x, y) : Ω × R → R be a Carathéodory function that is monotone increasing and locally Lipschitz in y with a(·, 0) ∈ L 2 (Ω). Let y ∈ H 1 0 (Ω) ∩ H 2 (Ω) and y h ∈ V h be the solutions to (2.1) and (4.2), respectively, with f ∈ L 2 (Ω). If h is sufficiently small, we thus have the following error estimates: Let Ω 1 Ω 0 Ω with Ω 0 smooth. If, in addition, f, a(·, 0) ∈ L ∞ (Ω 0 ), then In all estimates the hidden constant is independent of h.

Discrete adjoint equation.
Let u ∈ U ad and {y t } t∈D ⊂ R. We define the Galerkin approximation to the adjoint equation (3.6) by denotes the solution to (3.2) with u ∈ U ad , i.e., y = Su. Standard results yield the existence and uniqueness of q h ∈ V h .
We present the following error estimates.
Proof. Define a(x) = ∂a/∂y(x, y(x)), where y = Su and u ∈ U ad . Since a ∈ L ∞ (Ω) and a(x) ≥ 0 for a.e. x ∈ Ω, we can apply [10,Theorem 3] in combination with Theorem 2.1 to deduce (4.8). The proof of the estimate (4.9) follows similar arguments as the ones developed in [22] and [7,Lemma 5.3]. Let w be the solution to where we have used that p and q h solve (3.6) and (4.7), respectively. Since D Ω, similar arguments to the ones used in the proof of (4.6) yield This concludes the proof.
The next result will be of importance to obtain an error estimate for a solution technique of (3.1)-(3.2) based on the variational discretization approach.  Here, B t ⊂ Ω denotes a suitable open ball centered at t ∈ D of strictly positive radius.
Proof. The proof of (4.10) follows from a combination of the arguments elaborated in the proof of Theorem 4.2 and [7, Lemma 5.5]. For brevity, we skip details.
Let y h ∈ V h be the unique solution to the discrete problem (4.2) with f = u h ∈ U ad ⊂ L ∞ (Ω) and define the discrete variable p h ∈ V h as the unique solution to We present the following error estimate, which will be of importance to perform an a priori error analysis for a suitable discretization of our optimal control problem.
Theorem 4.4 (auxiliary error estimate). Let a be as in the statement of Theorem 4.2. Let u, u h ∈ L 2 (Ω) be such that u L 2 (Ω) ≤ C and u h L 2 (Ω) ≤ C, for every h > 0, where C > 0. Let p and p h be the solutions to (3.6) and (4.11) with y = y(u) and y h = y h (u h ), respectively. Then, we have the following error estimate: with a hidden constant that is independent of h.

Discretization of the control problem.
We propose two strategies to discretize (3.1)-(3.2): a semidiscrete approach where the control is not discretizedthe so-called variational discretization approach -and a fully discrete approach where the control variable is discretized with piecewise constant functions.
4.3.1. A fully discretization scheme. We propose the following fully discrete approximation of problem (3.1)-(3.2): Find min J(y h , u h ) subject to (4.20) The existence of a solution follows from the compactness of U ad,h and the continuity of J. Let us introduce the discrete control to state map S h : (4.20), and the reduced cost functional j h (u h ) := J(S h u h , u h ). With these ingredients at hand, first order optimality conditions for the fully discrete optimal control problem reads as follows: Ifū h denotes a local solution, then wherep h ∈ V h solves the discrete problem (4.11) with y h replaced byȳ h := S hūh .

The variational discretization approach.
In this section, we propose a semidiscrete scheme that is based on the so-called variational discretization approach [25]. This scheme discretizes only the state space (the control space is not discretized) and induces a discretization of optimal controls by projecting the optimal discrete adjoint state into the admissible control set U ad .
The scheme is defined as follows: Find min J(y h , u) subject to and the control constraints u ∈ U ad . The existence of a discrete solution and first order optimality conditions for the semidiscrete scheme follow standard arguments. In particular, ifū denotes a local solution, then wherep h ∈ V h solves the discrete problem (4.11) with y h replaced byȳ h . Here, y h denotes the solution to (4.22) with u replaced byū. Notice that, in view of the variational inequality (4.23), the following projection formula holds [36, section 4.6]: Sinceū implicitly depends on h, in what follows we will adopt the notationū h .

Convergence of discretizations.
We begin with the following error bounds, the proofs of which can be found in [14,Lemmas 37 and 38].
In addition, if u h u in L s (Ω) as h ↓ 0, with s > d/2, then y h → y in H 1 0 (Ω) ∩ C(Ω) as h ↓ 0 and j(u) ≤ lim inf h↓0 j h (u h ).

The fully discrete scheme: convergence of discretizations.
In what follows we provide a convergence result that, in essence, guarantees that a sequence of discrete global solutions {ū h } h>0 contains subsequences that converges, as h ↓ 0, to global solutions of problem (3.1)-(3.2).
Proof. Since, for every h > 0,ū h ∈ U ad,h ⊂ U ad , the sequence {ū h } h>0 is uniformly bounded in L ∞ (Ω). Then, there exists a nonrelabeled subsequence such that In what follows, we prove thatū ∈ U ad is a solution to the optimal control problem (3.1)-(3.2) and that the convergence results in (5.1) hold.
Let h ≤ h ε and letū h be a global solution of min{j h (u h ) : u h ∈ U ad,h ∩ B ε (ū)}. We proceed as in the proof of Theorem 5.2 to conclude the existence of a subsequence of {ū h } h≤hε such that it converges strongly in L 2 (Ω) to a solution of min{j(u) : u ∈ U ad ∩ B ε (ū)}. Since the latter problem admits a unique solutionū, we must havē u h →ū in L 2 (Ω) as h ↓ 0. This implies that the constraintū h ∈ B ε (ū) is not active for small h. As a result,ū h is a local solution of the fully discrete scheme and the convergence properties in (5.1) hold.
6. Error estimates. In this section, we derive error estimates for the fully and semidiscrete schemes introduced in sections 4.3.1 and 4.3.2, respectively. 6.1. Error estimates for the fully discrete scheme. Let {ū h } h>0 ⊂ U ad,h be a sequence of local minima of the fully discrete control problems such thatū h →ū in L 2 (Ω), as h ↓ 0, whereū ∈ U ad is a local solution of (3.1)-(3.2); see Theorems 5.2 and 5.3. The main goal of this section is to derive an estimate forū −ū h in L 2 (Ω).
We follow [17,11] and proceed by contradiction. Let us assume that {ū h } h>0 converges toū as h ↓ 0 and that (6.1) does not hold. We thus have, for every k ∈ N, h k > 0 such that ū −ū h k L 2 (Ω) > kh k | log h k |, and thus {h k } k∈N ⊂ R + such that To prove (6.1) we need some preparatory results. Let us assume that, in addition, (6.1) is false. Letū ∈ U ad satisfies the second order optimality condition (3.30). Then, there exists h † > 0 such that where α is the regularization parameter and µ denotes the constant appearing in (3.31).
Proof. Since (6.1) is false, there exists a sequence {h k } k∈N such that the limits in (6.2) hold. In an attempt to simplify the exposition of the material, in what follows, we will omit the subindex k, i.e., we denote u h k = u h . Observe that h ↓ 0 as k ↑ ∞.
Proof. We proceed on the basis of two steps.
Step 1. The goal of this step is to obtain (6.6). To accomplish this task, we begin with a basic computation which reveals that j (u 1 )v = (p u1 + αu 1 , v) L 2 (Ω) , where for all w h ∈ V h . In (6.8) the variableŷ h ∈ V h corresponds to the solution to (4.20) with u h replaced by u 1 . Definep ∈ W 1,r 0 (Ω) as the unique solution to (∇w, ∇p) L 2 (Ω) + ∂a ∂y (·,ŷ h )p, w Here, r ∈ [2d/(d+2), d/(d−1)). Notice thatp h ∈ V h corresponds to the finite element approximation ofp within V h . We also notice the following stability estimate forp: With all these continuous and discrete variables at hand, we can write To estimate the term I we define ζ := p u1 −p ∈ W 1,r 0 (Ω) and observe that (∇w, ∇ζ) L 2 (Ω) + ∂a ∂y (·, y u1 )ζ, w for all w ∈ W 1,r 0 (Ω). Here, g := [ ∂a ∂y (·,ŷ h ) − ∂a ∂y (·, y u1 )]p. An inf-sup condition, which follows from [10, Theorem 1], yields (6.11) ∇ζ L r (Ω) Let us concentrate on g L 2 (Ω) . Let Λ 1 , Ω 0 be smooth domains such that Ω 1 Λ 1 Ω 0 Ω and D ⊂ Ω 1 . The basic identity g 2 combined with the error bounds derived in Theorem 4.1 yield The Sobolev embedding W 1,r 0 (Ω) → L 2 (Ω) and the fact that u 1 ∈ U ad reveal that (6.12) g 2 where the hidden constant is independent of the involved continuous and discrete variables but depends on the continuous optimal control problem data. We now invoke Theorem 3.4 to conclude that p L ∞ (Ω\Λ1) is uniformly bounded. On the other hand, the stability estimate (6.9) and analogous arguments to the ones that lead to (4.15) allows us to conclude that ∇p L r (Ω) ≤ C, where C depends on {y t } t∈D , a, a, and b. We thus invoke (6.12) to arrive at g L 2 (Ω) h 2 | log h| 2 . This bound, estimate (6.11), and the local estimate (4.6), yield The control of II in (6.10) follows immediately from the error estimate (4.9): Upon combining (6.10), (6.13), and (6.14), we obtain the estimate (6.6).
Step 2. We now obtain (6.7). From the previous step, we have that j h (u 1 )v = (p h + αu 1 , v) L 2 (Ω) , wherep h ∈ V h solves (6.8). On the other hand, similar arguments yield j h (u 2 )v = (p h +αu 2 , v) L 2 (Ω) , wherep h ∈ V h is the unique solution to the problem Here,ỹ h ∈ V h corresponds to the solution to (4.20) with u h replaced by u 2 . Therefore, It thus suffices to bound p h −p h L 2 (Ω) . To accomplish this task, we define ξ ∈ W 1,r 0 (Ω) : (∇w, ∇ξ) L 2 (Ω) + ∂a ∂y (·,ŷ h )ξ, w We also define ξ h :=p h −p h ∈ V h and immediately observe that ξ h corresponds to the finite element approximation of ξ within V h . We thus invoke basic estimates, (4.8), and a stability estimate for the problem that ξ solves to arrive at . This estimate, in light of assumption (A.3), immediately yields We now bound ŷ h −ỹ h L ∞ (Ω) . Let us first recall that y ui solves (3.2) with u = u i , where i ∈ {1, 2}, and thatŷ h andỹ h correspond to the finite element approximations of y u1 and y u2 , respectively. Since a(·,ŷ h ) − a(·,ỹ h ) = ∂a ∂y (·, y h )(ŷ h −ỹ h ), where y h =ỹ h + θ h (ŷ h −ỹ h ) and θ h ∈ (0, 1), we deduce thatŷ h −ỹ h ∈ V h solves the problem Define η ∈ H 1 0 (Ω) as the solution to (∇η, ∇v) L 2 (Ω) + ∂a ∂y (·, y h )η, v Noticing thatŷ h −ỹ h ∈ V h corresponds to the finite element approximation of η within V h , we conclude, in view of estimate (4.5), that Replace this bound into (6.16) and the obtained one into (6.15) to conclude We observe that similar arguments to the ones used to derive (4.15) yield p h L 2 (Ω) u 2 − a(·, 0) L 2 (Ω) + t∈D |y t | C, where C > 0. This concludes the proof.
Proof. Define, for each T ∈ T h , I T :=´Tp(x)dx and u * h ∈ U h by We recall thatp =p + αū. In view of the fact thatū ∈ C 0,1 (Ω), which follows from Theorem 3.5, there exists h > 0 such that |ū(x 1 ) −ū(x 2 )| ≤ (b − a)/2 for every h < h and x 1 , x 2 ∈ T . This implies, in particular, that, for each T ∈ T h ,ū do not take both values a and b in T . Therefore, with (3.23) at hand, we deduce that either, for a.e x ∈ T ,p(x) ≥ 0 orp(x) ≤ 0. Consequently, we have that I T = 0 if and only ifp(x) = 0 for a.e. x ∈ T , and that, if I T = 0,p(x)/I T ≥ 0 for a.e. x ∈ T . From this fact, and in view of the generalized mean value theorem, we conclude the existence of x T ∈ T such that u * h | T =ū(x T ). Since u * h ∈ U h , we have thus obtained that u * h ∈ U ad,h . Now, let T ∈ T h . We estimate ū − u * h L 2 (T ) as follows: We finally observe that (6.18) immediately yields This concludes the proof.
Proof of Theorem 6.1. Adding and subtracting the term j h (ū h )(ū −ū h ) in the right hand side of inequality (6.3) yield Invoke inequality (6.6) in conjunction with that fact thatū,ū h ∈ U ad to arrive at (6.20) ū where C > 0. We now estimate [j (ū) − j h (ū h )](ū −ū h ). To accomplish this task, we set u =ū h in (3.5) and u h = u * h in (4.21) to obtain Adding these inequalities we arrive at . We utilize that u * h is such that j (ū)(u * h −ū) = 0, which follows from Lemma 6.4, to obtain We thus apply estimates (6.6) and (6.7) to obtain Invoke Young's inequality and the estimate of Lemma 6.4 to arrive at for every h < h . Here,C > 0. Finally, replacing estimate (6.22) into (6.20) we conclude (6.1). This, which contradicts (6.2), concludes the proof. We now improve, in two dimensions, the error estimate of Theorem 6.1.
Theorem 6.5 (improved error estimate). Let d = 2. In the framework of Theorem 6.1, we have the following optimal error estimate: where the hidden constant is independent of h.
Proof. We proceed, again, by contradiction and assume that {ū h } h>0 converges toū as h ↓ 0 and (6.23) does not hold. Hence, we can find, for every k ∈ N, h k > 0 such that ū −ū h k L 2 (Ω) > kh k , and thus a sequence {h k } k∈N ⊂ R + such that (6.24) lim Withing this setting, the proof of estimate (6.3) follows verbatim the arguments used in the proof of Lemma 6.2 upon replacing (6.2) by (6.24). The next ingredient is To obtain (6.25), we follow the arguments in Lemma 6.3: it suffices to bound the terms I and II; I being already controlled. Observe that, since we are in two dimensions, the error estimate (4.8) yields error bound To obtain (6.23) we thus proceed as in the proof of Theorem 6.1. Invoke (6.19) and (6.25) . We now utilize (6.21), (6.25), and (6.7) to derive the error estimate In view of Lemma 6.4, and Young's inequality, we arrive at ū −ū h L 2 (Ω) h for every h sufficiently small. This bound contradicts (6.24) and concludes the proof. Remark 6.6 (d = 3). We notice that, the use of estimate (4.8) in three dimensions leads to the deteriorated bound II ≤ p −p h L 2 (Ω) v L 2 (Ω) h 1 2 v L 2 (Ω) . 6.2. Error estimates for the variational discretization approach. Let {ū h } h>0 ⊂ U ad be a sequence of local minima of the semidiscrete optimal control problems such thatū h →ū in L 2 (Ω), as h ↓ 0, whereū ∈ U ad is a local solution of (3.1)-(3.2); see Theorems 5.2 and 5. 3. In what follows, we derive an error estimate for ū −ū h L 2 (Ω) .
The following result is instrumental.
where α is the regularization parameter and µ denotes the constant appearing in (3.31).
Since v satisfies (3.25), we obtainp(x)v(x) ≥ 0. Therefore, (6.27) allows us to conclude that´Ω |p(x)v(x)|dx =´Ωp(x)v(x)dx ≤ 0. Consequently, ifp(x) = 0, then v(x) = 0 for a.e. x ∈ Ω. This allows us to conclude that v ∈ Cū. The rest of the proof follows the same arguments developed in the proof of Lemma 6.4. For the sake of brevity, we skip details.
Lemma 6.9 (ū(x) =ū h (x) on ∪ t∈E B t ). Let d = 2. For any M > 0 there exists h M such that |p h (x)| > M for every x ∈ ∪ t∈E C t and h < h M . Here, E is defined in (3.13) and C t ⊂ Ω denotes a suitable open ball centered at t ∈ E of strictly positive radius. This, in particular, implies thatū(x) =ū h (x) for every x ∈ ∪ t∈E B t ; B t ⊂ Ω being defined similarly.
Theorem 6.10 (improved error estimate). Let d = 2. Assume that E = D. In the framework of Theorem 6.1, we have the following quasi-optimal error estimate: with a hidden constant that is independent of h.
In view of (6.29) and (6.31), we immediately arrive at the desired estimate.
Remark 6.11 (improved error estimate when E D). We notice that, if E D, the results of Lemma 6.9 allow us to control the term II, within the proof of Theorem 6.10, in a slightly different way: We would like now to apply (4.10). Unfortunately, estimate (4.10) yields a control on Ω \ ∪ t∈DBt and Ω \ ∪ t∈DBt Ω \ ∪ t∈EBt . We thus proceed as follows: where we have used estimate (4.10). To control e p :=p −p h on ∪ t∈D\EBt , we invoke similar arguments to the ones that yield (4.10), the fact that y t =ȳ(t) for t ∈ D \ E, the Lipschitz property (2.2), and the error estimate (4.5). These arguments yield e p 2 L 2 (Σ) h 2 | log h| 2 e p L 2 (Σ) + h e p L 2 (Σ) ȳ h −ŷ L ∞ (Ω) + ŷ −ȳ L ∞ (Ω) where Σ = ∪ t∈D\E B t . Here,ŷ ∈ H 1 0 (Ω) denotes the unique solution to (3.2) with u replaced byū h . In view of (6.28) we can thus deduce that II h 2 | log h| 2 ū−ū h L 2 (Ω) .
7. Numerical example. In this section, we conduct a numerical experiment that illustrates the performance of the fully and semidiscrete schemes of sections 4.3.1 and 4.3.2, respectively, when are used to approximate the solution to (1.1)-(1.3). The numerical experiment has been carried out with the help of a code that was implemented using C++. All matrices have been assembled exactly and global linear systems were solved using the multifrontal massively parallel sparse direct solver (MUMPS) [3]. The right hand sides and the approximation errors were computed by a quadrature formula which is exact for polynomials of degree nineteen (19).
For a given mesh T h , we seek (ȳ h ,p h ,ū h ) ∈ V h × V h × U ad,h and (ȳ h ,p h ,ū h ) ∈ V h × V h × U ad that solve the fully and semidiscrete problems, respectively. We solve the fully discrete scheme on the basis of a primal-dual active set strategy [36, section 2.12.4] combined with a fixed point algorithm. The semidiscrete problem is solved by using a semi-smooth Newton method.
The numerical example. We set Ω = (0, 1) 2 , a(·, y) = y In the absence of an exact solution, we calculate the error committed in the approximation of the optimal control variable by taking as a reference solution the discrete optimal control obtained on a fine triangulation T h : the mesh T h is such that h ≈ 10 −3 . In Figure 7.1, we observe that optimal experimental rates of convergence, in terms of approximation, are attained: O(h) for the fully discrete scheme and O(h 2 ) for the semidiscrete scheme. These rates are in agreement with Theorems 6.5 and 6.10.