Error analysis for global minima of semilinear optimal control problems

In [1] we consider an optimal control problem subject to a semilinear elliptic PDE together with its variational discretization, where we provide a condition which allows to decide whether a solution of the necessary first order conditions is a global minimum. This condition can be explicitly evaluated at the discrete level. Furthermore, we prove that if the above condition holds uniformly with respect to the discretization parameter the sequence of discrete solutions converges to a global solution of the corresponding limit problem. With the present work we complement our investigations of [1] in that we prove an error estimate for those discrete global solutions. Numerical experiments confirm our analytical findings.


Introduction
In this work we are concerned with the error analysis of a variational discretization of the control problem and the pointwise constraints where the precise assumptions on the data of the problem will be given in Section 2.1. In [1] the authors considered the same class of problems and established a sufficient condition for the global minima of (P) assuming particular types of growth conditions for the nonlinearity φ. The same result was established for the variational discrete counterpart of (P), and it was shown that a sequence of the computed discrete global minima converges to a global minimum of the continuous control problem but without discussing the corresponding rate of convergence. Hence, our aim in this study is to investigate this convergence rate. The organization of the paper is as follows: in § 2.1 we formulate the control problem and give the exact assumptions on the data. In § 2.2 and in § 2.3 we review the results concerning the state equation and the control problem (P), respectively. The variational discretization of (P) is considered in § 2.4 while § 3 is devoted to the error analysis. Finally, in § 4 we verify our theoretical findings by a numerical example.
Before starting, we give a short list of literature considering the problem (P). For a broad overview, we refer the reader to the references of the respective citations. In [4] the problem (P) is studied when the controls are of boundary type, and the necessary first order conditions are established. Compare [3] where the function φ is linear, and [8] where the pointwise constraints are imposed on the gradient of the state.
The regularity of the optimal controls of (P) and their associated multipliers are investigated in [12] and [11], where also the sufficient second order conditions are discussed. Compare [9,6,7] for second order conditions when the set K contains finitely/infinitely many points, and [13] for the role of those conditions in PDE constrained control problems.
Finite element discretization of problem (P) under more general setting is studied in [10], and in [19] where a wider class of perturbations are considered. The convergence of the discrete solutions to the continuous solutions is verified there but without rates. However, when the set K contains finitely many points, convergence rates are established in [23] for finite dimensional controls, and in [5] for control functions. Only in [25] error analysis is studied for general pointwise state constraints in K. There, Pfefferer at al. prove an error estimate for discrete solutions in the vicinity of a local solution which satisfies a quadratic growth condition. Error analysis for linear-quadratic control problems can be found in e.g. [11], [14] and [24]. A detailed discussion of discretization concepts and error analysis in PDE-constrained control problems can be found in [20,21] and [17,Chapter 3].
2 Problem Setting and discretization

Assumptions
• Ω ⊂ R 2 is a bounded, convex and polygonal domain.
• K is a (possibly empty) compact subset of Ω.
• φ : R → R is of class C 2 and monotonically increasing.
• There exist r > 1 and M ≥ 0 such that where φ and φ denote the first and second derivative of φ, respectively.

The State Equation
Recall that a function y ∈ H 1 0 (Ω) is called a weak solution of (1), (2) if Theorem 2.1 For every u ∈ L 2 (Ω) the boundary value problem (1), (2) admits a unique weak solution y ∈ H 1 0 (Ω) ∩ H 2 (Ω). Moreover, there exists c > 0 such that Proof. The existence and uniqueness of the solution y in H 1 0 (Ω) follows from the monotone operator theorem. Using the method of Stampacchia one can show, in addition, that y ∈ L ∞ (Ω). Utilizing the boundedness of y and the properties of the nonlinearity φ, one can show y ∈ H 2 (Ω) and the estimate (5) using the regularity results from [16,Chapter 4]. For a detailed proof compare for instance [4].
In the light of Theorem 2.1, we introduce the control-to-state operator such that y := G(u) is the solution to (4) for a given u ∈ L 2 (Ω).
Lemma 2.2 Let G be the mapping introduced in (6). Then there exists c > 0 depending only on Ω such that Proof. Given u, v ∈ L 2 (Ω) let y u := G(u) and y v := G(v). Using Poincaré's inequality, the monotonicity of φ and (4) we have which implies the result.
Proof. Defining again y u := G(u), y v := G(v) we infer from Theorem 2.1 and the continuous embedding Using a standard a-priori estimate, the Lipschitz continuity of φ on bounded sets and Lemma 2.2 we infer that where L(m) is a constant depending on m. This completes the proof.

The Optimal Control Problem (P)
Using the control-to-state operator G defined in (6), the reduced form of our optimal control problem reads It is well-known that (P) admits at least one solution provided that a feasible point exists (compare [4]). Moreover, if a solution of (P) satisfies some constraint qualification, then one can guarantee the existence of a multiplier associated with the pointwise state constraints and the necessary first order conditions can be established. A typical constraint qualification for a local solutionū of problem (P) is the linearized Slater condition which reads: there exist u 0 ∈ U ad and δ > 0 such that The next result is a consequence of [4, Theorem 5.2].
Theorem 2.4 Letū ∈ U ad be a local solution of problem (P) satisfying (7). Then there existp ∈ W 1,s 0 (Ω) for 1 < s < 2 and a regular Borel measure Note that in view of (10)ū is the L 2 -projection of − 1 αp onto U ad so that Sincep ∈ W 1,s (Ω) for 1 < s < 2 it follows from [22, Corollary A.6] that u ∈ W 1,s (Ω) for 1 < s < 2 as well. Furthermore, it is well known that the multiplierμ associated with the pointwise state constraints is concentrated at the points in K where the state constraints are active. We state this more precisely in the next proposition whose proof can be found in [11]. Compare also the proof in [3] when the bounds y a , y b are constant functions.
We note that the problem (P) is in general nonconvex since the state equation is not linear. In other words, the problem (P) can have several solutions. A decision of which of these solution is a global minimum proves difficult in general. However, it is shown in [1] that if the nonlinearity φ of the state equation enjoys certain growth conditions, namely (3), then one can establish a condition that helps to decide if a given point satisfying the first order conditions is a global minimum. We state this condition of global optimality in the next result, but before that we first need to introduce the following constant: Here, q := 3r−2 r−1 , ρ := r+q rq , while M and r appear in (3). Furthermore, C q is an upper bound on the optimal constant in the Gagliardo-Nirenberg inequality For sharp upper bounds for the constant C, see for instance [1,Theorem 7.3].
thenū is a global minimum for Problem (P). If the inequality (13) is strict, thenū is the unique global minimum.

Variational Discretization
Let T h be an admissible triangulation of the polygonal domain Ω ⊂ R 2 with Here h := max T ∈T h diam(T ) is the maximum mesh size, while diam(T ) stands for the diameter of the triangle T . We introduce the following spaces of linear finite elements: The Lagrange interpolation operator I h is defined by where {x 1 , . . . , x n } denote the nodes in the triangulation T h and {φ 1 , . . . , φ n } are the basis functions of the space X h which satisfy φ i (x j ) = δ ij . The finite element discretization of (4) reads: for a given u ∈ L 2 (Ω), find Using the monotonicity of φ and the Brouwer fixed-point theorem one can show that (14) admits a unique solution y h ∈ X h0 . Hence, analogously to (6), we introduce the discrete control-to-state operator such that y h := G h (u) is the solution of (14). The variational discretization (see [18]) of Problem (P) reads: with the set of nodes In an analogous way to that of problem (P), one can show that (P h ) admits at least one solution, denoted byū h , provided that a feasible point exists. In practice one calculates candidates for solutions of (P h ) by solving the system of necessary first order conditions which reads: As in the continuous case, there exist multipliersp h andμ j ∈ R, x j ∈ N h solving (16)- (19) provided that the local solutionū h satisfies the linearized Slater condition, that is, there exist u 0 ∈ U ad and δ > 0 such that It will be convenient in the upcoming analysis to associate with the multipliers (μ j ) xj ∈N h from the system (16)- (19) the measureμ h ∈ M(Ω) defined bȳ where δ xj is the Dirac measure at x j . We can easily deduce from (19) the following result about the support of the measureμ h .
Proposition 2.7 Letμ h ∈ M(Ω) be the measure introduced in (21) satisfying (19). Then there holds Analogously to Theorem 2.6, we have the next theorem about global solutions of problem (P h ). The proof can be found in [1].
thenū h is a global minimum for Problem (P h ). If the inequality (22) is strict, thenū h is the unique global minimum.

Error Analysis
Let {T h } 0<h≤h0 be a sequence of admissible triangulations of Ω. We assume that the sequence {T h } 0<h≤h0 is quasi-uniform in the sense that each T ∈ T h is contained in a ball of radius γ −1 h and contains a ball of radius γh for some γ > 0 independent of h. In addition we make the following assumption concerning the set K: In what follows we consider a sequence (ū h ,ȳ h ,p h ,μ h ) 0<h≤h1 of solutions of (16)- (19) satisfying for some κ ∈ (0, 1) that is independent of h. We immediately infer from Theorem 2.8 thatū h is the unique global minimum of (P h ) and we are interested in the convergence properties of these solutions as h → 0. It is shown in [1] (see Theorem 4.2 and its proof) that there existū ∈ U ad ,p ∈ L q (Ω) andμ ∈ M(K) such thatū and (ū,ȳ = G(ū),p,μ) is a solution of (8)- (11). Since Theorem 2.6 implies thatū is the unique global optimum of (P). The aim in the remaining part of this paper is to prove error estimates forū h −ū and the corresponding optimal statesȳ h −ȳ. Our main results read: Theorem 3.1 Suppose that (23) holds and letū h ,ū be the unique global minima of (P h ) and (P) respectively. Then we have for any 1 < s < 2 that Before we start presenting the proof of this result we collect some results concerning the uniform boundedness of the discrete optimal controlū h , its statē y h and the associated multipliersp h andμ h . Lemma 3.2 Letū h ∈ U ad ,ȳ h ,p h ∈ X h0 and (μ j ) xj ∈N h be a solution of (16)- (19) satisfying Then there exists a constant C > 0, which is independent of h, such that Next, let us introduce the auxiliary functionsỹ h ∈ H 2 (Ω)∩H 1 0 (Ω),ỹ h ∈ X h0 , p h ∈ X h0 as the solutions of Lemma 3.3 Letỹ h ,ỹ h andp h be as above and Ω 0 an open set such that Ω 0 ⊂ Ω and K ⊂ Ω 0 . Then we have Proof. The estimates (31) We see that from (16) and (29) where we utilized (17) and (30) with the test function v h =ỹ h −ȳ h to rewrite the term containing the gradients in the first equality. Consequently, adding the terms S 2 , S 3 , S 4 to S 1 in (35) gives Young's inequality together with (34) implies that In a similar way we deduce with the help of (32) Let us next consider the first integral in S 3 . Usingμ h =μ b h −μ a h , Proposition 2.7, the fact that y a ≤ȳ ≤ y b on K, Lemma 3.2 and (33) we have To estimate the second integral in S 3 we use Proposition 2.5, the fact that I h y a ≤ȳ h ≤ I h y b in K, a well-known interpolation estimate and (33) to obtain Combining (37) and (38) yields Let us next turn S 4 , which we rewrite as .
In order to estimate S 4.1 we first observe that S 4.1 = R h (u h ) for the choice y h =ỹ h , where R h (u h ) is defined at the bottom of page 266 in [1]. Retracing the steps in [1] leading to (3.11) we infer that where q and ρ are defined immediately after (12), while In view of the definition of η(α, r) and (23) this implies by (32), we finally obtain Using (24), (31) and (27), we derive in a similar way Since φ ∈ C 2 and ỹ h L ∞ (Ω) is uniformly bounded in h (in view of (31) and (27)) we infer with the help of Lemma 2.2 and (34) In a similar way we obtain using (32) and (31) where we note that p h L 2 (Ω) is uniformly bounded for sufficiently small h in view of (34). Collecting the estimates for S 4.1 , . . . , S 4.5 , we conclude that S 4 can be bounded by Inserting the estimates of the terms S 1 , . . . , S 4 into (36) yields Since κ < 1, choosing > 0 to be small enough in the above expression yields the existence of c > 0 independent of h such that Let us next establish an upper bound for ∇(ȳ h −ȳ) L 2 (Ω) . To this end we introduce R hȳ as the Ritz projection ofȳ, i.e Let us first derive an upper bound on ∇(ȳ h − R hȳ ) L 2 (Ω) . To begin, from the definition of R hȳ and the weak formulation ofȳ we have If we combine this relation with (16) we obtain for all w h ∈ X h0 that Using w h = R hȳ −ȳ h in the previous variational equation and observing that ȳ h L ∞ (Ω) is uniformly bounded in h we deduce that by Poincaré's inequality. Thus, which together with a standard error bound for the Ritz projection and (40) implies It remains to prove the uniform estimate forȳ h −ȳ. We obtain from (31), the continuous embedding H 2 (Ω) → C(Ω), Lemma 2.3 and (40) that This completes the proof of Theorem 3.1.

Remark 2
The choice K =Ω is in fact allowed for Problem (P) provided that the bounds y a , y b ∈ C(Ω) satisfy in addition to y a < y b inΩ the compatibility condition y a < 0 < y b on ∂Ω. In this case the set N h , which appears in the discrete optimal control problem, should be defined as We claim that the assertion of Theorem 3.1 remains valid in this setting. To see this, we note that the only change in the proof concerns the term S 3 which now reads However, using the fact that y a , y b ∈ C(Ω), y a < y b inΩ and y a < 0 < y b on ∂Ω, it can be shown that there exists Ω 0 ⊂⊂ Ω such that supp(μ) ⊂ Ω 0 and supp(μ h ) ⊂ Ω 0 for h small enough, see [11,Corollary 5.4]. We may then use again (33) and argue in the same way as before.

Numerical Example
We now examine numerically the error bounds established in Theorem 3.1. For this purpose, we consider the following example taken from [25,Section 7], in which Problem (P) is considered with the following choice for the data: Ω := (0, 1) × (0, 1), φ(s) = s 3 , α = 10 −2 , y 0 := −1, U ad = L 2 (Ω), y b = ∞ and It was shown in [1] that this example admits a unique global solution. In fact, it is easy to see that (3) holds for φ(s) = s 3 with r = 2 and M = 2 √ 3, hence q = 4 in (12). After applying the variational discretization, the numerical solution of the resulting discrete optimality system (16)- (19) is obtained by the semismooth Newton method proposed in [15] whose extension to semilinear elliptic control problems is straightforward. Consequently, the condition (22) from Theorem 2.8 now reads where C 4 ≈ 0.648027075 is an upper bound for the constant in Gagliardo-Nirenberg inequality, precisely it is the bound C It can be seen from this figure that the previous condition is satisfied strictly which in turn implies that the considered example admits a unique global solution. The global minimum of the considered example together with its state and the associated multipliers are presented graphically in Figure 2. We see that the state constraints are active at one point, namelỹ x := ( 1 2 , 1 2 ), and the corresponding multiplier is approximately given bȳ where δx is a Dirac measure atx. We can easily find a polygonal subdomain K ⊂⊂ Ω that contains the active pointx so that Assumption 1 holds. Consequently, we are expecting the bound | ln h|h 3 2 − 1 s , or equivalently h 1−ε for arbitrarily small ε > 0, for the computed errors according to Theorem 3.1.
To deduce the convergence rates numerically, we compute the experimental order of convergence (EOC) which is defined as where E is a given positive error functional and h i−1 , h i are two consecutive mesh sizes. For our experiment, we consider the error functionals and denote the corresponding experimental orders of convergence by EOC u L2 , EOC y H1 , EOC y L2 and EOC y L∞ , respectively. Furthermore, we consider the sequence of mesh sizes h i = 2 −i √ 2, for i = 1, . . . , 9. Since we don't have the exact solution at hand, we consider the numerical solution computed at mesh size h 10 = 2 −10 √ 2 to be the reference solution, that is, we defineū ref :=ū h10 andȳ ref :=ȳ h10 . Figure 3 shows the values of our error functionals in dependence of h, and also illustrates the order of convergence. The computed values of the associated EOC are presented in Table 1.
From the numerical findings we see that as the mesh size h decreases the errors E u L2 (h) and E y H1 (h) behave like O(h) which indicates that the convergence rate, namely O(h 1−ε ) for arbitrarily small ε > 0, predicted in Theorem 3.1 is optimal. On the other hand, for E y L2 (h) and E y L∞ (h) we see the behaviour O(h 2 ) and O(h 1.6 ), respectively, from which we conclude that the error bounds for the discrete optimal state in the spaces L 2 (Ω) and L ∞ (Ω) which are deduced from the error bound of the discrete optimal control via the Lipschitz continuity of the control-to-state map are not sharp.
In fact, the O(h 2 ) behaviour of E y L2 (h) could be explained in the light of the work [26] where it was shown that for an elliptic control problem with finitely many pointwise inequality state constraints the error of the discrete optimal state in L 2 (Ω) is of order h 4−d up to logarithmic factor in d = 2 or d = 3 space dimensions when the control problem is discretized by continuous, piecewise linear finite elements.