Error analysis for global minima of semilinear optimal control problems

Error analysis for global minima of semilinear optimal control problems

Ahmad Ahmad Ali111Schwerpunkt Optimierung und Approximation, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany., Klaus Deckelnick222Institut für Analysis und Numerik, Otto–von–Guericke–Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany & Michael Hinze333Schwerpunkt Optimierung und Approximation, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany.

Dedicated to Eduardo Casas on the occasion of his 60th birthday.

Abstract: 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.
Mathematics Subject Classification (2000): 49J20, 35K20, 49M05, 49M25, 49M29, 65M12, 65M60
Keywords: Optimal control, semilinear PDE, uniqueness of global solutions, error estimates

1 Introduction

In this work we are concerned with the error analysis of a variational discretization of the control problem

subject to

 −Δy+ϕ(y) =u in Ω, (1) y =0 on ∂Ω, (2)

and the pointwise constraints

 ua≤u(x)≤ub for a.e. x∈Ω, ya(x)≤y(x)≤yb(x) ∀x∈K⊂Ω,

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 assuming particular types of growth conditions for the nonlinearity . The same result was established for the variational discrete counterpart of , 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 , respectively. The variational discretization of 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 . For a broad overview, we refer the reader to the references of the respective citations. In [4] the problem is studied when the controls are of boundary type, and the necessary first order conditions are established. Compare [3] where the function is linear, and [8] where the pointwise constraints are imposed on the gradient of the state.

The regularity of the optimal controls of 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 contains finitely/infinitely many points, and [13] for the role of those conditions in PDE constrained control problems.

Finite element discretization of problem 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 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 . 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

2.1 Assumptions

• is a bounded, convex and polygonal domain.

• is a (possibly empty) compact subset of .

• and with .

• are given functions that satisfy , .

• and are given.

• is of class and monotonically increasing.

• There exist and such that

 |ϕ′′(s)|≤Mϕ′(s)1r for all s∈R, (3)

where and denote the first and second derivative of , respectively.

2.2 The State Equation

Recall that a function is called a weak solution of (1), (2) if

 ∫Ω∇y⋅∇v+ϕ(y)vdx=∫Ωuvdx∀v∈H10(Ω). (4)
Theorem 2.1

For every the boundary value problem (1), (2) admits a unique weak solution . Moreover, there exists such that

 ∥y∥H2(Ω)≤c(1+∥u∥L2(Ω)). (5)
Proof.

The existence and uniqueness of the solution in follows from the monotone operator theorem. Using the method of Stampacchia one can show, in addition, that . Utilizing the boundedness of and the properties of the nonlinearity , one can show 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

 G:L2(Ω)→H10(Ω)∩H2(Ω) (6)

such that is the solution to (4) for a given .

Lemma 2.2

Let be the mapping introduced in (6). Then there exists depending only on such that

 ∥G(u)−G(v)∥L2(Ω)≤c∥u−v∥L2(Ω)∀u,v∈L2(Ω).
Proof.

Given let and . Using Poincaré’s inequality, the monotonicity of and (4) we have

 ∥yu−yv∥2L2(Ω)≤c∫Ω|∇(yu−yv)|2dx ≤ ∫Ω|∇(yu−yv)|2+[ϕ(yu)−ϕ(yv)](yu−yv)dx = ∫Ω(u−v)(yu−yv)dx≤∥u−v∥L2(Ω)∥yu−yv∥L2(Ω),

which implies the result. ∎

Lemma 2.3

Let be the mapping introduced in (6). Then for any there exists such that

 ∥G(u)−G(v)∥H2(Ω)≤L(m)∥u−v∥L2(Ω)

for all , with , .

Proof.

Defining again we infer from Theorem 2.1 and the continuous embedding that , for some depending on . Clearly, belongs to and satisfies

 −Δ(yu−yv)=(u1−u2)−[ϕ(yu)−ϕ(yv)] in Ω.

Using a standard a–priori estimate, the Lipschitz continuity of on bounded sets and Lemma 2.2 we infer that

 ∥y1−y2∥H2(Ω) ≤c(∥u−v∥L2(Ω)+∥ϕ(yu)−ϕ(yv)∥L2(Ω)) ≤L(m)(∥u−v∥L2(Ω)+∥yu−yv∥L2(Ω)) ≤L(m)∥u−v∥L2(Ω),

where is a constant depending on . This completes the proof. ∎

2.3 The Optimal Control Problem (P)

Using the control-to-state operator defined in (6), the reduced form of our optimal control problem reads

where

It is well–known that admits at least one solution provided that a feasible point exists (compare [4]). Moreover, if a solution of 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 solution of problem is the linearized Slater condition which reads: there exist and such that

 ya(x)+δ≤G(¯u)(x)+G′(¯u)(u0−¯u)(x)≤yb(x)−δ∀x∈K. (7)

The next result is a consequence of [4, Theorem 5.2].

Theorem 2.4

Let be a local solution of problem satisfying (7). Then there exist for and a regular Borel measure such that with there holds

Note that in view of (10) is the –projection of onto so that

 ¯u(x)=min(max(ua,−1α¯p(x)),ub)∀x∈Ω,

Since for it follows from [22, Corollary A.6] that for as well. Furthermore, it is well known that the multiplier associated with the pointwise state constraints is concentrated at the points in 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 , are constant functions.

Proposition 2.5

Let and satisfy (11). Then there holds

 supp(¯μb)⊂{x∈K:¯y(x)=yb(x)}, supp(¯μa)⊂{x∈K:¯y(x)=ya(x)},

where with is the Jordan decomposition of .

We note that the problem is in general nonconvex since the state equation is not linear. In other words, the problem 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 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:

 η(α,r):=αρ2C2−2rrqM−1(r−12r−1)1−rrq1/qr1/rρρ/2(2−ρ)ρ2−1. (12)

Here, , while and appear in (3). Furthermore, is an upper bound on the optimal constant in the Gagliardo-Nirenberg inequality

 ∥f∥Lq≤C∥f∥2qL2∥∇f∥q−2qL2∀f∈H1(R2).

For sharp upper bounds for the constant , see for instance [1, Theorem 7.3].

Theorem 2.6

Suppose that , , (), is a solution of (8)–(11). If

 ∥¯p∥Lq(Ω)≤η(α,r), (13)

then is a global minimum for Problem . If the inequality (13) is strict, then is the unique global minimum.

2.4 Variational Discretization

Let be an admissible triangulation of the polygonal domain with

 ¯¯¯¯Ω=⋃T∈Th¯¯¯¯T.

Here is the maximum mesh size, while stands for the diameter of the triangle . We introduce the following spaces of linear finite elements:

 Xh :={vh∈C(¯Ω):vh|T is a linear % polynomial on each T∈Th}, Xh0 :={vh∈Xh:vh|∂Ω=0}.

The Lagrange interpolation operator is defined by

 Ih:C(¯Ω)→Xh,Ihy:=n∑i=1y(xi)ϕi,

where denote the nodes in the triangulation and are the basis functions of the space which satisfy .

The finite element discretization of (4) reads: for a given , find such that

 ∫Ω∇yh⋅∇vh+ϕ(yh)vhdx=∫Ωuvhdx∀vh∈Xh0. (14)

Using the monotonicity of and the Brouwer fixed-point theorem one can show that (14) admits a unique solution . Hence, analogously to (6), we introduce the discrete control–to–state operator

 Gh:L2(Ω)→Xh0 (15)

such that is the solution of (14).

The variational discretization (see [18]) of Problem  reads:

where we define

with the set of nodes

 Nh:={xj|xj is a vertex of T∈Th, where T∩K≠∅}.

We remark that provided that is small enough. This follows from the fact that and are continuous functions with .

In an analogous way to that of problem , one can show that admits at least one solution, denoted by , provided that a feasible point exists. In practice one calculates candidates for solutions of by solving the system of necessary first order conditions which reads: find and such that

As in the continuous case, there exist multipliers and solving (16)–(19) provided that the local solution satisfies the linearized Slater condition, that is, there exist and such that

 ya(xj)+δ≤Gh(¯uh)(xj)+G′h(¯uh)(u0−¯uh)(xj)≤yb(xj)−δ,xj∈Nh. (20)

It will be convenient in the upcoming analysis to associate with the multipliers from the system (16)–(19) the measure defined by

 ¯μh:=∑xj∈Nh¯μjδxj, (21)

where is the Dirac measure at . We can easily deduce from (19) the following result about the support of the measure .

Proposition 2.7

Let be the measure introduced in (21) satisfying (19). Then there holds

 supp(¯μbh)⊂{xj∈Nh:¯yh(xj)=yb(xj)}, supp(¯μah)⊂{xj∈Nh:¯yh(xj)=ya(xj)}.

where with is the Jordan decomposition of .

Analogously to Theorem 2.6, we have the next theorem about global solutions of problem . The proof can be found in [1].

Theorem 2.8

Suppose that , , , is a solution of (16)–(19). If

 ∥¯ph∥Lq(Ω)≤η(α,r), (22)

then is a global minimum for Problem . If the inequality (22) is strict, then is the unique global minimum.

3 Error Analysis

Let be a sequence of admissible triangulations of . We assume that the sequence is quasi-uniform in the sense that each is contained in a ball of radius and contains a ball of radius for some independent of . In addition we make the following assumption concerning the set :

Assumption 1

For every there exists a set of triangles such that

 K=⋃T∈Th¯T.

In what follows we consider a sequence of solutions of (16)–(19) satisfying

 ∥¯ph∥Lq(Ω)≤κη(α,r) for 0

for some that is independent of . We immediately infer from Theorem 2.8 that is the unique global minimum of and we are interested in the convergence properties of these solutions as . It is shown in [1] (see Theorem 4.2 and its proof) that there exist , and such that

 ¯uh→¯u in L2(Ω),¯ph⇀¯p in Lq(Ω),¯μh⇀¯μ in M(K)

and is a solution of (8)–(11). Since

 ∥¯p∥Lq(Ω)≤liminfh→0∥¯ph∥Lq(Ω)≤κη(α,r), (24)

Theorem 2.6 implies that is the unique global optimum of . The aim in the remaining part of this paper is to prove error estimates for and the corresponding optimal states . Our main results read:

Theorem 3.1

Suppose that (23) holds and let be the unique global minima of and respectively. Then we have for any that

 ∥¯uh−¯u∥L2(Ω) ≤ cs√|lnh|h32−1s (25) ∥¯yh−¯y∥H1(Ω)+∥¯yh−¯y∥L∞(Ω) ≤ cs√|lnh|h32−1s. (26)
Remark 1

In [25] Pfefferer at al. for problems in two and three dimensions present a similar error estimate for discrete (local) solutions in the vicinity of a local solution which satisfies a quadratic growth condition. Assuming (23) we here use different techniques to prove an error estimate for the unique global discrete solutions which converge to the unique global solution of our optimization problem.

Before we start presenting the proof of this result we collect some results concerning the uniform boundedness of the discrete optimal control , its state and the associated multipliers and .

Lemma 3.2

Let , , and be a solution of (16)–(19) satisfying

 ∥¯ph∥Lq(Ω)≤η(α,r),0

Then there exists a constant , which is independent of , such that

 ∥¯uh∥L2(Ω),∥¯yh∥H1(Ω),∥¯yh∥L∞(Ω),∥¯μh∥M(K)≤C. (27)
Proof.

The uniform boundedness of is shown in [1, Lemma 4.1] while the one of is a consequence of the uniform convergence [1, (4.16)]. ∎

Next, let us introduce the auxiliary functions , , as the solutions of

 ∫Ω∇~yh⋅∇v+ϕ(~yh)vdx=∫Ω¯uhvdx∀v∈H10(Ω), (28) ∫Ω∇~yh⋅∇vh+ϕ(~yh)vhdx=∫Ω¯uvhdx∀vh∈Xh0, (29) ∫Ω∇~ph⋅∇vh+ϕ′(¯y)~phvhdx =∫Ω(¯y−y0)vhdx+∫Kvhd¯μ∀vh∈Xh0. (30)
Lemma 3.3

Let and be as above and an open set such that and . Then we have

 ∥¯yh−~yh∥L2(Ω)+h∥¯yh−~yh∥L∞(Ω) ≤ch2(∥¯uh∥L2(Ω)+1), (31) ∥~yh−¯y∥L2(Ω)+h∥~yh−¯y∥L∞(Ω) ≤ch2(∥¯u∥L2(Ω)+1), (32) ∥~yh−¯y∥L∞(Ω0) ≤c|lnh|h3−2s(∥¯u∥W1,s(Ω)+1), (33) ∥~ph−¯p∥L2(Ω) ≤ch(∥¯y−y0∥L2(Ω)+∥¯μ∥M(K)). (34)
Proof.

The estimates (31) and (32) can be found as Theorems 1 and 2 in [10]. On the other hand, (33) follows from [25, Theorem 3.5]. Finally, the estimate (34) is a consequence of [2, Theorem 3]. ∎

Proof of Theorem 3.1: Testing (10) with and (18) with and adding the resulting inequalities gives

 ∫Ω(¯ph−¯p)(¯u−¯uh)−α(¯uh−¯u)2dx≥0

from which we obtain

 α∥¯u−¯uh∥2L2(Ω) ≤∫Ω(¯u−¯uh)(¯ph−¯p)dx =∫Ω(~ph−¯p)(¯u−¯uh)dxS1+∫Ω(¯ph−~ph)(¯u−¯uh)dx. (35)

We see that from (16) and (29) with the choice that

 ∫Ω(¯ph−~ph)(¯u−¯uh)dx=∫Ω[ϕ(~yh)−ϕ(¯yh)](¯ph−~ph)dx +∫Ω∇(~yh−¯yh)⋅∇(¯ph−~ph)dx =∫Ω(¯yh−¯y)(~yh−¯yh)dxS2+∫K(~yh−¯yh)d¯μh−∫K(~yh−¯yh)d¯μ%$S3$ +∫Ω[ϕ(~yh)−ϕ(¯yh)](¯ph−~ph)dx−∫Ω[ϕ′(¯yh)¯ph−ϕ′(¯y)~ph](~yh−¯yh)dxS4,

where we utilized (17) and (30) with the test function to rewrite the term containing the gradients in the first equality. Consequently, adding the terms , , to in (35) gives

 α∥¯u−¯uh∥2L2(Ω)≤4∑i=1Si. (36)

Young’s inequality together with (34) implies that

 S1 ≤∥~ph−¯p∥L2(Ω)∥¯u−¯uh∥L2(Ω)≤12αϵ∥~ph−¯p∥2L2(Ω)+αϵ2∥¯u−¯uh∥2L2(Ω) ≤cαϵh2(∥¯y−y0∥2L2(Ω)+∥¯μ∥2M(K))+αϵ2∥¯u−¯uh∥2L2(Ω) =αϵ2∥¯u−¯uh∥2L2(Ω)+cϵh2.

In a similar way we deduce with the help of (32)

 S2 =−∥¯yh−¯y∥2L2(Ω)+∫Ω(¯yh−¯y)(~yh−¯y)dx ≤−∥¯yh−¯y∥2L2(Ω)+∥¯yh−¯y∥L2(Ω)∥~yh−¯y∥L2(Ω)