Recent Advances in the Analysis of State-constrained ...math.ecnu.edu.cn/~jypan/Research/papers/2009...

International Series of Numerical Mathematics, Vol. 158, 57–72c© 2009 Birkhauser Verlag Basel/Switzerland

Recent Advances in the Analysisof State-constrained EllipticOptimal Control Problems

Eduardo Casas and Fredi Troltzsch

Abstract. Pointwise state-constrained control problems associated with semi-linear elliptic equations are studied. Theoretical results are derived, whichare necessary to carry out the numerical analysis of the control problem. Inparticular, sufficient second-order optimality conditions, some new regularityresults on optimal controls and a sufficient condition for the uniqueness of theLagrange multiplier associated with the state constraints are presented.

Mathematics Subject Classification (2000). 49J20, 49K20, 35J65.

Keywords. Optimal control, pointwise state constraints, first- and second-order optimality conditions, Lagrange multipliers, Borel measures.

1. Introduction

In this paper, we consider several aspects of state-constrained optimal controlproblems for semilinear elliptic equations, which seem to be important for a re-lated numerical analysis. For instance, due to the non-convex character of suchproblems, a reasonable error analysis should be based on second-order sufficientoptimality conditions at locally optimal controls. It is known that such conditionsare fairly delicate under the presence of state constraints. In [2], second-order suf-ficient conditions were established, which are, in some sense, closest to associatednecessary ones and admit a form similar to the theory of nonlinear programmingin finite-dimensional spaces. Here, we briefly discuss this result and show its equiv-alence to an earlier form stated in [5] that was quite difficult to explain.

Another important pre-requisite for the numerical analysis of control prob-lems is the smoothness that can be expected of optimal controls. We show that

The first author was partially supported by the Spanish Ministry of Education and Science underprojects MTM2005-06817 and “Ingenio Mathematica (i-MATH)” CSD2006-00032 (ConsoliderIngenio 2010).

58 E. Casas and F. Troltzsch

the optimal control is Lipschitz, if the state constraints are only active at finitelymany points. We also present a counterexample that this result is not true for in-finitely many active points. On the other hand, we prove the somehow surprisingresult that optimal controls belong to H1(Ω) no matter how large the active setis. Moreover, we also discuss the uniqueness of the Lagrange multiplier associatedwith the state-constraints.

2. The control problem

Let Ω be an open, connected and bounded domain in Rn, n = 2, 3, with a Lipschitzboundary Γ. In this domain we consider the state equation{

−Δy + a0(y) = u+ e in Ω,y = 0 on Γ, (1)

where a0 and e are fixed functions specified below. In (1), the function u denotesthe control and we will denote by yu the solution associated with u. We will statelater the conditions leading to the existence and uniqueness of a solution of (1) inC(Ω) ∩H1

0 (Ω).The optimal control problem is formulated as follows

(P)

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

min J(u) =12

∫Ω

(yu(x) − yd(x))2 dx+N

2

∫Ω

u(x)2 dx

subject to (yu, u) ∈ (C(Ω) ∩H10 (Ω))× L∞(Ω),

α ≤ u(x) ≤ β for a.e. x ∈ Ω,a ≤ yu(x) ≤ b ∀x ∈ K.

We impose the following assumptions on the data of the control problem:(A1) In the whole paper, fixed functions yd, e ∈ L2(Ω), and real numbers N > 0,α, β, a, b are given with α ≤ β and a < b. The fixed function a0 : R → R ismonotone non-decreasing and locally of class C2,1.

We introduce the sets of admissible controls Uα,β and the feasible set Uad ofthe problem by

Uα,β = {u ∈ L∞(Ω) : α ≤ u(x) ≤ β a.e. in Ω}Uad = {u ∈ Uα,β : a ≤ yu(x) ≤ b ∀x ∈ K}.

(A2) K is a compact subset of Ω. We also assume that either K ∩ Γ = ∅ ora < 0 < b. Moreover, we define the set

Yab = {z ∈ C(K) : a ≤ z(x) ≤ b ∀x ∈ K}.By the formal setting a = −∞ (b = +∞) we can include the case, where only

upper (or lower) bounds on the state are given.We are able to deal with more general problems, for instance with a more

general elliptic differential operator, a nonlinearity a0 of the form a0(x, y), a non-quadratic objective functional of integral type, bounds depending on x etc. These

Analysis of State-constrained Elliptic Optimal Control Problems 59

generalizations will be included in a forthcoming paper. Here, to keep the presen-tation simple, we consider the setting introduced above.

Let us state first the existence and uniqueness of the solution correspondingto the state equation (1).

Theorem 2.1. Under assumption (A1), the equation (1) has a unique solutionyu ∈ H1

0 (Ω) ∩ C(Ω) for every u ∈ L2(Ω). Moreover, yu ∈ H2(Ω) if Ω is convexand yu ∈ W 2,p(Ω) if Γ is of class C1,1 and u, e ∈ Lp(Ω), with p > n.

It is well known that equation (1) has a unique solution yu ∈ H10 (Ω) ∩ C(Ω)

for every u ∈ Lp(Ω). A proof of this result can be obtained by the usual cut offprocess applied to a0, then applying a Schauder’s fix point theorem combined withthe monotonicity of a0 and L∞ estimates for the state; cf. Stampacchia [11]. Thecontinuity of yu is proven in [6]. The W 2,p(Ω) and H2(Ω) estimates can be foundin Grisvard [7].

Now the existence of an optimal control can be proved by using standardarguments.

Theorem 2.2. Assume that the set of controls Uad is not empty. Then the controlproblem (P) has at least one solution.

In the rest of the paper, u denotes a local minimum of (P) in the sense of theL∞(Ω)-topology and y will be its associated state. At such a local minimizer, wewill assume the linearized Slater condition.(A3) There exists u0 ∈ Uα,β such that

a < y(x) + z0(x) < b ∀x ∈ K, (2)

where z0 ∈ H10 (Ω) ∩ C(Ω) is the unique solution of{

−Δz + a′0(y)z = u0 − u in Ω

z = 0 on Γ.(3)

Since K is compact and y, z0 ∈ C(K), we deduce that (2) is equivalent tothe existence of real τ1, τ2 ∈ R such that

a < τ1 < y(x) + z0(x) < τ2 < b ∀x ∈ K. (4)

3. First- and second-order optimality conditions

Before deriving the first-order optimality conditions satisfied by the local minimizeru, we recall some results about the differentiability of the mappings involved inthe control problem. For the proofs, the reader is referred to Casas and Mateos[3], where a Neumann boundary condition was considered instead of the Dirichletcondition posed in this paper. The method of proof is very similar and the necessarychanges are obvious.


Theorem 3.1. If (A1) holds, then the mapping G : L2(Ω) −→ C(Ω) ∩ H10 (Ω),

defined by G(u) = yu is of class C2. Moreover, for all u, v ∈ L2(Ω), zv = G′(u)vis defined as the solution of{

−Δzv + a′0(yu)zv = v in Ω

zv = 0 on Γ.(1)

Finally, for every v1, v2 ∈ L2(Ω), zv1v2 = G′′(u)v1v2 is the solution of{−Δzv1v2 + a′0(yu)zv1v2 + a′′0 (yu)zv1zv2 = 0 in Ω

zv1v2 = 0 on Γ,(2)

where zvi = G′(u)vi, i = 1, 2.

Remark 3.1. Let us remark that the assumption n ≤ 3 is required to apply thesecond-order optimality conditions because the differentiability of G from L2(Ω) toC(Ω) is needed for the proof. This result holds true only for n ≤ 3.

Theorem 3.2. Suppose that (A1) holds. Then J : L2(Ω) → R is a functional ofclass C2. Moreover, for every u, v, v1, v2 ∈ L2(Ω)

J ′(u)v =∫Ω

(ϕ0u +Nu) v dx (3)

andJ ′′(u)v1v2 =

∫Ω

[zv1zv2 +Nv1v2 − ϕ0u a′′0(yu)zv1zv2 ] dx, (4)

where yu = G(u) and ϕ0u ∈ W 1,s0 (Ω), for all s < n/(n− 1), is the unique solution

of the adjoint problem{−Δϕ+ a′0(yu)ϕ = yu − yd in Ω

ϕ = 0 on Γ,(5)

and zvi = G′(u)vi, i = 1, 2.

The previous theorem and the next one follow easily from Theorem 3.1 andthe chain rule.

Theorem 3.3. If (A1) is satisfied, then the mapping F : L2(Ω) → C(K), definedby F (u) = yu(·)|K , is of class C2. Moreover, for every u, v, v1, v2 ∈ L2(Ω)

F ′(u)v = zv(·) (6)

andF ′′(u)v1v2 = zv1v2(·), (7)

where zvi = G′(u)vi, i = 1, 2, and zv1v2 = G′′(u)v1v2.

Before stating the first-order optimality conditions, let us fix some notation.We denote by M(K) the Banach space of all real and regular Borel measures inK, which is identified with the dual space of C(K). The following result is wellknown.


Theorem 3.4. Let u be a local solution of (P) and suppose that the assumptions(A1)–(A2) are satisfied. Then there exist a measure μ ∈ M(K) and a functionϕ ∈ W 1,s

0 (Ω), for all 1 ≤ s < n/(n− 1), such that{−Δϕ+ a′0(y(x))ϕ = y − yd + μ in Ω,

ϕ = 0 on Γ,(8)

∫K

(z(x)− y(x))dμ(x) ≤ 0 ∀z ∈ Yab, (9)∫Ω

(ϕ+Nu)(u − u) dx ≥ 0 ∀u ∈ Uα,β. (10)

Remark 3.2. It is well known that, in view of a ≤ y(x) ≤ b, inequality (9) impliesthat the support of μ is in the set K0 = Ka ∪Kb with

Ka = {x ∈ K : y(x) = a} and Kb = {x ∈ K : y(x) = b}.Moreover the Lebesgue decomposition of μ = μ+ − μ− implies that suppμ+ ⊂ Kb

and suppμ− ⊂ Ka. Because of this property and by assumption (A2) we have thatsupp μ ∩ Γ = ∅. Notice that that Ka and Kb are closed subsets.

Remark 3.3. From (10), it follows

u(x) = Proj[α,β](−1Nϕ(x)) = max{α,min{ϕ(x), β}} for a.e. x ∈ Ω. (11)

Let us formulate the Lagrangian version of the optimality conditions (8)–(10). We define the Lagrange function L : L2(Ω) ×M(K) −→ R associated withthe problem (P) by

L(u, μ) = J(u) +∫

K

yu(x) dμ(x).

Using (3) and (6) we find that

∂L∂u

(u, μ)v =∫Ω

(ϕu(x) +Nu(x)) v(x) dx, (12)

where ϕu with ϕu ∈ W 1,s0 (Ω), for all 1 ≤ s < n/(n − 1), is the solution of the

Dirichlet problem {−Δϕ+ a′0(yu)ϕ = yu + μ in Ω

ϕ = 0 on Γ.(13)

Now the inequality (10) along with (12) lead to

∂L∂u

(u, μ)(u − u) ≥ 0 ∀u ∈ Uα,β . (14)


Before stating the sufficient second-order optimality conditions, we providethe expression of the second derivative of the Lagrangian with respect to thecontrol. From (7), we get

∂2L∂u2

(u, μ)v1v2 = J ′′(u)v1v2 +∫

K

zv1v2(x) dμ(x).

By (2) and (4), this is equivalent to

∂2L∂u2

(u, μ)v1v2 =∫Ω

[zv1zv2 +Nv1v2 − ϕu a′′0 (yu)zv1zv2 ] dx, (15)

where ϕu is the solution of (13).Associated with u, we define the cone of critical directions by

Cu = {v ∈ L2(Ω) : v satisfies (16), (17) and (18)},

v(x) =

⎧⎨⎩ ≥ 0 if u(x) = α,≤ 0 if u(x) = β,= 0 if ϕ(x) +Nu(x) �= 0,

(16)

zv(x) ={≥ 0 if x ∈ Ka

≤ 0 if x ∈ Kb,(17)∫

K

zv(x) dμ(x) = 0, (18)

where zv ∈ H10 (Ω) ∩ C(Ω) satisfies{

−Δzv + a′0(y)zv = v in Ωzv = 0 on Γ.

The relation (17) expresses the natural sign conditions, which must be fulfilledfor feasible directions at active points x ∈ Ka or Kb, respectively. On the otherhand, (18) states that the derivative zv must be zero whenever the correspondingLagrange multiplier is non-vanishing. This restriction is needed for second-ordersufficient conditions. Compared with the finite-dimensional case, this is exactlywhat we can expect. Therefore the relations (17)–(18) provide a convenient exten-sion of the usual conditions of the finite-dimensional case.

We should mention that (18) is new in the context of infinite-dimensionaloptimization problems. In earlier papers on this subject, other extensions to theinfinite-dimensional case were suggested. For instance, Maurer and Zowe [10] usedfirst-order sufficient conditions to account for the strict positivity of Lagrangemultipliers. Inspired by their approach, in [5] an application to state-constrainedelliptic boundary control was suggested by the authors. In terms of our problem,equation (18) was relaxed by∫

K

zv(x) dμ(x) ≥ −ε∫{x:|ϕ(x)+Nu(x)|≤τ}

|v(x)| dx


for some ε > 0 and τ > 0, cf. [5, (5.15)]. In the next theorem, which was provenin [2, Theorem 4.3], we will see that this relaxation is not necessary. We obtain asmaller cone of critical directions that seems to be optimal. However, the readeris referred to Theorem 3.6 below, where we consider the possibility of relaxing theconditions defining the cone Cu.

Theorem 3.5. Let us assume that (A1) holds. Let u be a feasible control of problem(P), y the associated state and (ϕ, μ) ∈ W 1,s

0 (Ω)×M(K), for all 1 ≤ s < n/(n−1),satisfying (8)–(10). Assume further that

∂2L∂u2

(u, μ)v2 > 0 ∀v ∈ Cu \ {0}. (19)

Then there exist ε > 0 and δ > 0 such that it holds

J(u) +δ

2‖u− u‖2L2(Ω) ≤ J(u) if ‖u− u‖L2(Ω) ≤ ε and u ∈ Uad. (20)

The condition (19) seems to be natural. In fact, under some regularity as-sumption, we can expect the inequality

∂2L∂u2

(u, μ)v2 ≥ 0 ∀v ∈ Cu

as a necessary condition for local optimality. At least, this is the case when thestate constraints are of integral type, see [3], or when K is a finite set of points,see [1]. In the general case, the necessary second-order optimality conditions forproblem (P) remain open for us.

We finish this section by establishing an equivalent condition to (19) thatis more convenient for the numerical analysis of problem (P). Let us introduce acone Cτ

u of critical directions that is bigger than Cu. Given τ > 0, we denote byCτ

u the set of elements v ∈ L2(Ω) satisfying

v(x) =

⎧⎨⎩≥ 0 if u(x) = α,≤ 0 if u(x) = β,= 0 if |ϕ(x) +Nu(x)| > τ,

(21)

zv(x) ={≥ −τ‖v‖L2(Ω) if x ∈ Ka

≤ +τ‖v‖L2(Ω) if x ∈ Kb,(22)∫

K

zv(x) dμ(x) ≥ −τ‖v‖L2(Ω). (23)

Theorem 3.6. Under the assumption (A1), (19) holds if and only if there existτ > 0 and ρ > 0 such that

∂2L∂u2

(u, μ)v2 ≥ ρ‖v‖2L2(Ω) ∀v ∈ Cτu . (24)

Proof. Since Cu ⊂ Cτu , it is clear that (24) implies (19). Let us prove by contra-

diction that (24) follows from (19). Assume that (19) holds but not (24). Then for


any positive integer k there exists an element vk ∈ C1/ku such that

∂L∂u

(u, μ)v2k <1k‖vk‖2L2(Ω). (25)

Redefining vk as vk/‖vk‖L2(Ω) and taking, if necessary, a subsequence denoted inthe same way, we can assume that

‖vk‖L2(Ω) = 1, vk ⇀ v weakly in L2(Ω) and∂L∂u

(u, μ)v2k <1k, (26)

and from (21)–(23)

vk(x) =

⎧⎨⎩ ≥ 0 if u(x) = α,≤ 0 if u(x) = β,= 0 if |ϕ(x) +Nu(x)| > 1/k,

(27)

zvk(x) =

{≥ −1/k if x ∈ Ka

≤ +1/k if x ∈ Kb,(28)∫

K

zvk(x) dμ(x) ≥ −1/k. (29)

Since zvk→ zv strongly inH1

0 (Ω)∩C(Ω), we can pass to the limit in (26)–(29)and get that v ∈ Cu and

∂L∂u

(u, μ)v2 ≤ 0. (30)

This is only possible if v = 0; see (19). Let us note that the only delicate pointto prove that v ∈ Cu is to establish (18). Indeed, (16) and (17) follow easily from(27) and (28). Passing to the limit in (29) we obtain∫

K

zv(x) dμ(x) ≥ 0.

This inequality, along with (17) and the structure of μ, implies (18).Therefore we have that vk ⇀ 0 weakly in L2(Ω) and zvk

→ 0 strongly inH10 (Ω) ∩ C(Ω). Hence, using the expression (15) of the second derivative of the

Lagrangian we deduce

N = lim infk→∞

N‖vk‖2L2(Ω) ≤ lim infk→∞

∂L∂u

(u, μ)v2k = 0,

which is a contradiction. �

4. Regularity of the optimal control

In this section, the existence of the second derivative a′′0 is not needed. We startwith the following regularity result for the optimal control, which is well known.We recall that K0 denotes the set of points, where the state constraints are active.

Theorem 4.1. If (y, u, ϕ) ∈ (H10 (Ω) ∩ C(Ω)) × L∞(Ω) ×W 1,s

0 (Ω) satisfies the op-timality system (8)–(10), (y, u) being a feasible pair for problem (P), then u ∈W 1,s(Ω) for all s < n/(n− 1) and u ∈ C(Ω \K0).


The regularity u ∈W 1,s(Ω) follows immediately from (11) and the continuityu ∈ C(Ω \K0) is deduced in the same way. This regularity result on the control ucan be improved if there is a finite number of points, where the state constraintsare active. More precisely, assume that K0 = {xj}m

j=1 ⊂ Ω. Then Remark 3.2implies that

μ =m∑

j=1

λjδxj , with λj ={≥ 0 if y(xj) = b,≤ 0 if y(xj) = a,

(1)

where δxj denotes the Dirac measure concentrated at xj . If we denote by ϕj ,1 ≤ j ≤ m, and ϕ0 the solutions of{

−Δϕj + a′0(y(x))ϕj = δxj in Ω,

ϕj = 0 on Γ,(2)

and {−Δϕ0 + a′0(y(x)) ϕ0 = y − yd in Ω,

ϕ0 = 0 on Γ,(3)

then the adjoint state associated to u is given by

ϕ = ϕ0 +m∑

j=1

λj ϕj . (4)

Theorem 4.2. Assume that Γ is of class C1,1, yd, e ∈ Lp(Ω), p > n, and let(y, u, ϕ) ∈ (H1

0 (Ω) ∩C(Ω))× L∞(Ω)×W 1,s0 (Ω), for all 1 ≤ s < n/(n− 1), satisfy

the optimality system (8)–(10). If the active set consists of finitely many points inΩ, i.e., K0 = {xj}m

j=1 ⊂ Ω, then u belongs to C0,1(Ω) and y to W 2,p(Ω).

Since p > n, it holds that W 2,p(Ω) ⊂ C1(Ω) and therefore ϕ0 ∈ C1(Ω). Onthe other hand, ϕj(x)→ +∞ when x→ xj , hence ϕ has singularities at the pointsxj where λj �= 0. Consequently ϕ cannot be Lipschitz.

Surprisingly, this does not lower the regularity of u: Notice that (11) impliesthat u is identically equal to α or β in a neighborhood of xj , depending on thesign of λj . This implies the desired result; see Casas [1] for the details.

Now the question arises if this Lipschitz property remains also valid for aninfinite number of points where the pointwise state constraints are active. Unfor-tunately, the answer is negative. In fact, the optimal control can even fail to becontinuous if K0 is an infinite and numerable set. Let us present an associated

Counterexample. We set

Ω = {x ∈ R2 : ‖x‖ <√2}, y(x) =

{1 if ‖x‖ ≤ 1

1− (‖x‖2 − 1)4 if 1 < ‖x‖ ≤√2,

K = {xk}∞k=1 ∪ {x∞}, where xk = (1k, 0) and x∞ = (0, 0), μ =

∞∑k=1

1k2

δxk .


Now we define ϕ ∈ W 1,s0 (Ω) for all 1 ≤ s < n/(n − 1) as the solution of the

equation {−Δϕ = y + μ in Ω,

ϕ = 0 on Γ.(5)

The function ϕ can be decomposed in the form

ϕ(x) = ψ(x) +∞∑

k=1

1k2[ψk(x) + φ(x − xk)],

where φ(x) = −(1/2π) log ‖x‖ is the fundamental solution of −Δ and ψ, ψk ∈C2(Ω) satisfy{

−Δψ(x) = y(x) in Ω,

ψ(x) = 0 on Γ,

{−Δψk(x) = 0 in Ω,

ψk(x) = −φ(x− xk) on Γ.

Finally, we set

M =

∣∣∣∣∣ψ(0) +∞∑

k=1

1k2

ψk(0)

∣∣∣∣∣+∞∑

k=1

1k2

φ(xk) + 1, u(x) = Proj[−M,+M ](−ϕ(x)) (6)

and e = −(u(x) + Δy(x)) and a0 = 0. Then u is the unique global solution of thecontrol problem

(Q)

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

min J(u) =12

∫Ω

(y2u(x) + u2(x)) dx


−M ≤ u(x) ≤ +M for a.e. x ∈ Ω,−1 ≤ yu(x) ≤ +1 ∀x ∈ K,

where yu is the solution of{−Δy = u+ e in Ω,

y = 0 on Γ. (7)

As a first step to prove that u is a solution of problem, let us check that Mis a real number. Since {φ(x− xk)}∞k=1 is bounded in C2(Γ), we get that {ψk}∞k=1is also bounded in C2(Ω). Therefore, the convergence of the first series of (6) isobvious. The convergence of the second series follows immediately from

∞∑k=1

1k2

φ(xk) =12π

∞∑k=1

1k2

log k <∞.

Problem (Q) is strictly convex and u is a feasible control with associated state ysatisfying the state constraints. Therefore, there exists a unique solution charac-terized by the optimality system. More precisely, the first-order optimality con-ditions are necessary and sufficient for a global minimum. Let us check that(y, u, ϕ, μ) ∈ H1

0 (Ω) ∩ C(Ω) × L∞(Ω) ×W 1,s0 (Ω) ×M(K) satisfies the optimal-


ity system (8)–(10). First, it is clear that y is the state associated with u becauseof the definition of e. On the other hand, ϕ is the solution of (5), which is thesame as (8) for our example. Relation (10) follows directly from the definition ofu given in (6). Finally, because of the definition of μ and K, (9) can be written inthe form

∞∑k=1

1k2

z(xk) ≤∞∑

k=1

1k2

∀z ∈ C(K) such that − 1 ≤ z(x) ≤ +1 ∀x ∈ K,

which obviously is satisfied.Now we prove that u is not continuous at x = 0. Notice that ϕ(xk) = +∞

for every k ∈ N, because φ(0) = +∞. Therefore, (6) implies that u(xk) = −M forevery k. Since xk → 0, the continuity of u at x = 0 requires that u(x) → −M asx→ 0. However, we have for ξj = (xj + xj+1)/2 that

limj→∞

u(ξj) = ψ(0) +∞∑

k=1

1k2

ψk(0) +∞∑

k=1

1k2

φ(xk) > −M. (8)

Nevertheless, we are able to improve the regularity result of Theorem 4.1.

Theorem 4.3. Suppose that u is a strict local minimum of (P) in the sense of theL2(Ω) topology. We also assume that Assumptions (A1)–(A3) hold. Then u belongsto H1

0 (Ω).

Let us remark that any global solution of (P) is a local solution of (P) in thesense of L2(Ω), but we can expect to have more local or global solutions in thesense of L2(Ω). Theorem 3.5 implies that u is at least a strict local minimum in thesense of L2(Ω), if the sufficient second-order optimality conditions are satisfied atu. This guarantees that u is the unique global solution in an L2(Ω)-neighborhood,but it does not exclude the case that u is an accumulation point of different localminima.Proof of Theorem 4.3. Let εu > 0 be chosen such that u is a strict global minimumof (P) in the closed ball Bεu(u) ⊂ L2(Ω). This implies that u is the unique globalsolution of the problem

(P0)

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

min J(u)


α ≤ u(x) ≤ β for a.e. x ∈ Ω, ‖u− u‖L2(Ω) ≤ εu

a ≤ yu(x) ≤ b ∀x ∈ K,

where yu is the solution of (1).


Now we take a sequence {xk}∞k=1 that is dense in K and consider the familyof control problems

(Pk)

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

min J(u)


α ≤ u(x) ≤ β for a.e. x ∈ Ω, ‖u− u‖L2(Ω) ≤ εu

a ≤ yu(xj) ≤ b, 1 ≤ j ≤ k.

It is obvious that u is a feasible control for every problem (Pk), hence theexistence of a global minimum uk of (Pk) follows easily by a standard argumenta-tion. The proof is split into three steps: First, we prove that the sequence {uk}∞k=1converges to u strongly in L2(Ω); in a second step we will check that the linearizedSlater condition corresponding to problem (Pk) holds for all sufficiently large k.Finally, we deduce the boundedness of {uk}∞k=1 in H1

0 (Ω).Step 1 – Convergence of {uk}∞k=1. By taking a subsequence, if necessary, we

can suppose that uk ⇀ u weakly in L2(Ω). This implies that yk = yuk→ y = yu

strongly in H10 (Ω) ∩ C(Ω). In view of the density of {xk}∞k=1 in K and using the

fact thata≤y(xj) = lim

k→∞yk(xj) ≤ b ∀ j ≥ 1,

we get a ≤ y(x) ≤ b for every x ∈ K. Clearly, u is a feasible control for problem(P0). Since u is the solution of (P0), uk is a solution of (Pk) and u is a feasiblecontrol for every problem (Pk). Therefore, J(uk) ≤ J(u) and we get

J(u) ≤ J(u) ≤ lim infk→∞

J(uk) ≤ lim supk→∞

J(uk) ≤ lim infk→∞

J(u) = J(u),

which implies that u = u and lim J(uk) = J(u). Hence the strong convergenceuk → u in L2(Ω) follows from the last equality.

Step 2 – The linearized Slater condition for (Pk) is satisfied at uk. This followsfrom a standard idea: With some ε > 0, one takes the control u0,ε = ε(u0− u)+ u,where u0 is taken from the Slater condition (A3). Then it is not difficult to showthat the linearized Slater condition for (Pk) is satisfied for all sufficiently large k.

Step 3 – {uk}∞k=1 is bounded in H10 (Ω). The strong convergence uk → u in

L2(Ω) implies that ‖uk − u‖L2(Ω) < εu holds for all sufficiently large k. Therefore,uk is a local minimum of the problem

(Qk)

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

min J(u)


α ≤ u(x) ≤ β for a.e. x ∈ Ω,a ≤ yu(xj) ≤ b, 1 ≤ j ≤ k.

Next, we apply Theorem 3.4 and deduce

uk(x) = Proj[α,β]

(− 1Nϕk(x)

)= min

{max

{α,− 1

Nϕk(x)

}, β}, (9)


with

ϕk = ϕk,0 +k∑

j=1

λk,jϕk,j . (10)

Above, {λk,j}kj=1 are the Lagrange multipliers, more precisely

μk =k∑

j=1

λk,jδxj , with λk,j ={≥ 0 if yk(xj) = b,≤ 0 if yk(xj) = a.

(11)

Finally, ϕk,0 and {ϕk,j}kj=1 are given by{

−Δϕk,0 + a′0(yk(x))ϕk,0 = yk − yd in Ω,

ϕk,0 = 0 on Γ,(12)

{−Δϕk,j + a′0(yk(x))ϕk,j = δxj in Ω,

ϕk,j = 0 on Γ.(13)

It is known that the linearized Slater condition implies the boundedness ofthe Lagrange multipliers

∃C > 0 such that ‖μk‖M(K) =k∑

j=1

|λk,j | ≤ C ∀k. (14)

We suppress the associated proof. Now (10), (12) and (13) lead to{−Δϕk + a′0(yk(x))ϕk = yk − yd + μk in Ω,

ϕk = 0 on Γ.(15)

Define

Cα,β = |α|+ |β|+ 1 and vk(x) = Proj[−Cα,β,+Cα,β ]

(− 1Nϕk(x)

).

From the last relation and (9) it follows that

uk(x) = Proj[α,β](vk(x)).

The goal is to prove that {vk}∞k=1 is bounded inH10 (Ω), which implies the bounded-

ness of {uk}∞k=1 in the same space. The last claim is an immediate consequence of

|∇uk(x)| ≤ |∇vk(x)| for a.e. x ∈ Ω.If {uk}∞k=1 is bounded in H1

0 (Ω), then obviously u belongs to H10 (Ω).

Let us prove the boundedness of {vk}∞k=1 in H10 (Ω). Notice that the solution

of the Dirichlet problem for −Δ and a Lipschitz boundary Γ belongs to W 1,r0 (Ω)

if the right-hand side belongs to W−1,r(Ω) for any n < r < n + εn, with εn > 0depending on n and n = 2 or 3; see Jerison and Kenig [8] and Mateos [9]. SinceL2(Ω) ⊂W−1,6(Ω) for n ≤ 3, we have that ϕk,0 ∈ W 1,r

0 (Ω) holds for all r ≤ 6 in therange indicated above. From this, it follows ϕk ∈W 1,s

0 (Ω)∩W 1,r(Ω\Sk), where Sk

is the set of points xj such that λk,j �= 0. Taking into account that vk is constant


in a neighborhood of every point xj ∈ Sk, we obtain vk ∈ W 1,r0 (Ω) ⊂ C(Ω).

Multiplying equation (15) by −vk and integrating by parts, we get

−∫Ω

(∇vk · ∇ϕk + a′0(yk)vkϕk) dx = −∫Ω

(yk − yd)vk dx−k∑

j=1

λk,jvk(xj). (16)

The definition of vk implies for almost all x ∈ Ω

∇vk(x) =

⎧⎨⎩ − 1N∇ϕk(x) if −Cα,β ≤ −

1Nϕk(x) ≤ +Cα,β

0 otherwise.(17)

Invoking this property in (16) along with the boundedness of {yk}∞k=1 in C(Ω),the estimate ‖vk‖L∞(Ω) ≤ Cα,β and assumptions (A3), (A2) we deduce

λAN

∫Ω

|∇vk|2 dx ≤ ‖ψM‖L2(Ω)‖vk‖L2(Ω) + C

k∑j=1

|λk,j |‖vk‖L∞(Ω) ≤ C′,

which implies that {vk}∞k=1 is bounded in H10 (Ω) as required.

5. On the uniqueness of the Lagrange multiplier μ

Finally, we provide a sufficient condition for the uniqueness of the Lagrange mul-tiplier associated to the state constraints. We also analyze some situations underwhich the condition is satisfied.

Theorem 5.1. Assume (A1)–(A3) and the existence of some ε > 0 such that

T : L2(Ωε) −→ C(K0), v �→ zv, has a dense range , (1)

whereΩε = {x ∈ Ω : α+ ε < u(x) < β − ε},

zv ∈ H10 (Ω) ∩ C(Ω) satisfies{

−Δzv + a′0(y)zv = v in Ωzv = 0 on Γ, (2)

and v is extended by zero to the whole domain Ω. Then there exists a uniqueLagrange multiplier μ ∈M(K) such that (8)–(10) holds.

Proof. Assume to the contrary that μi, i = 1, 2, are two Lagrange multipliersassociated with the state constraints corresponding to the optimal control u. Then(14) holds for μ = μi, i = 1, 2. Take an arbitrary v ∈ L∞(Ωε) \ {0}. Then we have

α ≤ uρ(x) = u(x) + ρv(x) ≤ β for a.e. x ∈ Ω, ∀|ρ| < ε

‖v‖L∞(Ωε),


where v is extended by zero to the whole domain Ω. Taking u = uρ in (14), with ρpositive and negative respectively, and remembering that supp μi ⊂ K0 (Remark3.2), we deduce

J ′(u)v +∫

K0

zv(x) dμi(x) =∂L∂u

(u, μi)v = 0, i = 1, 2,

which leads to

〈μ1, T v〉 = −J ′(u)v = 〈μ2, T v〉 ∀v ∈ L∞(Ωε).

Since L∞(Ωε) is dense in L2(Ωε) and T (L2(Ωε)) is dense in C(K0), this identityimplies μ1 = μ2. �

Remark 5.1. For a finite set K = {xj}nj=1, assumption (1) is equivalent to the

independence of the gradients {G′j(u)}j∈I0 in L2(Ωε), where the functions Gj :L2(Ωε) −→ R are defined by Gj(u) = g(xj , yu(xj)) and I0 is the set of indexesj corresponding to active constraints. It is a regularity assumption on the controlproblem at u. This type of assumption was introduced by the authors in [4] to ana-lyze control constrained problems with finitely many state constraints. The first au-thor proved in [1] that, under very general hypotheses, this assumption is equivalentto the Slater condition in the case of a finite number of pointwise state constraints.

Remark 5.2. We are able to show that (1) holds under some assumptions on uand the set of points K0 where the state constraint is active. For instance, assumein addition to (A1)–(A3) that also

1. the Lebesgue measure of K0 is zero and that2. there exists ε > 0 such that, for every open connected component A of Ω\K0,

the set A ∩ Ωε has a nonempty interior.

Then the regularity assumption (1) is satisfied, as it will be proven in a forthcomingpaper. We are also able to show that the regularity hypothesis (1) is stronger thanthe linearized Slater assumption (A3).

Remark 5.3. We know u ∈ C(Ω \K0) from Theorem 4.1, hence the assumption 2of the theorem holds if u is not identically equal to α or β in any open connectedcomponent A ⊂ Ω \ K0. Indeed, since u ∈ C(A) and u �≡ α and u �≡ β in A,there exists x0 ∈ A such that α < u(x0) < β. Then the continuity of u implies theexistence of ε > 0 such that A ∩ Ωε contains a ball Bρ(x0).

References

[1] E. Casas, Necessary and sufficient optimality conditions for elliptic control problemswith finitely many pointwise state constraints, ESAIM:COCV, (to appear).

[2] E. Casas, J. de los Reyes, and F. Troltzsch, Sufficient second order optimalityconditions for semilinear control problems with pointwise state constraints, SIAM J.Optim., (to appear).


[3] E. Casas and M. Mateos, Second order optimality conditions for semilinear ellipticcontrol problems with finitely many state constraints, SIAM J. Control Optim., 40(2002), pp. 1431–1454.

[4] E. Casas and F. Troltzsch, Second order necessary optimality conditions forsome state-constrained control problems of semilinear elliptic equations, App. Math.Optim., 39 (1999), pp. 211–227.

[5] E. Casas, F. Troltzsch, and A. Unger, Second order sufficient optimality con-ditions for some state-constrained control problems of semilinear elliptic equations,SIAM J. Control Optim., 38 (2000), pp. 1369–1391.

[6] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of SecondOrder, Springer-Verlag, Berlin-Heidelberg-New York, 1977.

[7] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston-London-Melbourne, 1985.

[8] D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz do-mains, J. Funct. Anal., 130 (1995), pp. 161–219.

[9] M. Mateos, Problemas de control optimo gobernados por ecuaciones semilinealescon restricciones de tipo integral sobre el gradiente del estado, PhD thesis, Universityof Cantabria, 2000.

[10] H. Maurer and J. Zowe, First- and second-order conditions in infinite-dimensionalprogramming problems, Math. Programming, 16 (1979), pp. 98–110.

[11] G. Stampacchia, Le probleme de Dirichlet pour les equations elliptiques du secondordre a coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), pp. 189–258.

Eduardo CasasDepartmento de MatematicaAplicada y Ciencias de la ComputacionE.T.S.I. Industriales y de TelecomunicacionUniversidad de Cantabria39005 Santander, Spaine-mail: [email protected]

Fredi TroltzschInstitut fur MathematikTechnische Universitat BerlinD–10623 Berlin, Germanye-mail: [email protected]

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