Document [original]

Technische Universit¨at Berlin

Institut f¨ur Mathematik

Optimal control problems with convex control

constraints

Preprint 35-2005

Preprint-Reihe des Instituts f¨ur Mathematik

Technische Universit¨at Berlin

Report 35-2005 January 2006

Optimal control problems with convex control

constraints

Daniel Wachsmuth

Abstract. We investigate optimal control problems with vector-valued con-

trols. As model problem serve the optimal distributed control of the instation-

ary Navier-Stokes equations. We study pointwise convex control constraints,

which is a constraint of the form u(x, t)∈U(x, t) that has to hold on the do-

main Q. Here, Uis an set-valued mapping that is assumed to be measurable

with convex and closed images. We establish first-order necessary as well as

second-order sufficient optimality conditions. And we prove regularity results

for locally optimal controls.

1. Introduction

In fluid dynamics the control can be brought into the system by blowing or suction

on the boundary. Then the control is a velocity, which is a directed quantity,

hence it is a vector in R2respectively R3. That is, the optimal control problem is

to find a vector-valued function u∈Lp((0, T )×Ω)n. Distributed control can be

realized for instance as a force induced by an outer magnet field in a conducting

fluid, see e.g. Kunisch and Griesse [14]. There, the control uis a function of class

L2(Q)2=L2(Q;R2). This illustrates that the control is a directed quantity: it

consists of a direction and an absolute value. Or in other words, the control uat

a point (x, t) is a vector in R2.

The optimization has to take into account that one is not able to realize

arbitrarily large controls. To this end, control constraints are introduced. If the

control u(x, t) is only a scalar variable such as heating or cooling then there is only

one choice of a convex pointwise control constraint: the so-called box constraints

(1a) ua(x, t)≤u(x, t)≤ub(x, t).

For the analysis of optimal control of non-stationary Navier-Stokes equations using

this particular type of control constraints, we refer to Hinze and Hinterm¨uller [16],

Roub´ıˇccek and Tr¨oltzsch [23], Tr¨oltzsch and Wachsmuth [25], and Wachsmuth [28].

But these box constraints are not the only choice for vector-valued controls. For

instance, if one wants to bound the R2-norm of the control, one gets a nonlinear

constraint

(1b) |u(x, t)|=pu1(x, t)2+u2(x, t)2≤ρ(x, t).

2 D. Wachsmuth

What happens if the control is not allowed to act in all possible directions but

only in directions of a segment with an angle less than π? Using polar coordinates

ur(x, t) and uφ(x, t) for the control vector u(x, t), this can be formulated as

(1c) 0 ≤ur(x, t)≤ψ(uφ(x, t), x, t),

where the function ψmodels the shape of the set of allowed control actions.

Here, we will use another — and more natural — representation of the con-

straints. Let us denote by Uthe set of admissible control vectors. Then we can

write the control constraints (1a)–(1c) as an inclusion

u(x, t)∈U.

The advantage of this approach is that the analysis is based on rather elementary

say geometrical arguments, hence there is no need of any constraint qualification.

We will impose assumptions on Uthat allow to apply the common theory of

existence and optimality condition: non-emptyness, convexity, and closedness, but

no boundedness or further regularity of the boundary. We have to admit that the

assumption of convexity gives some inherent regularity, the boundary of convex

sets is locally Lipschitz. However, even in the convex case, there can be very

irregular situations: one can construct convex sets in R2with countably many

corners, which lie dense on the boundary, see [10].

The formulation of the control constraint as an inclusion has a further benefit:

the set of admissible control vectors can vary over time and space by simply writing

u(x, t)∈U(x, t),

without causing any additional problems. The main difficulty appears already in

the non-varying case, see the discussion in Section 7.1 below.

Optimal control problems with such control constraints are rarely investi-

gated in literature. Second-order necessary conditions for problems with the con-

trol constraint u(ξ)∈U(ξ) were proven by P´ales and Zeidan [20] involving second-

order admissible variations. Second-order necessary as well as sufficient conditions

were established in Bonnans [5], Bonnans and Shapiro [8], and Dunn [12]. How-

ever, the set of admissible controls has to be polygonal and independent of ξ, i.e.

U(ξ)≡U. This results were extended by Bonnans and Zidani [9] to the case of

finitely many convex contraints gi(u(ξ)) = 0, i= 1,...,l. As already mentioned,

we will follow another approach and treat the control constraint as an inclusion

u(x, t)∈U(x, t). State constraints of the form y(x, t)∈Care considered in the

recent research paper by Griesse and de los Reyes [13].

As a model problem serves the optimal distributed control of the instationary

Navier-Stokes equations in two dimensions. We emphasize that the restriction to

two dimensions, i.e. u∈L2(Q)2, is only due to the limitation of the analysis of

instationary Navier-Stokes equations. As long as there exists an applicable theory

of a state equation in Rn, all results regarding convex control constraints are ready

for an extension to the n-dimensional case.

Convex control constraints 3

To be more specific, we want to minimize the following quadratic objective

functional:

(2) J(y, u) = αT

2ZΩ

|y(x, T)−yT(x)|2dx+αQ

2ZQ

|y(x, t)−yQ(x, t)|2dxdt

+αR

2ZQ

|curl y(x, t)|2dxdt+γ

2ZQ

|u(x, t)|2dxdt

subject to the instationary Navier-Stokes equations

(3)

yt−ν∆y+ (y· ∇)y+∇p=uin Q,

div y= 0 in Q,

y(0) = y0in Ω,

and to the control constraints u∈Uad with set of admissible controls defined by

(4) Uad ={u∈L2(Q)2:u(x, t)∈U(x, t) a.e. on Q}.

Here, Ω is a bounded domain in R2,Qdenotes the time-space cylinder Q:=

Ω×(0, T). Let us underline the fact that for (x, t)∈Qthe control u(x, t) is a

vector in R2.

The conditions imposed on the various ingredients of the optimal control

problem are specified in Sections 2.1 and 4.1, see assumptions (A) and (AU).

For the optimal control of the non-stationary Navier-Stokes equations there

are several articles about existence of solution and necessary optimality conditions,

for instance Abergel and Temam [1], Gunzburger and Manservisi [15]. Sufficient

optimality conditions and second-order optimization methods were investigated by

Hinze [17], Hinze and Kunisch [18], Ulbrich [26], and Tr¨oltzsch and Wachsmuth

[25]. However, in these articles only the box constraints (1a) or even no control

constraints are considered.

The plan of the article is as follows. At first we introduce some notation and

results concerning the state equation in Section 2. Set-valued mappings are the

subject of Section 3. The exact statement of our model problem can be found in

Section 4 together with first-order necessary optimality conditions in Section 5. We

prove regularity results for locally optimal controls in Section 6. Finally, we discuss

sufficient optimality conditions and stability of optimal controls in Sections 7 and 8

respectively.

2. Notations and preliminary results

At first, we introduce some notations and results that we will need later on. To

begin with, we define the spaces of solenoidal or divergence-free functions

H:= {v∈L2(Ω)2: div v= 0}, V := {v∈H1

0(Ω)2: div v= 0}.

These spaces are Hilbert spaces with scalar products (·,·)Hand (·,·)Vrespectively.

The dual of Vwith respect to the scalar product of Hwe denote by V0with the

duality pairing h·,·iV0,V .

4 D. Wachsmuth

We will work with the standard spaces of abstract functions from [0, T] to a

real Banach space X,Lp(0, T;X), endowed with its natural norm,

kykLp(X):= kykLp(0,T ;X)= ZT

|y(t)|p

Xdt!1/p

1≤p < ∞,

kykL∞(X):= ess sup

t∈(0,T )

|y(t)|X.

In the sequel, we will identify the spaces Lp(0, T;Lp(Ω)2) and Lp(Q)2for 1 < p <

∞, and denote their norm by kukp:= |u|Lp(Q)2.The usual L2(Q)2-scalar product

we denote by (·,·)Qto avoid ambiguity.

In all what follows, k · k stands for norms of abstract functions, while | · |

denotes norms of ”stationary” spaces like Hand V.

To deal with the time derivative in (3), we introduce the common spaces of

functions ywhose time derivatives ytexist as abstract functions,

Wα(0, T;V) := {y∈L2(0, T;V) : yt∈Lα(0, T;V0)}, W(0, T) := W2(0, T ;V),

where 1 ≤α≤2. Endowed with the norm

kykWα(0,T ;V):= kykL2(V)+kytkLα(V0),

these spaces are Banach spaces. Every function of W(0, T) is, up to changes on

sets of zero measure, equivalent to a function of C([0, T ], H), and the imbedding

W(0, T),→C([0, T], H) is continuous, cf. [2, 19].

2.1. The state equation

Before we start with the discussion of the state equation, we specify the require-

ments for the various ingredients describing the optimal control problem. In the

sequel, we assume that the following conditions are satisfied:

(A)









1. Ω has Lipschitz boundary Γ := ∂Ω,

2. y0, yT∈H,yQ∈L2(Q)2,

3. αT, αQ, αR≥0,

4. γ, ν > 0.

The assumptions on the set-valued mapping Uare given in the next section. Now,

we will briefly summarize known facts about the solvability of the instationary

Navier-Stokes equations (3). First, we define the trilinear form b:V×V×V7→ R

b(u, v, w) = ((u· ∇)v, w)2=ZΩ

i,j=1

∂vj

∂xi

wjdx.

Its time integral is denoted by bQ,

bQ(y, v, w) = ZT

b(y(t), v(t), w(t)) dt.

Convex control constraints 5

To specify the problem setting, we introduce a linear operator A:L2(0, T;V)7→

L2(0, T;V0) by

h(Ay)(t), v(t)iV0,V dt:= ZT

(y(t), v(t))Vdt,

and a nonlinear operator Bby

0B(y)(t), v(t)V0,V dt:= ZT

b(y(t), y(t), v(t)) dt.

For instance, the operator Bis continuous and twice Frech´et-differentiable as op-

erator from W(0, T ) to L2(0, T;V0).

Now, we concretize the notation of weak solutions for the instationary Navier-

Stokes equations (3) in the Hilbert space setting.

Definition 2.1 (Weak solution).Let f∈L2(0, T;V0)and y0∈Hbe given. A

function y∈L2(0, T;V)with yt∈L2(0, T;V0)is called weak solution of (3) if

(5) yt+νAy +B(y) = f,

y(0) = y0.

Results concerning the solvability of (5) are standard, cf. [24] for proofs and

further details.

Theorem 2.2 (Existence and uniqueness of solutions).For every source term f∈

L2(0, T;V0)and initial value y0∈H, the equation (5) has a unique solution

y∈W(0, T). Moreover, the mapping (y0, f)7→ yis locally Lipschitz continuous

from H×L2(0, T ;V0)to W(0, T).

It is well-known that the control-to-state mapping is Fr´echet-differentiable.

The first derivative can be computed as the solution of a linearized equation,

cf. [15, 17, 18].

Remark 2.3 (Linearized state equation).We consider the linearized equation

(6) yt+νAy +B0(¯y)y=f,

y(0) = y0,

for a given state ¯y, which is usually the solution of the nonlinear system (5).

Following the lines of Temam, existence and uniqueness of a weak solution yin

the space W(0, T)was proven for instance in [18, Prop. 2.4]. See also the discussion

in [15].

3. Set-valued functions

Before we begin with the formulation of the optimal control problem with inclusion

constraints, we will provide some background material. Here, we will specify the

notation and assumptions for the admissible set U(·). It is itself a mapping from the

control domain Qto the set of subsets of R2, it is a so-called set-valued mapping.

We will use the notation U:Q R2.

6 D. Wachsmuth

The optimal control problem is the minimization of the objective functional

subject to the state equations and to the control constraint

(7) u(x, t)∈U(x, t).

The controls are taken from the space L2(Q)2, so it is natural to require the

fulfillment of (7) for (only) almost all (x, t)∈Q. And we have to impose at least

some measurability conditions on the mapping U. In the sequel, we will work

with measurable set-valued mappings. For an excellent — and for our purposes

complete — introduction we refer to the textbook by Aubin and Frankowska [4].

Definition 3.1. A set-valued mapping F:Q Xwith closed images is called

measurable, if the inverse of each open set is measurable. In other words, for

every open subset O ⊂ Xthe inverse image

F−1(O) = {ω∈Q:F(ω)∩ O 6=∅}

has to be measurable.

Observe, that for a single-valued function fthe definition of measurability

coincides with the definition of measurability for the set-valued function ˜

fgiven

by ˜

f(ω) = {f(ω)}.

However, this definition does not imply the existence of a measurable se-

lection, which is a single-valued function fsatisfying f(x, t)∈U(x, t) almost

everywhere on Q. The existence is guaranteed under additional assumptions on U.

Theorem 3.2. [4, Th. 8.1.4] Let F:Q R2be a set-valued mapping with non-

empty closed images. Then the following two statements are equivalent:

1. Fis measurable

2. There exists a sequence of measurable selections {fn}∞

n=1 of Fsuch that

for all (x, t)∈Qit holds

F(x, t) = [

n≥1

{fn(x, t)}.

The theorem gives not only the existence of a measurable selection but also

a tool to prove measurability of set-valued mappings based on countable approxi-

mations.

It is well-known that every optimal control is the projection of its associated

state on the admissible set. Such a characterization is also valid in the set-valued

constraint case. But as a first step, we have to make sure that the pointwise

projection on the set-valued mapping Upreserves measurability.

Theorem 3.3. [4, Cor. 8.2.13] Let F:Q R2be a set-valued measurable mapping

with closed, non-empty, and convex images, and f:Q7→ R2a measurable (single-

valued) mapping. Then the projection

g(x, t) = ProjF(x,t)(f(x, t))

is a single-valued measurable function too.

Convex control constraints 7

4. The optimal control problem

Here, we will investigate the optimal control problem with the control constraint (7).

At first, we have to specify the assumptions to ensure existence of solutions.

4.1. Set of admissible controls

In this section, we want to investigate the convex control constraint, which has to

hold pointwise

u(x, t)∈U(x, t) a.e. on Q.

We recall the definition of the set of admissible controls Uad,

Uad ={u∈L2(Q)2:u(x, t)∈U(x, t) a.e. on Q}.

Once and for all, we specify the requirements for the function U, which defines the

control constraints.

(AU)











The set-valued function U:Q R2satisfies:

1. Uis a measurable set-valued function.

2. The images of Uare non-empty, closed, and convex a.e.

on Q. That is, the sets U(x, t)are non-empty, closed and

convex for almost all (x, t)∈Q.

3. There exists a function fU∈L2(Q)2with fU(x, t)∈

U(x, t)a.e. on Q.

Please note, we did not impose any conditions on the sets U(x, t) that are beyond

convexity such as boundedness or regularity of the boundaries ∂U(x, t). Assump-

tions (i) and (ii) guarantee that there exists a measurable selection of U, i.e. a

measurable single-valued function fMwith fM(x, t)∈U(x, t) a.e. on Q. However,

no measurable selection needs to be square-integrable as the following example

shows.

Example 4.1. Set U(t) = [t−1/2,1 + t−1/2],0< t ≤1. Assumptions (i) and (ii)

are fulfilled. But every function fwith f(t)∈U(t)for almost all 0< t ≤1cannot

be in L2(0,1), since the function g(t) = t−1/2is not square integrable on [0,1].

The existence of a square integrable, admissible function is then ensured by

the third assumption. This implies that the set of admissible control is non-empty.

Corollary 4.2. The set of admissible controls Uad defined by

Uad ={u∈L2(Q)2:u(x, t)∈U(x, t)a.e. on Q}

is non-empty, convex and closed in L2(Q)2.

The assumption (AU) is as general as the analysis of the second-order con-

dition allows it. In the case that the set-valued function Uis a constant function,

i.e. U(x, t)≡U0, we can give a simpler characterization.

Corollary 4.3. Let the set-valued function Ube a constant function, i.e. U(x, t) =

U0a.e. on Qfor some U0⊂R2. Then the assumption (AU) is fulfilled if the set

U0is non-empty, closed, and convex.

8 D. Wachsmuth

Assuming (AU) we can derive another interesting result. Conditon (iii) allows

us to prove that the pointwise projection on Uad of a L2-function is itself a L2-

function.

Corollary 4.4. Let be given a function u∈L2(Q)2. Then the function vdefined

pointwise a.e. by

v(x, t) = ProjU(x,t)(u(x, t))

is also in L2(Q)2. Further, if for some p≥2the functions uand fUare in Lp(Q)2,

then the projection vis in Lp(Q)2as well.

Proof. By assumption (AU), the set-valued function Uis measurable with closed

and convex images, and uis a measurable single-valued function. Then by The-

orem 3.3 the function vis measurable as well. By Lipschitz continuity of the

pointwise projection, it holds

|v(x, t)−fU(x, t)|=|ProjU(x,t)(u(x, t)) −ProjU(x,t)(fU(x, t))|

≤ |u(x, t)−fU(x, t)|

almost everywhere on Q. Thus, squaring and integrating gives

kv−fUk2

2≤ ku−fUk2

2<∞,

which implies v∈L2(Q)2. If in addition, uand fUare in Lp(Q)2for some p > 2,

then we can prove analogously that the projection is also in Lp, i.e. v∈Lp(Q)2.

4.2. Existence of optimal controls

Before we can think about existence of solution, we have to specify which problem

we want to solve. We will assume that conditions (A) of Section 2.1 are satified.

Moreover, we assume that U(·) fulfills the pre-requisite (AU). So we end up with

the following optimization problem

(8a) min J(y, u)

subject to the state equation

yt+νAy +B(y) = uin L2(0, T;V0),(8b)

y(0) = y0in H,(8c)

and the control constraint

(8d) u∈Uad,

where Uad is given by (7).

Under the assumptions above, the optimal control problem (8) is solvable.

We recall that in Section 2.1 the regularization parameter γis supposed to be

greater than zero. One can prove existence even with γ= 0 under the additional

condition of boundedness of Uad in L2.

Theorem 4.5. The optimal control problem admits a - global optimal - solution

¯u∈Uad with associated state ¯y∈W(0, T ).

Convex control constraints 9

5. First-order necessary conditions

The necessary optimality conditions for the optimal control problem discussed

in the present chapter differ slightly from the conditions that can be found in the

literature, see e.g. [25]. However, we will repeat the exact statement for convenience

of the reader.

Theorem 5.1 (Necessary condition).Let ¯ube locally optimal in L2(Q)2with as-

sociated state ¯y=y(¯u). Then there exists a unique Lagrange multiplier ¯

λ∈

W4/3(0, T;V), which is the weak solution of the adjoint equation

(9) −¯

λt+νA¯

λ+B0(¯y)∗¯

λ=αQ(¯y−yQ) + αRcurl∗curl ¯y

λ(T) = αT(¯y(T)−yT).

Moreover, the variational inequality

(10) (γ¯u+¯

λ, u −¯u)Q≥0∀u∈Uad

is satisfied.

Similar as in the box-constrained case, we can reformulate the variational

inequality (10). The projection representation of the optimal control is now realized

using the admissible sets U(·)

(11) ¯u(x, t) = ProjU(x,t)−1

γ¯

λ(x, t)a.e. on Q.

Here, it will be a little bit more difficult to prove regularity results for the optimal

control using the regularity of the adjoint state. The projection formula is used in

connection with Lipschitz stability of optimal controls [16, 23, 28]. The ideas there

cannot be transferred to the case of set-valued constraints, see the discussion in

Section 8 below.

Necessary optimality conditions of second order for optimal control problems

with set-valued constraints were developped in [20]. It involves the use of the

concept of second-order tangent, see e.g. [11].

6. Regularity of optimal controls

Let us comment on the regularity of a locally optimal control ¯u. By (11), it inherits

some regularity from the associated adjoint state ¯

λ. Here, we will show, how the

regularities ¯

λ∈Lp(Q)2respectively ¯

λ∈C(¯

Q)2can be carried over to the control

¯u. However, it is not clear whether and how it is possible to prove ¯u∈W1,p(Q)2

if ¯

λ∈W1,p(Q)2, and what assumptions on Uare needed.

6.1. Optimal controls in Lp

Corollary 4.4 gives a hint, how we can prove the regularity ¯u∈Lp(Q)2provided

λ∈Lp(Q)2holds. We have to assume only the existence of an admissible Lp-

function.

10 D. Wachsmuth

Theorem 6.1. Let ¯ube a locally optimal control of the optimal control problem (8)

with associated adjoint state λ∈Lp(Q)2,p≤ ∞. If there is an admissible function

fp∈Lp(Q)2∩Uad for p≤ ∞ then the optimal control ¯uis in that Lp(Q)2, too.

Proof. The proof follows immediately from the projection representation (11) and

Corollary 4.4.

We will complete this short section with the following corollary, which states

the precise regularity assumptions on the problem data, such that the pre-requisites

of the previous theorem are fulfilled, see also [28].

Corollary 6.2. Let be given y0, yT∈V,yQ∈L2(Q)2. Let the set-valued mapping

Usatisfy the assumption (AU). Further, we assume the existence of an admissible

Lp-function fp∈Lp(Q)2∩Uad for 2≤p < ∞.

Then every locally optimal control of problem (8) is in Lp(Q)2,2≤p < ∞.

The method of proof applied here does not work to obtain continuity of an

optimal control. This is investigated in the next section.

6.2. Continuity of optimal controls

Now, we are going to prove continuity of an locally optimal control. We will rely

in our considerations again on the projection formula (11), which says that the

optimal control is the pointwise projection of a continuous function on the admis-

sible sets. Hence, this admissible sets U(x, t) vary over space and time. Here, we

have to impose some continuity assumptions on the set-valued mapping U.

There are two equivalent characterizations of continuous single-valued functions:

1. the image of a converging sequence is also a converging sequence,

2. the preimages of open sets are open sets.

In the set-valued case, however both definitions of a continuous function are

no longer equivalent. They define two independent kinds of semicontinuity.

Definition 6.3. A set-valued mapping F:D⊂X Yis called lower semicontinu-

ous, if for all x∈D,y∈F(x), and any sequence {xn} ⊂ Dconverging to xthere

is a sequence of elements yn∈F(xn)converging to y.

Definition 6.4. A set-valued mapping F:D⊂X Yis called upper semicontin-

uous, if for all x∈Dand all open sets O⊃F(x)there exists δ=δ(O)such that

F(x0)⊂Ofor all x0with |x−x0| ≤ δ.

Both definition are not equivalent and are independent. There are set-valued

mappings, which are lower semicontinuous but not upper and vice-versa. It is

natural to define a continuous mapping to have both semicontinuous properties.

Definition 6.5. A set-valued mapping F:D⊂X Yis called continuous, if U

is both lower and upper semicontinuous.

Convex control constraints 11

The assumption (AU) on the set-valued mapping Ucontains the condition

that U(x, t) is non-empty, closed and convex almost everywhere on Q. Do these

properties of the images hold everywhere provided Uis continuous? At first, we

want to show the improvement from ’non-empty almost everywhere’ to ’non-empty

everywhere’ in the continuous case.

Lemma 6.6. Let Ufulfill (AU). Further let U:¯

Q R2be upper semicontinuous.

Then U(x, t)is non-empty for all (x, t)∈¯

Proof. We will proof it by contradiction. Let ξ= (x, t)∈¯

Qsuch that U(ξ) is

empty. Then we take a sequence of points ξn= (xn, tn)∈Qwith U(ξn)6=∅

converging to ξ. Now, take points un∈U(ξn) and set O=R2\∪∞

n=1un. Then un∈

U(ξn)6⊂ Oholds for all n, which is a contradiction to upper semicontinuity.

Unfortunately, the property ’closedness of the images’ cannot be transferred

from ’almost everywhere’ to ’everywhere’ for continuous Uas the following coun-

terexample shows.

Example 6.7. Define F: [0,1] Rby

F(t) = ((0,1] if t= 0

[t, 1 + t]otherwise.

Clearly, Fis lower semicontinuous. It is also upper semicontinuous: every open

set that contains F(0) contains also F(ε)for sufficiently small ε. Hence Fis

continuous. It has closed images almost everywhere but not everywhere.

Now, let us prove an lemma, which will help us later on.

Lemma 6.8. Let Ufulfill (AU). In addition, we assume that Uis upper semicontin-

uous on ¯

Qwith closed images U(x, t)for all (x, t)∈¯

Q. Then for given sequences

(xn, tn)converging to (x, t)∈¯

Qand yn∈U(xn, tn)converging to ythe limit y

lies in U(x, t),y∈U(x, t).

Proof. We will use again the notation ξ= (x, t) and ξn= (xn, tn). Let us assume

y6∈ U(ξ). Set ε= dist(y, U(ξ)), which is positive since U(ξ) is closed. Then there

exists Nsuch that for all n > N it holds yn6∈ U(ξ) and dist(yn, U(ξ)) ≥2

3ε. Now,

we construct an open set by O:= {v: dist(v, U(ξ)) <1

3ε}. It implies yn6∈ Oand

U(ξn)6⊂ Ofor n≥N. This yields a contradiction to upper semicontinuity, since

we have O⊃U(ξ). Hence it holds y∈U(ξ).

Furthermore, it turns out that the assumption of closed images is essential

to prove the convexity of the images of U.

Lemma 6.9. Let Ufulfill (AU). In addition, let Ube continuous on ¯

Qwith closed

images U(x, t)for all (x, t)∈¯

Q. Then U(x, t)is convex for all (x, t)∈¯

Proof. Let ξ= (x, t)∈¯

Qbe given with y1, y2∈U(ξ), λ∈(0,1). We have to show

that λy1+ (1 −λ)y2is in U(ξ). We take a sequence of points ξn= (xn, tn)∈Q,

for which U(ξn) is non-empty and convex, converging to ξ.

12 D. Wachsmuth

By lower semicontinuity there exists sequences of points yn

1, yn

2∈U(ξn) con-

verging to y1respectively y2. The points yn:= λyn

1+ (1 −λ)yn

2are in U(ξn) and

converge to y:= λy1+(1−λ)y2for n→ ∞. The previous Lemma 6.8 implies that

the limit y=λy1+ (1 −λ)y2is in U(ξ). Hence U(ξ) is convex.

Assuming the continuity of the set-valued mapping Uand the adjoint state

λwe can prove continuity of an optimal control.

Theorem 6.10. Let Usatisfies the assumption (AU). Furthermore, let U:¯

Q Rn

be continuous with closed images everywhere. Suppose ¯usatisfies the first-order

necessary optimality conditions together with the state ¯yand adjoint ¯

λ. If the

adjoint is continuous, ¯

λ∈C(¯

Q), so is the control as well, ¯u∈C(¯

Q).

Proof. We will show that the projection

ProjU(x,t)−1

γ¯

λ(x, t)= ¯u(x, t)

results in a continuous function. We abbreviate v(x, t) := −¯

λ(x, t)/γ, which is a

continuous function by assumption.

Let ξ= (x, t)∈¯

Qbe given. Take a sequence ξn= (xn, tn)∈Qthat converges

to ξ. We have to show the convergence ¯u(ξn)→¯u(ξ). We will give the proof in

several steps.

Step 1: U(x, t)is non-empty, closed and convex everywhere on ¯

Q.This follows by

the preceding Lemmata 6.6 and 6.9.

Step 2: Uad contains a continuous function. Define the function m:¯

Q→Rnas

m(x, t) = arg min {|v|:v∈U(x, t)},

which gives the elements of U(x, t) with the smallest norm. Since U(x, t) is non-

empty, closed and convex, the function mis well-defined. By [3, Chapt. 3, Sect.

1, Prop. 23, p. 120], the single-valued function mis continuous. It is also called a

continuous selection of U.

Step 3: Boundedness of {¯u(ξn)}.Using Lipschitz continuity of the projection,

we can estimate

|¯u(ξn)−m(ξn)|=|ProjU(ξn)(v(ξn)) −ProjU(ξn)(m(ξn))|

≤ |v(ξn)−m(ξn)| ≤ kv−mkC(¯

Q)<∞,

which proves boundedness of the set {¯u(ξn)}.

Step 4: Every accumulation point of {¯u(ξn)}is in U(ξ).Since {¯u(ξn)}is

bounded in Rn, we can select a subsequence {¯u(ξn0)}converging to some element

˜u. By Lemma 6.8, we find that ˜uis in U(ξ).

Step 5: There is exacty one accumulation point of {¯u(ξ)}.Take an arbitrary

element z∈U(ξ). By lower semicontinuity, there is a sequence of elements zn0∈

F(ξn0) converging to z∈U(ξ). Since u(ξn0) = ProjU(ξn0)v(ξn0), we find

u(ξn0)−v(ξn0), zn0−u(ξn0)≥0∀n0.

Convex control constraints 13

Hence

u(ξn0)−v(ξn0), zn0−z+u(ξn0)−v(ξn0), z −u(ξn0)≥0∀n0.

Letting n0→ ∞, we find

˜u−v(ξ), z −˜u≥0.

Since z∈U(x) was arbitrary, it holds

˜u= ProjU(ξ)v(ξ)

for every accumulation point of {¯u(ξn)}. The projection is unique hence the set

{u(ξn)}has exactly one accumulation point.

Conclusion. By the previous step, we find the convergence ¯u(ξn)→¯u(ξ).

Hence the prove is complete, and ¯uis a continuous function on ¯

For an exact statement, which regularity of the data is sufficient for λ∈

C(¯

Q)2, we refer to [28].

Remark 6.11. The projection formula (11) remains true if one replaces U(x, t)by

its closure ¯

U(x, t), provided U(·)is closed almost everywhere on Q. Furthermore,

one can show that for continuous U:Q R2the closure ¯

U:Q R2is also a

continuous set-valued mapping. In this way, we can construct a continuous repre-

sentation of a locally optimal control ¯uwithout the assumption of closedness of the

images of U.

7. Second-order sufficient optimality conditions

7.1. Normal directions

Before we start with the formulation of the sufficient optimality conditions, let

us recall some notations of convex set theory. Let be given a convex set C. Then

NC(u) and TC(u) are the normal and tangent cones of Cat some point u. The

space of normal directions is written NC(u) = span NC(u) with its orthogonal

complement TC(u).

Now, we want to use these notations with C=Uad. Let be given an admissible

control u∈Uad. It is well-known, that the sets NUad (u), TUad (u), NUad (u), and

TUad (u) admit a pointwise representation as Uad itself, cf. [4, 22]. For instance, for

u∈L2(Q)2the set NUad (u) is given by

NUad (u) = v∈L2(Q)2:v(x, t)∈ NU(x,t)(u(x, t)) a.e. on Q.

In a while, we will need the projection of a test function won the space of normal

directions and its complement. We will denote the resulting functions by wNand

wTrespectively. They are defined pointwise by

(12) wN(x, t) = Proj[NU(x,t)(u(x,t))](w(x, t))

14 D. Wachsmuth

and

(13) wT(x, t) = Proj[TU(x,t)(u(x,t))](w(x, t)).

It is not easy to prove that the functions wNand wTare measurable. At this

point the method of Dunn [12] requires that the admissible set Uis polyhedric

and independent of (x, t). However, these restrictions can be overcome using the

results for set-valued mappings. Let us sketch the method of the measurability

proof. Behind the projections there are the following mappings:

1. Q3(x, t)7→ NU(x,t)(u(x, t)) =: N(x, t)

2. Q3(x, t)7→ span{N(x, t)}= span NU(x,t)(u(x, t))=: N(x, t)

3. Q3(x, t)7→ ProjN(x,t)(w(x, t)) =: wN(x, t).

Here, one can see, what happens if U(x, t) is constant over Q: the mapping Nis

even in this case a set-valued mapping, which is not constant. Even the dimension

of N(x, t) varies. So we would not have any advantage if we assume constant

admissible sets U(x, t) = U0.

Now, the measurability can be proven as follows. The mapping (x, t)

TU(x,t)(u(x, t)) is measurable if Uhas closed and convex images, cf. [4, Cor. 8.5.2].

The normal cone Nis then the dual of T, and one can prove that the dual cone

operation does preserve measurability. The same can be done for the linear hull

N. Here, the proof is based on the countability argument already stated in Theo-

rem 3.2.

Now, the projection of measurable function on measurable set-valued map-

pings — here N— results in a measurable function, see Theorem 3.3. Altogether,

both of the projections (12) and (13) results in measurable function. Let us remark

that it is very difficult to prove this by hand: here one has to go step by step from

regular convex sets and constant Uto more irregular ones.

As a second point here, let us define some more notations in connection to

convex sets. The relative interior of a convex set is defined by

ri C={x∈aff C:∃ε > 0, Bε(x)∩aff C⊂C},

its complement in Cis called the relative boundary

rb C=C\ri C.

The distance of a point u∈Rnto a set C⊂Rnis defined by

dist(u, C) = inf

x∈C|u−x|.

7.2. Sufficiency

Let us come back to optimization with convex constraints. To motivate the follow-

ing, we will investigate the finite-dimensional problem minx∈Cf(x) with C⊂Rn

first. It is known that a local minimizer xof the function fover a convex set C

fulfills

−∇f(x)∈ NC(x).

Convex control constraints 15

A sufficient condition is then given by

(14) −∇f(x)∈ri NC(x)

and

(15) f00(x)[y,y]>0∀y∈TC(x).

It consists of a first-order part: strict complementarity and a second-order part:

coercivity. Now we want to adapt this formulation to the optimal control problem

considered here. Condition (14) would become

(16) −(γ¯u(x, t) + ¯

λ(x, t)) ∈ri NU(x,t)(¯u(x, t)) a.e. on Q.

However, this is not enough for optimal control problems, we need the satis-

faction of this condition in a uniform sense. We have to assume not only that

−(γ¯u(x, t) + ¯

λ(x, t)) lies in the relative interior of the normal cone, we need more-

over that −(γ¯u(x, t) + ¯

λ(x, t)) has a positive distance to the relative boundary

of NU(x,t)(¯u(x, t)). But this cannot be assumed for all (x, t): if ¯u(x, t) is in the

interior of the admissible set U(x, t) then the normal cone consists only of the ori-

gin and has no relative interior. Therefore, we introduce the set of strongly active

constraints as the set of points, where this condition is fulfilled,

(17) Qε=(x, t)∈Q: dist −(γ¯u(x, t) + ¯

λ(x, t)),rb NU(x,t)(¯u(x, t))> ε.

That is, we assume that (16) is only fulfilled on a subset of the domain Q. Con-

sequently, we have to require the coercivity assumption for more directions than

included in TUad (¯u). Furthermore, the inequality >0 in (15) has to be replaced

by a norm-square, since the proof in finite dimensions that ’>0’ suffices is tied to

compactness of the unit sphere, which does not hold in the infinite dimensional

case.

Altogether, we require that the following is fulfilled. We assume that the

reference pair (¯y, ¯u) satisfies the coercivity assumption on L00(¯y, ¯u, ¯

λ), in the sequel

called second-order sufficient condition:

(SSC)











There exist ε > 0and δ > 0such that

(18a) L00(¯y, ¯u, ¯

λ)[(z, h)]2≥δkhk2

holds for all pairs (z, h)∈W(0, T )×L2(Q)2with

(18b) h∈ TUad (¯u), hN= 0 on Qε,

and z∈W(0, T)being the weak solution of the linearized equation

(18c) zt+Az +B0(¯y)z=h

z(0) = 0.

In (18b) hNdenotes the pointwise projection of hon the subspaces N⊂R2,

compare (12). We required in (SSC) the coercivity of L00 for more test functions

than in (15). The set TUad (¯u), which was user there, is only a subset of TUad (¯u).

However, the space of test functions in (SSC) can be reformulated as: h∈ TUad (¯u)

with h(x, t)∈TU(x,t)(¯u(x, t)) on Qε. We can use test functions with values in the

spaces Tonly on the strongly active set, due to the strong complementarity, which

16 D. Wachsmuth

holds there. On the rest of the domain, the values of the test function has to lie

in the tangent cones T.

Now, the next theorem states the sufficiency of (SSC).

Theorem 7.1. Let (¯y, ¯u)be admissible for the optimal control problem and suppose

that (¯y, ¯u)fulfills the first order necessary optimality conditions with associated

adjoint state ¯

λ. Assume further that (SSC) is satisfied at (¯y, ¯u). Then there exist

α > 0and ρ > 0such that

J(y, u)≥J(¯y, ¯u) + αku−¯uk2

holds for all admissible pairs (y, u)with ku−¯uk∞≤ρ.

The proof can be found in [27].

There are a number of sufficient second-order optimality conditions for finite-

dimensional optimization problems with convex constraints, see for instance [7, 8,

21]. They all use the second-order tangent sets, and it is not clear how those

results are related to the condition presented here. Also, the extension of the

finite-dimensional results to optimal control problems is not a trivial exercise and

requires further research.

Remark 7.2. Let us comment on the definition of the strongly active set in (17)

if Uis formed by box constraints. Since this particular constraint is formed by

two independent inequalities, one can refine the definition of strongly active sets,

see [25], to contain more points than the active set introduced here.

8. Stability of optimal controls

Usually, the fulfillment of a second-order sufficient condition implies stability of

locally optimal controls under small perturbations. This is demonstrated in a great

variety of articles for optimal control problems with box constraints. However, in

the case of general convex control constraints the sufficient condition (SSC) is too

weak to get stability of optimal controls. This is due to the fact, that tangent

variations of the control are not necessarily admissible directions, which is an

essential ingredient in the proofs for the box-constrained case.

In finite-dimensional optimization, there are a few publications concerning

stability of solutions to convex constrained optimization problems, see [6, 8]. They

use the assumption of second-order regular sets. The extension of that conditions

to the infinite-dimensional case considered here is not obvious, since the proofs

argue by contradiction and rely on the finite-dimensionality i.e. on compactness of

the unit sphere. That means, one has to use methods which differ from the indirect

methods of [6] as well as from the direct proofs from e.g. [16, 23, 25].

Obviously, if we assume coercivity of L00 for all test directions, we can prove

such stability results. Since this would be only a technical exercise, we do not

proceed in this direction.

Convex control constraints 17

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Daniel Wachsmuth,

Institut f¨ur Mathematik,

Technische Universit¨at Berlin,

Str. des 17. Juni 136,

D-10623 Berlin, Germany.

E-mail address:[email protected]n.de