Second order sufficient optimality conditions for parabolic optimal control problems with pointwise state constraints

Weierstraß-Institut

für Angewandte Analysis und Stochastik

Leibniz-Institut im Forschungsverbund Berlin e. V.

Preprint

ISSN 0946 – 8633

Second order sufficient optimality conditions for parabolic

optimal control problems with pointwise state constraints

Klaus Krumbiegel, Joachim Rehberg

submitted: February 29, 2012 Weierstrass Institute Mohrenstr. 39 10117 Berlin Germany E-Mail: [email protected] [email protected] No. 1700 Berlin 2012

2010Mathematics Subject Classification. 35K20, 35B65, 47F05, 49J20, 49K20.

Key words and phrases. Parabolic equation, maximal parabolic regularity, optimal control, sufficient optimality condi-tions.

Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS) Leibniz-Institut im Forschungsverbund Berlin e. V.

Mohrenstraße 39 10117 Berlin Germany

Fax: +49 30 2044975

E-Mail:

[email protected]

ABSTRACT. In this paper we study optimal control problems governed by semilinear parabolic

equa-tions where the spatial dimension is two or three. Moreover, we consider pointwise constraints on the control and on the state. We formulate first order necessary and second order sufficient optimality con-ditions. We make use of recent results regarding elliptic regularity and apply the concept of maximal parabolic regularity to the occurring partial differential equations.

1. INTRODUCTION

In this paper we extend the theory of second order sufficient optimality conditions for state constrained optimal control problems, governed by semilinear parabolic partial differential equations, to spatial domains up to dimension three. It is well known that second order sufficient optimality conditions for nonlinear optimal control problems are essential both, in the numerical analysis, and for reliable opti-mization algorithms. For instance, the strong convergence of optimal controls and states for numerical discretizations of the problem heavily rests on second order sufficient optimality conditions. Moreover, one can show that numerical algorithms such as SQP methods are locally convergent if second order sufficient optimality conditions hold true.

Meanwhile, there are several contributions regarding second order sufficient optimality condition for state constrained optimal control problems governed by partial differential equations. For the case of elliptic equations we mention Casas, Tröltzsch and Unger [9], where especially boundary control prob-lems where considered. Furthermore, we refer to Casas and Mateos in [8], where new second order conditions for problems with finitely many state constraints were introduced. This theory was extended to the case of pointwise state constraints in the contribution of Casas, de los Reyes and Tröltzsch, see [7]. Furthermore, we refer to Bonnans and Zidani [5] where beside finitely many state constraints also polyhedric constraints on the control were discussed. In the case of parabolic equations second order sufficient conditions were investigated by Raymond and Tröltzsch [35] and again in [7].

Let us emphasize the differences of our work to former contributions: in [35] and [7] it was necessary that the parabolic solution operator for the state equation of the optimal control problem is continuous from

L

2 _{on the space-time cylinder to the space of (uniformly) continuous functions – due to the}

presence of pointwise state constraints. Furthermore, this is also caused by the well known two-norm discrepancy, see for instance Ioffe [27] and Maurer [31]. Thus, those authors were restricted to distributed control problems and spatial dimension one. Since the above mentioned mapping property of the parabolic solution operator does not hold in dimensions

d >

1

we introduce here a quite different approach, heavily resting on maximal parabolic regularity and optimal embedding results. Moreover, the parabolic state equation is discussed under mild assumptions concerning the spatial domain and the coefficient function in the elliptic differential operator. Furthermore, we are able to deal with mixed boundary conditions. In order to do so, we will use very recent results from elliptic/parabolic regularity theory.

The paper is organized as follows: in the next section we introduce the optimal control problem, and we specify assumptions on given the quantities of the problem. Furthermore, we will make use of the concept of maximal parabolic regularity in order to analyze the state equation and a linearized version of it. The third section is devoted to the formulation of first order necessary optimality condition using an adjoint state equation. Since our state space will be chosen as a reflexive Banach space, the for-mulation of the respective adjoint state equation can be carried out intuitively. Here we mainly follow the ideas from [26, Ch.6.2-6.3]. The fourth section concerns the elaboration of second order sufficient optimality conditions adapting ideas in [9] to the parabolic case. The appendix is devoted to the proof

of the elliptic regularity result which is one of the cornerstones in the foregoing chapters.

2. SEMILINEAR PARABOLIC PROBLEMS

We will consider the following parabolic optimal control problem with distributed control. min

J

(

y, u

) :=

ZZ

Q

L

(

x, t, y, u

)

dxdt

+

Z

T 0

Z

∂Ω

l

(

x, t, y

)

dσ dt

y

t

− ∇ ·

κ

∇

y

+

d

(

x, t, y

) =

u

in

Q

:= Ω

×

(0

, T

)

ν

·

κ

∇

y

= 0

on

Σ

N

= Γ

×

(0

, T

)

y

= 0

on

Σ

D

= (

∂

Ω

\

Γ)

×

(0

, T

)

y

(

x,

0) = 0

in

Ω

u

a

≤

u

(

x, t

)

≤

u

b a.e. in

Q.

y

(

x, t

)

≥

y

c

(

x, t

)

for all

(

x, t

)

∈

K

⊂

Q.































































(P)

Moreover, we incorporate pointwise constraints on the control and on the state. In this setting,

Ω

is a subset of

R

d

, d

∈ {

2

,

3

}

_{with boundary}

∂

Ω

_{, where}

Γ

_{denotes an open subset of the boundary.}

Moreover,

ν

defines the outward unit normal at the boundary part

Γ

. In the sequel we abbreviate

∂

Ω

×

(0

, T

) =: Σ

. K is a non-empty compact subset of

Q

.

2.1. Notations and assumptions. Throughout the paper, all occurring spatial domains are generally supposed to be Lipschitzian (see [32, Ch. 1.1.9]), in particular,

Ω

⊂

R

d

, d

∈ {

2

,

3

}

_{always denotes}

a bounded Lipschitz domain.

Remark 2.1. The Lipschitz property of the domain assures (besides other usefull things) a continuous extension operator

W

1,q

(Ω)

→

W

1,q

(

R

d

)

_{– universal for all}

q

∈

[1

,

∞

]

_{. This yields immediately the}

usual Sobolev embeddings

W

1,q

(Ω)

֒

→

L

r

(Ω)

.

Γ

is an open subset of the boundary

∂

Ω

and is supposed to satisfy throughout the whole paper the following general assumption:

Assumption 2.2. i) If

d

= 2

, then

∂

Ω

\

Γ

is the finite union of closed arc pieces – non of which is degenerated to a single point.

ii) If

d

= 3

, then

∂

Ω

\

Γ

is the closure of its interior (within

∂

Ω

). Moreover, the boundary of

Γ

within

∂

Ω

is locally bi-Lipschitz diffeomorphic to the unit interval

(0

,

1)

.

Remark 2.3. In fact, the conditions in Assumption 2.2 are an equivalent (but much more intuitive) for

Ω

∪

Γ

to be a regular set in the sense of[21] (see[23, Ch. 5] for a proof). In this spirit, this latter property is always implicitly understood when quoting results concerning this geometrical setting.

In all what follows, the space

C

δ

(

Q

)

_{denotes the space of Hölder continuous functions with index}

δ >

0

.

W

1,q

(Ω)

_{denotes the usual (real) Sobolev space on}

Ω

_{. We use the symbol}

W

1,q

Γ

(Ω)

for the

closure of

C

Γ∞

(Ω) :=

in

W

1,q

(Ω)

_{. We write (as usual)}

W

1,q

0

(Ω)

instead of

W

1,q

∅

(Ω)

.

W

−1,q

′

(Ω)

denotes the space of continuous linear forms on

W

₀1,q

(Ω)

and

W

_Γ−1,q′

(Ω)

denotes the dual to

W

_Γ1,q

(Ω)

when 1

q

+

1

q′

= 1

holds. In particular, the dual of

W

1,q′

(Ω)

is denoted by

W

_∂−_Ω1,q

(Ω)

. If the domain

Ω

is understood, we will abbreviate

W

_Γ1,q and

W

_Γ−1,q, respectively. For two Banach spaces

X

and

Y

we denote the space of linear, bounded operators from

X

into

Y

by

L

(

X

;

Y

)

. If

X

=

Y

, then we abbreviate

L

(

X

)

. As usual, we define for a (matrix valued) coefficient function

κ

the operator

−∇ ·

κ

∇

:

W

_Γ1,2

(Ω)

7−→

W

_Γ−1,2

(Ω)

by

h−∇ ·

κ

∇

v, w

i

:=

Z

Ω

κ

∇

v

· ∇

w dx v, w

∈

W

_Γ1,2

(Ω)

,

(2.1) where here and in the sequel

h·

,

·i

denotes the dual pairing between

W

_Γ−1,2

(Ω)

and

W

_Γ1,2

(Ω)

. The restriction of the operator to spaces

W

_Γ1,q

(Ω)

, q

≥

2

will also be denoted by

−∇ ·

κ

∇

.

Remark 2.4. When restricting the range space of the operator

−∇ ·

κ

∇

to

L

2

(Ω)

_{, one obtains an}

operator for which the elements

ψ

of its domain satisfy the conditions

ψ

|

∂Ω\Γ

= 0

in the sense of

traces and

ν

·

µ

∇

ψ

= 0

on

Γ

in a generalized sense –

ν

being the outward unit normal of the domain, compare[10, Ch. 1.2]or[16, Ch II.2]).

In this spirit, the operator

−∇ ·

κ

∇

is to be understood as one with mixed boundary conditions.

In all what follows, we will make the following general assumption:

Assumption 2.5. Let

κ

be a measurable, bounded function on

Ω

, taking its values in the set of real symmetric

d

×

d

matrices. Moreover,

κ

has to fulfil the usual strong ellipticity condition

ξ

T

κ

(

x

)

ξ

≥

c

κ

|

ξ

|

2

,

∀

ξ

∈

R

d

for almost all

x

∈

Ω

and some constant

c

κ

>

0

.

Assumption 2.6. There is a

q

0

> d

, such that

−∇

κ

· ∇

+ 1 :

W

_Γ1,q

(Ω)

7−→

W

_Γ−1,q

(Ω)

(2.2)

is a topological isomorphism for all

q

∈

[

q

0′

, q

0

]

, where

q

0′

=

q0q−01 is the conjugate exponent to

q

0.

Remark 2.7. The appendix of the paper is devoted to specify assumptions on the domain

Ω

, the boundary part

Γ

and the coefficient function

κ

such that Assumption 2.6 holds true. Since this is a black box in view of the rest of the paper, and its discussion is of quite different character, we postponed this to the appendix.

Remark 2.8. In view of Shamir’s famous counterexample[37]one cannot expect that (2.2)is a topo-logical isomorphism for a

q

≥

4

in general, if mixed boundary conditions are imposed. In this spirit we assume that

q

0 is taken from the interval

(

d,

4)

in the sequel, see also Proposition 2.10 below.

Next, we will formulate assumptions on the given quantities of the optimal control problem:

(A1) For each pair

(

x, t

)

∈

Q

or

(

x, t

)

∈

Σ

, respectively, the functions

L

(

x, t, y, u

)

,

l

(

x, t, y

)

and

d

(

x, t, y

)

are of Carathèodory type, i.e. for all fixed

(

y, u

)

∈

R

2_or

y

∈

R

_{they are measurable}

with respect to

(

x, t

)

∈

Q

or

(

x, t

)

∈

Σ

, respectively. The function

L

(

x, t, y, u

)

is twice partially differentiable w.r.t. to

(

y, u

)

for almost all

(

x, t

)

∈

Q

. Analogously, the functions

d

and

l

are twice partially differentiable w.r.t.

y

.

(A2) There is a constant

C

dand, for all

M >

0

, a constant

C

d,M such that

|

d

(

x, t,

0)

|

+

∂d

∂y

(

x, t,

0)

+

∂

2

d

∂y

2

(

x, t,

0)

≤

C

d

∂

2

d

∂y

2

(

x, t, y

1

)

−

∂

2

d

∂y

2

(

x, t, y

2

)

≤

C

d,M

|

y

1

−

y

2

|

hold for almost all

(

x, t

)

∈

Q

and all

|

y

i

| ≤

M, i

= 1

,

2

.

(A3) We assume that there is

C

land, for all

M >

0

, a constant

C

l,M such that

|

l

(

x, t,

0)

|

+

∂y

∂l

(

x, t,

0)

+

∂

2

l

∂y

2

(

x, t,

0)

≤

C

l

∂

2

l

∂y

2

(

x, t, y

1

)

−

∂

2

l

∂y

2

(

x, t, y

2

)

≤

C

l,M

|

y

1

−

y

2

|

hold for almost all

(

x, t

)

∈

Σ

and all

|

y

i

| ≤

M, i

= 1

,

2

.

(A4) There is a constant

C

Land, for all

M >

0

, a constant

C

L,M such that

|

L

(

x, t,

0)

|

+

|

L

′

(

x, t,

0)

|

+

|

L

′′

(

x, t,

0)

| ≤

C

L

|

L

′′

(

x, t, y

1

, u

1

)

−

L

′′

(

x, t, y

2

, u

2

)

| ≤

C

L,M

(

|

y

1

−

y

2

|

+

|

u

1

−

u

2

|

)

for almost all

(

x, t

)

∈

Q

and all

|

y

i

| ≤

M,

|

u

i

| ≤

M, i

= 1

,

2

, where

L

′ and

L

′′ denote the

gradient and the Hessian matrix of

L

with respect to

(

y, u

)

. (A5)

u

a

, u

bare assumed to be real numbers satisfying

u

a

< u

b

(A6)

y

c

(

x, t

)

denotes a continuous function on the compact subset

K

⊂

Q

with

y

c

(

x,

0)

<

0

for

almost all

x

∈

∂

Ω

\

Γ

2.2. Discussion of the state equation. This subsection is devoted to the analysis of the state equa-tion from problem (P). Let us first recall the concept of maximal parabolic regularity and point out some basis facts on this: Let

X

be a Banach space and let

A

be a closed operator with dense do-main

D ⊂

X

, the latter equipped with the associated graph norm. Moreover, let

J

= (

T

0

, T

)

⊂

R

be a bounded interval. Suppose

r

∈

(1

,

∞

)

, then

A

is said to satisfy maximal parabolic

L

r

(

J

;

X

)

-regularity if for any

f

∈

L

r

(

J

;

X

)

_{there is a unique function}

w

∈

W

1,r

(

J

;

X

)

∩

L

r

(

J

;

D

)

_such

that

w

t

+

Aw

=

f,

w

(

T

0

) = 0

.

This means, the mapping

W

1,r

(

J

;

X

)

∩

L

r

(

J

;

D

)

∩{

w

:

w

(

T

0

) = 0

} ∋

w

7→

w

t

+

Aw

∈

L

r

(

J

;

X

)

is a continuous bijection. Hence the inverse is continuous by the open mapping theorem, and the solution

w

admits an estimate

k

w

k

W1,r_(J;X)

+

k

w

k

_Lr_(J;D)

≤

c

k

f

k

_Lr_(J;X) (2.3) for some constant

c >

0

independent of

f

. Note that by

W

1,r

(

J

;

X

)

_{we denote the set of those}

functions from

L

r

(

J

;

X

)

_{whose distributional derivate also belongs to}

L

r

(

J

;

X

)

_.

Remark 2.9. The following things on maximal parabolic

L

r

(

J

;

X

)

_{-regularity are known:}

(i) If A satisfies maximal parabolic

L

r

(

J

;

X

)

_{-regularity, then it does for any other bounded interval}

ˆ

J

, see[13]

(ii) If A satisfies maximal parabolic

L

r

(

J

;

X

)

_{-regularity, then it satisfies maximal parabolic}

L

s

(

J

;

X

)

(iii) There is a continuous embedding[2]

W

1,r

(

J

;

X

)

∩

L

r

(

J

;

D

)

֒

→

C

β

( ¯

J

; (

X,

D

)

ς,1

)

if

0

≤

β <

1

−

1

r

−

ς,

(2.4)

where

(

X,

D

)

ς,1denotes the real interpolation space between

X

and

D

, see[39, Ch. 1].

We will apply this concept to our problem. To this end, we need the following result (see [25, Theorem 5.4.]).

Proposition 2.10. Let, under Assumption 2.2,

q

∈

[2

,

∞

(

in case of

d

= 2

or

q

∈

[2

,

6]

in case of

d

= 3

. Then, for every

r >

1

and any bounded interval

J

the operator

−∇ ·

κ

∇

satisfies maximal parabolic

L

r

(

J

;

W

−1,q

Γ

)

-regularity.

Remark 2.11. If, additionally, Assumption 2.6 is satisfied, then the domain

D

of

−∇ ·

κ

∇

equals

W

_Γ1,q.

Theorem 2.12. Let the Assumptions 2.2/2.6 be satisfied and

q

∈

(

d, q

0

]

be given. Moreover, we

assume

r >

_q2₋q_dand

p

≥

_ddq_+q. Then, there is a

T

>0, such that for all

u

∈

L

r

(0

, T

;

L

p

(Ω))

the state

equation of problem (P) admits a unique solution

y

=

y

(

u

)

∈

W

1,r

(0

, T

;

W

_Γ−1,q

)

∩

L

r

(0

, T

;

W

_Γ1,q

)

, which belongs to

C

γ

(

Q

)

_{with some}

γ >

₀

_{. The number}

T

_{may be taken uniformly for all functions}

u

with

u

a

≤

u

≤

u

b.

Proof. According to Proposition 2.10, the operator

−∇ ·

κ

∇

in the state equation enjoys maximal parabolic

L

r

(

J

;

W

−1,q

Γ

)

-regularity. Furthermore, the control

u

∈

L

r

(0

, T

;

L

p

(Ω))

is an element in

L

r

(0

, T

;

W

−1,q

Γ

)

due to

p

≥

ddq+q and the resulting Sobolev embedding. We refer to [24, Theorem

6.17] that gives the existence of a

T

>0 such that the state equation of problem (P) admits a unique

solution

y

=

y

(

u

)

on

(0

, T

)

satisfying

y

∈

W

1,r

(0

, T

;

W

−1,q

Γ

)

∩

L

r

(0

, T

;

W

1,q

Γ

)

. Let us prove

now the Hölder continuity of the solution. Due to the supposition

r >

_q2₋q_d and

q > d

the interval

(

1₂

+

₂d_q

,

1

−

1

r

)

is not empty. Assume now

ς

∈

(

1 2

+

d 2q

,

1

−

1 r

)

. Then (2.4) gives

W

1,r

(0

, T

;

W

_Γ−1,q

)

∩

L

r

(0

, T

;

W

_Γ1,q

)

֒

→

C

β

([0

, T

]; (

W

_Γ−1,q

, W

_Γ1,q

)

ς,1

)

(2.5)

as long as

β <

1

−

1_r

−

ς

. On the other hand we have, thanks to Corollary 5.28 and Remark 5.29, for

ς >

1₂

+

₂d_q

(

W

_Γ−1,q

(Ω)

, W

_Γ1,q

(Ω))

ς,1

֒

→

(

W

Γ−1,q

(Ω)

, W

1,q

Γ

(Ω))

ς,q

֒

→

W

Γ2ς−1,q

(Ω)

֒

→

C

α

(Ω)

(2.6)

with

α

:= 2

ς

−

1

−

d_q

>

0

. Finally, (2.5) in combination with (2.6) give the asserted Hölder continuity. The uniformity in

u

can also be obtained from [24, Theorem 6.17.], see also [33].

Remark 2.13. In this article we do not care how large the interval is where the solution exists. In general, these are highly nontrivial problems even in the case of smooth data, compare e.g.[1]. In this spirit, the occurring time intervals below shall be always understood as the existence interval

(0

, T

)

from Theorem 2.12.

Remark 2.14. We will note here, that it is also possible to treat boundary conditions of Robin-type instead of pure Neumann-type boundary conditions, see[24, Ch. 5.3]. Furthermore, we are also able to deal with inhomogeneous Dirichlet boundary data, provided that they are smooth enough.

2.3. Analysis of the linearized state equation. In this section we discuss the following linear initial boundary value problem that covers a linearized version of the state equation of problem (P).

w

t

− ∇ ·

κ

∇

w

+

c

0

w

=

f

in

Q

ν

·

κ

∇

w

= 0

on

Σ

N

w

= 0

on

Σ

D

w

(

x,

0) = 0

in

Ω

,

(2.7)

where

c

0

∈

L

∞

(

Q

)

is given. We note that

c

0 represents the partial derivative ∂d_∂y. Analogously to the

previous section, we will use the concept of maximal parabolic regularity for the solvability of (2.7). Theorem 2.15. Let Assumption 2.6 be satisfied and

q

∈

(

d, q

0

)

be given. Moreover, we assume

r >

_q2₋q_d and

p

≥

_ddq_+q. Then, for all

f

∈

L

r

(0

, T

;

L

p

(Ω))

_{the solution}

w

∈

W

1,r

(0

, T

;

W

−1,q

Γ

)

∩

L

r

(0

, T

;

W

1,q

Γ

)

of (2.7)belongs to

C

α

(

Q

)

with some

α >

0

. Furthermore, the mapping

f

7→

w

is

continuous from

f

∈

L

r

(0

, T

;

L

p

(Ω))

to

C

α

(

Q

)

.

Proof. Again, according to Proposition 2.10, the operator

−∇ ·

κ

∇

in the state equation enjoys max-imal parabolic

L

r

(0

, T

;

W

−1,q

Γ

)

-regularity and the right hand side

f

belongs to

L

r

(0

, T

;

W

−1,q

Γ

)

by

Sobolev embedding. Then a perturbation result, see e.g. [4, Prop. 1.3], gives the maximal parabolic

L

r

(0

, T

;

W

−1,q

Γ

)

-regularity of the operator

−∇·

κ

∇

+˜

c

0, where

˜

c

0denotes the multiplication operator

induced by

c

0. Thus, we have

k

w

k

_W1,r_(0,T;W−1,q

Γ )

+

k

w

k

Lr(0,T;W 1,q

Γ )

≤

c

k

f

k

L

r_(0,T;Lp_(Ω))

.

Moreover, we have the continuous embedding

W

1,r

(0

, T

;

W

_Γ−1,q

)

∩

L

r

(0

, T

;

W

_Γ1,q

)

֒

→

C

α

(

Q

)

.

for some

α >

0

, see the proof of Theorem 2.12.

Within the discussion of second order sufficient optimality conditions for problem (P), it is necessary to consider the solution of a linearized state equation with respect to right hand sides being elements from

L

2

(

Q

)

_{. In order to obtain optimal a priori estimates for the solution in this case we prove the}

following result, which is another keypoint for all what follows: Theorem 2.16. Let

Ω

and

Γ

be as above.

i) The restriction of the operator

−∇ ·

κ

∇

to

L

2

(Ω)

_{satisfies maximal parabolic regularity, i.e.}

for every

f

∈

L

2

(

Q

)

_{the equation(2.7)admits a unique solution}

w

∈

W

1,2

(0

, T

;

L

2

(Ω))

∩

L

2

(0

, T

;

D

L2

)

, where

D

_L2 denotes the domain of the operator

−∇ ·

κ

∇

, when considered on

L

2

(Ω)

_.

ii) Let

w

∈

W

1,2

(0

, T

;

L

2

(Ω))

∩

L

2

(0

, T

;

D

L2

)

be the solution of (2.7)with respect to the right

hand side

f

∈

L

2

(

Q

)

_{. If}

d

= 2

_{, then}

w

_{even belongs to every space}

C

([0

, T

];

L

p

(Ω))

_{, if}

p

∈

[1

,

∞

)

.

If

d

= 3

, then

w

belongs to the space

L

2ζ

(0

, T

;

L

6

1−ζ

(Ω))

_{for all}

ζ

∈

(0

,

1)

_{, and one has the}

estimate

k

w

k

L 2 ζ_(0,T;L 6 1−ζ_(Ω))

≤

c

k

f

k

L 2_(Q)

.

Proof. i) Due to the symmetry of the coefficient function

κ

, the operator

−∇ ·

κ

∇|

L2 is

But this implies maximal parabolic regularity since

L

2

(Ω)

_{is a Hilbert space, see [12]. This yields}

an a priori estimate like (2.3) – there taken

r

= 2

. It is well-known ([3, Thm. 4.10.2]) that the space

W

1,2

(0

, T

;

L

2

(Ω))

∩

L

2

(0

, T

;

D

L2

)

continuously embeds into

C

([0

, T

]; (

L

2

(Ω)

,

D

_L2

)

1 2,2

)

.

But

(

L

2

(Ω)

,

dom

L2_(Ω)

(

∇ ·

κ

∇

))

1

2,2equals

dom

L

2_(Ω)

(

−∇ ·

κ

∇

+ 1)

1/2including the equivalence

of norms, cf. [39, Ch. 1.18.10]. Moreover,

dom

L2_(Ω)

(

−∇ ·

κ

∇

+ 1)

1/2 is nothing else but the form

domain of

−∇ ·

κ

∇

(see [29, Ch. VI.2.6]); and this form domain equals

W

_Γ1,2

(Ω)

. Thus, the space

C

([0

, T

]; (

L

2

(Ω)

,

D

L2

)

1

2,2

)

is identical with

C

([0

, T

];

W

1,2

Γ

(Ω))

. Hence, the assertion for the

2

d

-case results from the embedding

W

_Γ1,2

(Ω))

֒

→

L

p

(Ω)

_{, whenever}

p

∈

[1

,

∞

)

_.

In the case of

d

= 3

one has, on one hand, the Sobolev embedding

W

_Γ1,2

(Ω)

֒

→

L

6

(Ω)

_{. Thus,}

w

∈

C

([0

, T

];

W

_Γ1,2

(Ω))

֒

→

L

∞

(0

, T

;

L

6

(Ω))

.

(2.8) On the other hand, one knows the continuous embedding

D

L2

֒

→

L

∞

(Ω)

, if

d

≤

3

, cf. [19] or [38].

This gives

w

∈

L

2

(0

, T

;

L

∞

(Ω))

,

(2.9)

together with a corresponding estimate with respect to

f

. Let us show that (2.8) together with (2.9) implies

w

∈

L

2ζ

(0

, T

;

L

6

1−ζ

(Ω))

, for all

ζ

∈

(0

,

1)

. For doing so, one first observes that (2.9) implies the (Bochner-) measurability of

w

, when the function is considered as

L

1−6ζ

(Ω)

-valued. Secondly, one has for any

ψ

∈

L

∞

(Ω)

k

ψ

k

L 6 1−ζ_(Ω)

=

Z

Ω

|

ψ

|

1−ζ

|

ψ

|

ζ

6 1−ζ

d

x

1−ζ 6

≤ k

ψ

k

1_L−6_(Ω)ζ

k

ψ

k

ζ L∞_(Ω)

.

Thus, one obtains

Z

T 0

k

w

(

t

)

k

2 ζ L 6 1−ζ_(Ω)

dt

ζ 2

≤

Z

T 0

k

w

(

t

)

k

_L1−6_(Ω)ζ

k

w

(

t

)

k

ζ L∞_(Ω)

2 ζ

dt

ζ 2

≤ k

w

k

1_L−∞ζ (0,T;L6_(Ω))

k

w

k

ζ L2_(0,T;L∞ (Ω))

,

and the right hand side is finite, according to (2.8)/(2.9).

3. NECESSARY OPTIMALITY CONDITIONS

Let us first recall that here and in the following chapter we always impose the Assumptions 2.2, 2.5 and 2.6 without any further comment. We start with the introduction of a control-to-space map-ping based on Theorem 2.12. For the subsequent we define

J

:= (0

, T

)

and

W

01,r

(

J

;

W

−1,q

Γ

) :=

W

1,r

(

J

;

W

−1,q

Γ

)

∩ {

ψ

:

ψ

(0) = 0

}

. Further, we introduce the state space

Y

:=

W

01,r

(

J

;

W

−1,q

Γ

)

∩

L

r

(

J

;

W

1,q

Γ

)

,

thereby assuming

r >

_q2₋q_d and

q

∈

(

d, q

0

)

.

Definition 3.1. Based on Theorem 2.12, we introduce for

p >

_qdq_+d the control-to-state operator

G

:

L

r

(

J

;

L

p

(Ω))

→

Y

_by

y

=

G

(

u

)

_{, which assigns to}

u

∈

L

r

(

J

;

L

p

(Ω))

_{the unique solution}

y

(

u

)

∈

Y

of the state equation of problem (P).

Following the lines of the proof of Theorem 2.12, we obtain that the unique state

y

=

G

(

u

)

is Hölder continuous. By means of the analysis for the linearized state equation and the implicit function theorem one can easily prove the Frèchet-differentiability of the solution operator

G

:

Theorem 3.2. Let assumptions (A1)-(A2) be satisfied. Then, the control-to-state operator

G

is twice Frèchet differentiable from

L

r

(

J

;

L

p

(Ω))

_to

Y

_{. Its derivative}

y

h

:=

G

′

(

u

)

h

at point

u

in direction

h

∈

L

r

(

J

;

L

p

(Ω))

_{is given by the solution of}

(

y

h

)

t

− ∇ ·

κ

∇

y

h

+

d

y

(

x, t, y

(

u

))

y

h

=

h

in

Q

ν

·

κ

∇

y

h

= 0

on

Σ

N

y

h

= 0

on

Σ

D

y

h

(

x,

0) = 0

in

Ω

,

(3.1) where

y

(

u

) =

G

(

u

)

. Furthermore,

y

h1h2

=

G

′′

(

u

)

h

1

h

2

, h

i

∈

L

r

(

J

;

L

p

(Ω))

, i

= 1

,

2

is the solution of

(

y

h1h2

)

t

− ∇ ·

κ

∇

y

h1h2

+

d

y

(

x, t, y

(

u

))

y

h1h2

=

−

d

yy

(

x, t, y

(

u

))

y

h1

y

h2 in

Q

ν

·

κ

∇

y

h1h2

= 0

on

Σ

N

y

h1h2

= 0

on

Σ

D

y

h1h2

(

x,

0) = 0

in

Ω

(3.2) with

y

hi

=

G

′

(

u

)

h

i

, i

= 1

,

2

.

The following Proposition shows that the remainder term of the first derivative of the solution operator

G

can also be estimated with respect to the

L

2

(

Q

)

_{-norm of the directions, which will be essential for}

second order sufficient conditions.

Proposition 3.3. Let

u,

¯

u

ˆ

∈

U

ad be given with associated states

y

¯

=

G

(¯

u

)

and

y

ˆ

=

G

(ˆ

u

)

,

respectively. Moreover, we introduce

y

=

G

′

(¯

u

)(ˆ

u

−

u

¯

)

as the solution of the linearized state equation in direction

u

ˆ

−

u

¯

. Then, there is a constant

c >

0

depending on

u,

¯

u

ˆ

∈

U

adsuch that

k

y

ˆ

−

y

¯

−

y

k

Y

≤

c

k

u

ˆ

−

u

¯

k

2L2_(Q)

.

(3.3)

Proof. Introducing the difference

z

:= ˆ

y

−

y

¯

−

y

and using the first order expansion of

d

at

(

x, t,

y

¯

)

,

one can easily see that

z

solves the following system

z

t

− ∇ ·

κ

∇

z

+

d

y

(

x, t,

y

¯

)

z

=

−

r

d in

Q

ν

·

κ

∇

z

= 0

on

Σ

N

z

= 0

on

Σ

D

z

(

x,

0) = 0

in

Ω

,

where

r

ddenotes the remainder term of the first order expansion of

d

. According to Assumption (A2),

we obtain for

r

d

|

r

d

(

x, t

)

| ≤

c

M

|

y

ˆ

(

x, t

)

−

y

¯

(

x, t

)

|

2

,

where

c

M depends on

k

y

ˆ

k

C(Q)and

k

y

¯

k

C(Q), respectively. By means of Theorem 2.15, we conclude

z

∈

Y

and the a priori estimate

k

z

k

Y

≤

c

k

r

d

k

Lr_(J;Lp_(Ω))

for

r >

_q2₋q_d and

p

≥

_qdq_+d. Note that

q > d

implies directly

p > d/

2

. Thus we end up with

k

z

k

Y

≤

c

k

y

ˆ

−

y

¯

k

2Lr˜_(J;Lp˜_(Ω))

for

r >

˜

_q4₋q_d and

p > d

˜

and a constant depending on

u,

¯

u

ˆ

∈

U

ad due to Theorem 2.12. According

to differentiability of the solution operator

G

, there is a real number

0

< η <

1

such that

y

ˆ

−

y

¯

=

of a linear initial boundary value problem (2.7) with

c

0

:=

d

y

(

x, t, y

(

u

))

and the right hand side

f

:= ˆ

u

−

u

¯

. Thus we can exploit Theorem 2.16 and we derive the estimate

k

y

ˆ

−

y

¯

k

Lr˜_(J;Lp˜_(Ω))

≤

c

k

u

ˆ

−

u

¯

k

_L2_(Q)

taking

ζ

in Theorem 2.16(ii) sufficiently small. By means of this estimate the assertion is proven.

It is known that Lagrange multipliers associated to pointwise state constraints are in general only regular Borel measures, see e.g. [6] and [34]. Let us define the Lagrange function associated to our problem (P) as follows:

L

:

L

∞

(

Q

)

× M

(

Q

)

→

R

,

L

(

u, µ

) =

J

(

G

(

u

)

, u

)+

< y

_c

−

G

(

u

)

, µ >

C(Q),M(Q)

,

where the space of regular Borel measures on

Q

is denoted by

M

(

Q

)

. Note, that by Riesz represen-tation theorem

M

(

Q

)

can be identified with the dual space of

C

(

Q

)

.

One can easily see that the Lagrange function is continuously differentiable w.r.t. to the control variable

u

from

L

∞

(

Q

)

_to

R

_{by Theorem 3.2, the chain rule and assumption (A1). Its derivative at point}

u

∈

L

∞

(

Q

)

_{in direction}

h

∈

L

∞

(

Q

)

_{is given by}

L

′

(

u, µ

)

h

=

ZZ

Q

∂L

∂u

(

x, t, G

(

u

)

, u

)

h

+

∂L

∂y

(

x, t, G

(

u

)

, u

)

y

h

dxdt

+

Z Z

Σ

∂l

∂y

(

x, t, G

(

u

))

y

h

dσdt

−

< y

h

, µ >

C(Q),M(Q)

,

(3.4)

where

y

h

=

G

′

(

u

)

h

is the solution of (3.1) w.r.t.

h

∈

L

∞

(

Q

)

. We continue with the definition of

Lagrange multipliers associated to the state constraints in (P). Note that the control constraints were handled by a convex set of admissible controls:

U

ad

=

{

u

∈

L

∞

(

Q

) :

u

a

≤

u

≤

u

b

}

.

Definition 3.4. Let

u

¯

∈

U

ad be a locally optimal control of (P), then

µ

∈ M

(

Q

)

is said to be a

Lagrange multiplier associated to the state constraints of (P), if

L

′

(¯

u, µ

)(

u

−

u

¯

)

≥

₀

∀

u

∈

U

ad (3.5)

< y

c

−

G

(¯

u

)

, µ >

C(Q),M(Q)

= 0

(3.6)

Z Z

Q

ϕdµ

≥

0

,

∀

ϕ

∈

C

(

Q

)

, ϕ

≥

0

(3.7) hold true.

By means of an adjoint equation, we will reformulate the previous conditions into a optimality system. Moreover, it is well known that a certain constraint qualification is needed to ensure the existence of such a Lagrange multiplier, see e.g. [41]. We require a linearized Slater condition, which guarantees the existence of Lagrange multipliers in the sense of the previous definition.

Assumption 3.5. Let

u

¯

∈

U

ad satisfy the following condition: there exist a feasible control

u

ˆ

∈

U

ad

, G

(ˆ

u

)

≥

y

cand a real number

δ >

0

such that

Definition 3.6. Let

q

0

> d

be the number of Assumption 2.6 and let

q

∈

(

d, q

0

)

be given. Further,

we assume

r >

_q2₋q_d. We denote, for

s

∈

(1

,

∞

)

, by

s

′ the conjugated exponent _s−s₁ and observe

q

′

∈

(

q

₀′

, d/

(

d

−

1))

and

r

′

∈

(1

,

(2

q

)

/

(

q

+

d

))

.

Regarding the numbers

q, q

′

, r

and

r

′of the previous definition, we set

W

r,0

=

W

01,r

(

J

;

W

−1,q Γ

)

∩

L

r

(

J

;

W

1,q Γ

)

W

r′

=

W

1,r ′

(

J

;

W

_Γ−1,q′

)

∩

L

r′

(

J

;

W

_Γ1,q′

)

The associated dual space are denoted by the superscript

∗

.

Theorem 3.7. Let

u

¯

∈

U

ad be local optimal control with associated state

y

¯

=

G

(¯

u

)

∈

Y

=

W

r,0

satisfying Assumption 3.5. Then there exist a Lagrange multiplier

µ

¯

∈ M

(

Q

)

and an adjoint state

¯

p

∈

L

r′

(

J

;

W

_Γ1,q′

)

such that the following optimality system is satisfied:

¯

y

t

− ∇ ·

κ

∇

y

¯

+

d

(

x, t,

y

¯

) = ¯

u

in

Q

ν

·

κ

∇

y

¯

= 0

on

Σ

N

¯

y

= 0

on

Σ

D

¯

y

(

x,

0) = 0

in

Ω

(3.8)

Z

J

< z

t

,

p >

¯

_W−1,q Γ ,W 1,q′ Γ

dt

+

Z Z

Q

κ

∇

z

· ∇

pdxdt

¯

=

Z Z

Q

∂L

∂y

(

x, t,

y,

¯

u

¯

)

zdxdt

+

ZZ

Σ

∂l

∂y

(

x, t,

y

¯

)

zdσdt

+

ZZ

Q

zd

µ

¯

∀

z

∈

W

r,0 (3.9)

Z Z

Q

(¯

p

+

∂L

∂u

(

x, t,

y,

¯

u

¯

))(

u

−

u

¯

)

dxdt

≥

0

∀

u

∈

U

ad (3.10)

ZZ

Q

(

y

c

−

y

¯

)

d

µ

¯

= 0

,

ZZ

Q

ϕdµ

≥

0

,

∀

ϕ

∈

C

(

Q

)

+ (3.11)

Proof. Follow the lines of [26, Ch.6.2-6.3].

Please note, that the optimal adjoint state can be splitted into a regular part

p

1 and a irregular part

p

2,

where on the one hand

p

2

∈

L

r

′

(

J

;

W

_Γ1,q′

)

is the solution of (3.9) with respect to the measure

µ

¯

and on the other hand

p

1

∈

W

r′ solves the following equation in

L

r

′

(

J

;

W

_Γ−1,q′

)

−

(

p

1

)

t

− ∇ ·

κ

∇

p

1

+

d

y

(

x, t,

y

¯

)

p

1

= ˜

f

p

1

(

T

) = 0

,

where the right hand side

f

˜

∈

L

r′

(

J

;

W

_Γ−1,q′

)

is induced by ∂L

∂y

(

x, t,

y,

¯

u

¯

)

and ∂l ∂y

(

x, t,

y

¯

)

.

Before we start discussing second order sufficient optimality conditions for problem (P), we will derive the second derivative of the Lagrangian w.r.t. the control

u

that will be widely used. From (3.4), we

obtain with the help of the introduced adjoint state:

L

′′

(

u, µ

)

h

1

h

2

=

ZZ

Q

∂

2

L

∂y

2

(

x, t, y

(

u

)

, u

)

y

h1

y

h2

+

∂

2

L

∂y∂u

(

x, t, y

(

u

)

, u

)(

y

h1

h

2

+

y

h2

h

1

)

+

∂

2

L

∂u

2

(

x, t, y

(

u

)

, u

)

h

1

h

2

−

p

(

u

)

∂

2

d

∂y

2

(

x, t, y

(

u

))

y

h1

y

h2

dxdt

+

ZZ

Σ

∂

2

l

∂y

2

(

x, t, y

(

u

))

y

h1

y

h2

dσdt,

(3.12)

where

p

(

u

)

is the solution of (3.9) with

u

,

y

(

u

)

and

µ

taken for

u

¯

,

y

¯

and

µ

¯

, respectively. The following proposition shows, that the second derivative of the Lagrangian w.r.t.

u

is bounded, when considered as quadratic form on

L

2

(

Q

)

.

Proposition 3.8. Let the assumptions (A1)-(A6) be satisfied. Moreover, let

u, h

∈

L

∞

(

Q

)

_and

µ

∈

M

(

Q

)

be given. Then there is a constant

c >

0

such that

L

′′

(

u, µ

)

h

2

≤

c

k

h

k

2_L2_(Q)

.

Proof. The result mainly depends on the estimate of the term of the form

Z Z

Q

p

(

u

)

∂

2

d

∂y

2

(

x, t, y

(

u

))

y

2 h

dxdt

,

(3.13)

where

p

(

u

)

is the solution of (3.9) and

y

h

=

G

′

(

u

)

h

. The other parts of

L

′′

(

u, µ

)

are discussed by

means of assumptions (A1)-(A6), the continuity of the state

y

(

u

)

and standard a priori estimates for the solution of the linearized state equation. Let us come back to the estimation of (3.13). According to Theorem 3.7 and usual embedding results, we expect for the adjoint state

p

(

u

)

∈

L

r′

(

J

;

W

_Γ1,q′

)

֒

→

L

r′

(

J

;

L

s′

(Ω))

,

for

r

′

<

2

q

q

+

d

, s

′

<

dq

′

d

−

q

′

<

d

d

−

2

due to Definition 3.6 for the choice of

q

′_{. By means of Hölder inequality, assumptions on}

d

yy and the

continuity of the state, we find

ZZ

Q

p

(

u

)

∂

2

d

∂y

2

(

x, t, y

(

u

))

y

2 h

≤

c

k

p

(

u

)

k

Lr′ (J;Ls′ (Ω))

k

y

h

k

2 Lr_(J;Ls_(Ω))

≤

c

k

p

(

u

)

k

_Lr′ (J;WΓ1,q′(Ω))

k

y

h

k

2Lr_(J;Ls_(Ω))

,

with

s > d

and

r >

_q4₋q_d. Relying on the regularity result given in Proposition 2.16, the linearized state

y

h belongs to a space

L

2

θ

(

J

;

L

6

1−θ

(Ω))

for all

θ

∈

(0

,

1)

considering the direction

h

as an element in

L

2

(

Q

)

_{. Due to the estimate above, we have to require}

2

θ

>

4

q

q

−

d

⇒

θ <

q

−

d

2

q

that can be satisfied by a

θ >

0

since

q > d

. Fixing such

θ

, we of course fulfil the second assumption

6

1−θ

> d

of the estimation for the case of

d

= 2

,

3

. Thus, we continue the estimate with some

θ <

q−d 2q

ZZ

Q

p

(

u

)

∂

2

d

∂y

2

(

x, t, y

(

u

))

y

2 h

≤

c

k

p

(

u

)

k

Lr′ (J;WΓ1,q′(Ω))

k

y

h

k

2Lr_(J;Ls_(Ω))

≤

c

k

p

(

u

)

k

_Lr′ (J;W1,q′ Γ (Ω))

k

y

h

k

2 L2θ(J;L 6 1−θ_(Ω))

≤

c

k

p

(

u

)

k

_Lr′ (J;WΓ1,q′(Ω))

k

h

k

2 L2_(Q)

,

see Proposition 2.16.

Remark 3.9. Please note that the estimate of Proposition 3.8 can only be proven, if the state variable

y

appears linearly in the pointwise state constraints. In order to deal with more general constraints like

g

(

y

(

x, t

))

≤

0

, where

g

is induced by a Nemytski operator, the continuity of solution operator

G

from

L

2

(

Q

)

_to

C

(

Q

)

_{becomes necessary, since the term}

< g

yy

(¯

y

)(

y

h

)

2

,

µ >

¯

C(Q),M(Q)occurs in the

sec-ond derivative of the Lagrangian. However, in many real world applications pointwise state constraints are directly claimed to the state variable

y

, see e.g. the temperature variable in the thermistor problem discussed in[26].

Based on the estimate in Proposition 3.8, we are able to discuss the remainders of the first and second Taylor expansion of the Lagrangian. A Taylor expansion of

L

is given by

L

(

u,

µ

¯

)

− L

(¯

u,

µ

¯

) =

L

′

(¯

u,

µ

¯

)(

u

−

u

¯

) +

r

1,L

=

L

′

(¯

u,

µ

¯

)(

u

−

u

¯

) +

1

2

L

′′

(¯

u,

µ

¯

)(

u

−

u

¯

)

2

+

r

2,L Taking into account Assumptions (A1)-(A4), we are able to verify

|

r

1,L

| ≤

c

L

k

u

−

u

¯

k

2L2_(Q)

|

r

2,L

| ≤

c

L

k

u

−

u

¯

k

L∞

(Q)

k

u

−

u

¯

k

2L2_(Q)

.

(3.14) The constant

c

L

>

0

depends in particular on

u,

¯

y

¯

=

G

(¯

u

)

and

p

¯

as the solution of the adjoint equation, see (3.9). Note, that the

L

∞

(

Q

)

_{-norm in the second estimate can be weakened to a norm} in

L

r

(

J

;

L

p

(Ω))

_with

r >

2q

q−dand

p

≥

dq d+q.

4. SECOND ORDER SUFFICIENT OPTIMALITY CONDITIONS

Our aim is to establish sufficient optimality conditions for problem (P), where we will follow the approach of Casas, Tröltzsch and Unger [9]. In the sequel

(¯

u,

y

¯

=

G

(¯

u

))

satisfies together with the dual variables

(¯

p,

µ

¯

)

the first order necessary conditions given in Theorem 3.7. We associate to

u

¯

the following cone of critical directions

C

(¯

u

) =

{

h

∈

L

∞

(

Q

) :

h

satisfies (4.2) and (4.3)

}

(4.1)

h

(

x, t

)

≥

₀

,

_if

u

¯

(

x, t

) =

u

a

≤

0

,

if

u

¯

(

x, t

) =

u

b (4.2)

y

h

=

G

′

(¯

u

)

h

≥

0

if

y

¯

(

x, t

) =

y

c

(

x, t

)

(4.3)

Note, that

C

(¯

u

)

can be interpreted as the linearized cone at the point

u

¯

, see e.g. Maurer and Zowe [30]. The following theorem tells us that the difference of every feasible point

u

to

u

¯

can be approxi-mated by an element from the cone

C

(¯

u

)

.

Theorem 4.1. Suppose that Assumption 3.5 is satisfied. Then for all feasible controls

u

ˆ

∈

U

ad of

problem (P) there is a

h

∈

C

(¯

u

)

such that the difference

r

h

= ˆ

u

−

u

¯

−

h

can be estimated by

k

r

h

k

L∞_(Q)

≤

c

k

u

ˆ

−

u

¯

k

2

L2_(Q) (4.4)

Moreover, the respective distance in the states

r

y,h

= ˆ

y

−

y

¯

−

y

hwith

y

h

=

G

′

(¯

u

)

h

can be estimated

by

k

r

y,h

k

Y

≤

c

k

u

ˆ

−

u

¯

k

2L2_(Q)

.

(4.5)

Proof. We will follow the lines of [9, Theorem 4.2]. Let

u

ˆ

∈

U

adbe a feasible control with associated

state

y

ˆ

=

G

(ˆ

u

)

. Moreover, we introduce

y

˜

=

G

′

(ˆ

u

)(ˆ

u

−

u

¯

)

_{as the solution of the linearized state} equation in direction

u

ˆ

−

u

¯

. By Proposition 3.3, the estimate

k

y

ˆ

−

y

¯

−

y

˜

k

Y

≤

c

k

u

ˆ

−

u

¯

k

2L2_(Q) (4.6)

is valid. Let us introduce the mapping

Φ(

u

) =

G

(

u

)

−

y

c. Then the pointwise state constraints of

problem (P) can be written in an abstract way by

Φ(

u

)

∈

K

or

Φ(

u

)

≥

K

0

, where

K

is the positive

convex cone

K

:=

{

z

∈

C

(

Q

) :

z

(

x, t

)

≥

0

∀

(

x, t

)

∈

Q

}

. It is known that the linearized Slater condition in Assumption 3.5 is sufficient for the famous regularity condition by Zowe and Kurcyusz [41], see e.g. [40, Ch.6.1.2.]. This regularity condition can be written by

Φ

′

(¯

u

)

C

(¯

u

)

− K

(Φ(¯

u

)) =

C

(

Q

)

,

with

C

(¯

u

) :=

{

α

(

u

−

u

¯

) :

α

≥

0

, u

∈

U

ad

}

and

K

(¯

z

) :=

{

β

(

z

−

z

¯

) :

β

≥

0

, z

∈

K

}

,

respectively. Due to feasibility of

u

ˆ

, we have

Φ(ˆ

u

)

∈

K

and a Taylor expansion gives

Φ(ˆ

u

) = Φ(¯

u

) + Φ

′

(¯

u

)(ˆ

u

−

u

¯

) +

r

Φ₁

where the remainder term can be estimated by

k

r

₁Φ

k

_C(Q)

≤

c

k

u

ˆ

−

u

¯

k

2

L2_(Q)

due to (4.6) and the definition of

Φ

. Since

Φ(¯

u

)

and

Φ(ˆ

u

)

belong to

K

, the Taylor expansion implies

Φ

′

(¯

u

)(ˆ

u

−

u

¯

)

∈ K

(Φ(¯

u

))

−

r

₁Φ

.

Hence, we have

u

ˆ

−

u

¯

∈ C

(¯

u

)

and

Φ

′

(¯

u

)(ˆ

u

−

u

¯

)

≥

K(Φ(¯u))

−

r

Φ1. According to [36], the last inequality

is regular in the sense of Robinson, and we can thus apply the linear version of the Robinson-Ursescu theorem. Hence, we obtain the existence of a constant

c >

0

and an element

h

∈ C

(¯

u

)

satisfying

k

u

ˆ

−

u

¯

−

h

k

L∞_(Q)

≤

c

k

r

Φ

1

k

C( ¯Q)

≤

c

k

u

ˆ

−

u

¯

k

2L2_(Q)

.

Moreover, we have

Φ

′

(¯

u

)

h

∈ K

(Φ(¯

u

))

_{and thus}

h

∈

C

(¯

u

)

_{, see (4.1). The difference}

r

y,h

:=

ˆ

y

−

y

¯

−

y

hwith

y

h

=

G

′

(¯

u

)

h

can be estimated as follows

k

r

y,h

k

Y

≤ k

y

ˆ

−

y

¯

−

y

˜

k

Y

+

k

G

′

(¯

u

)(ˆ

u

−

u

¯

−

h

)

k

Y

.

Hence, the estimate (4.5) immediately follows from (4.6) and (4.4).

Let us define, for a fixed

τ >

0

, the set

Q

τ

=

{

(

x, t

)

∈

Q

:

|

p

¯

(

x, t

) +

∂L

∂u

(

x, t,

y,

¯

u

¯

)

| ≥

τ

}

.

Hence,

Q

τ is a subset of so called strongly active control constraints. Moreover, we introduce the

Q

τ. Following [9], it is also possible to incorporate “strongly active“ state constraints. Therefore, we

introduce the following subset of

C

(¯

u

)

C

β,τ

(¯

u

) =

h

∈

C

(¯

u

) :

< y

h

,

µ >

¯

C(Q),M(Q)

≤

β

ZZ

Q\Qτ

|

h

(

x, t

)

|

dxdt

for fixed constants

τ >

0

and

β >

0

. Now, we are in the position to require second order sufficient optimality conditions:

(SSC) There exist positive numbers

β, τ

and

δ

such that

L

′′

(¯

u,

µ

¯

)

h

2₂

≥ k

h

2

k

2L2_(Q) (4.7)

holds for all

h

2 constructed in the following way: For every

h

∈

C

β,τ

(¯

u

)

we split the direction

h

in

h

1

=

h

−

P

τ

h

and

h

2

=

P

τ

h

.

By means of this condition, we can formulate:

Theorem 4.2. Let the feasible control

u

¯

∈

U

adwith associated state

y

¯

satisfy Assumption 3.5 and the

first order necessary optimality conditions (3.8)-(3.11). Moreover, we require (SSC). Then there are constants

ρ >

0

and

δ

′

>

0

such that

J

(ˆ

y,

u

ˆ

)

≥

J

(¯

y,

u

¯

) +

δ

′

k

u

ˆ

−

u

¯

k

2

L2_(Q)

holds for all feasible controls

u

ˆ

∈

U

ad with respective state

y

ˆ

=

G

(ˆ

u

)

satisfying

k

u

ˆ

−

u

¯

k

L∞_(Q)

≤

ρ

.

Proof. Let

u

ˆ

be a feasible control with associated state

y

ˆ

=

G

(ˆ

u

)

and

k

u

ˆ

−

u

¯

k

L∞_(Q)

≤

ρ

. Due to definition of the Lagrangian we have

J

(ˆ

y,

u

ˆ

)

−

J

(¯

y,

u

¯

) =

L

(ˆ

u,

µ

¯

)

− L

(¯

u,

µ

¯

)

−

< G

(¯

u

)

−

G

(ˆ

u

)

,

µ >

¯

_C(Q),M(Q)

.

(4.8) Next, we approximate

u

ˆ

−

u

¯

by an element

h

∈

C

(¯

u

)

according to Theorem 4.1. In this way, we have the following remainder estimates for

r

h

:= ˆ

u

−

u

¯

−

h

and

r

y,h

=

G

(ˆ

u

)

−

G

(¯

u

)

−

G

′

(¯

u

)

h

:

k

r

h

k

L∞_(Q)

≤

c

k

u

ˆ

−

u

¯

k

2

L2_(Q) (4.9)

k

r

y,h

k

Y

≤

c

k

u

ˆ

−

u

¯

k

2L2_(Q)

,

(4.10)

where we recall that the state space

Y

is embedded in

C

(

Q

)

. Now, we have to consider two different cases for the approximative direction

h

.

Case I:

h

∈

C

β,τ

(¯

u

)

Here, we make use of first order and second order sufficient conditions. Due to complementary slack-ness conditions (3.11) and the feasibility of

u

ˆ

, we have

< G

(¯

u

)

−

G

(ˆ

u

)

,

µ >

¯

_C(Q),M(Q)

≤

0

.

Hence, we neglect this term in (4.8) and a Taylor expansion of

L

yields

J

(ˆ

y,

u

ˆ

)

−

J

(¯

y,

u

¯

)

≥L

(ˆ

u,

µ

¯

)

− L

(¯

u,

µ

¯

)

=

L

′

(¯

u,

µ

¯

)(

u

−

u

¯

) +

1

2

L

′′

(¯

u,

µ

¯

)(

u

−

u

¯

)

2

+

r

2,L

.

Using the variational inequality (3.10), one can easily find

J

(ˆ

y,

u

ˆ

)

−

J

(¯

y,

u

¯

)

≥

τ

Z Z

Qτ

|

u

ˆ

−

u

¯

|

dxdt

+

1

2

L

′′

(¯

u,

µ

¯

)(

u

−

u

¯

)

2

+

r

2,L

.

(4.11)

Let us introduce the abbreviation

B

:=

L

′′

(¯

u,

µ

¯

)

_{for the bilinear form associated to the second} derivative of the Lagrangian. We proceed with

B

[ˆ

u

−

u

¯

]

2

=

B

[

h

]

2

+ 2

B

[

h, r

h

] +

B

[

r

h

]

2

since

u

ˆ

−

u

¯

=

h

+

r

h. In this case we consider

h

∈

C

β,τ

(¯

u

)

such that (SSC) applies to

B

[

h

]

2. By

means of the splitting of

h

=

h

1

+

h

2as described in (SSC), we obtain

B

[

h

]

2

=

B

[

h

1

]

2

+ 2

B

[

h

1

, h

2

] +

B

[

h

2

]

2

≥

δ

k

h

2

k

2L2_(Q)

−

c

k

h

₁

k

_L2_(Q)

k

h

₂

k

_L2_(Q)

−

c

k

h

₁

k

2

L2_(Q)

using the estimates of Proposition 3.8. Forthcoming, Young’ inequality implies

B

[

h

]

2

≥

δ

2

k

h

2

k

2 L2_(Q)

−

c

k

h

₁

k

2_L2_(Q)

=

δ

2

Z Z

Q\Qτ

|

u

ˆ

−

u

¯

−

r

h

|

2

dxdt

−

c

ZZ

Qτ

|

u

ˆ

−

u

¯

−

r

h

|

2

dxdt

≥

δ

2

ZZ

Q\Qτ

|

u

ˆ

−

u

¯

|

2

dxdt

−

c

ZZ

Q\Qτ

|

u

ˆ

−

u

¯

||

r

h

|

dxdt

−

c

ZZ

Qτ

|

u

ˆ

−

u

¯

|

2

dxdt

−

c

ZZ

Qτ

|

u

ˆ

−

u

¯

||

r

h

|

dxdt

−

c

ZZ

Qτ

|

r

h

|

2

dxdt.

Assume

ρ <

1

is given and

k

u

ˆ

−

u

¯

k

L∞_(Q)

< ρ

. Using

ZZ

Qτ

|

u

ˆ

−

u

¯

|

2

dxdt

≤ k

u

ˆ

−

u

¯

k

L∞_(Q)

ZZ

Qτ

|

u

ˆ

−

u

¯

|

dxdt

and (4.9), we derive

B

[

h

]

2

≥

δ

2

Z Z

Q\Qτ

|

u

ˆ

−

u

¯

|

2

dxdt

−

cρ

Z Z

Qτ

|

u

ˆ

−

u

¯

|

dxdt

−

cρ

k

u

ˆ

−

u

¯

k

2 L2_(Q)

.

Similar estimates for the terms

B

[

h, r

h

]

and

B

[

r

h

]

2yields altogether

L

′′

(¯

u,

µ

¯

)(ˆ

u

−

u

¯

)

2

≥

δ

2

Z Z

Q\Qτ

|

u

ˆ

−

u

¯

|

2

dxdt

−

cρ

Z Z

Qτ

|

u

ˆ

−

u

¯

|

dxdt

−

cρ

k

u

ˆ

−

u

¯

k

2 L2_(Q)

.

Substituting this estimate in (4.11) and choosing

ρ

sufficiently small, we derive

J

(ˆ

y,

u

ˆ

)

−

J

(¯

y,

u

¯

)

≥

(

τ

−

cρ

)

ZZ

Qτ

|

u

ˆ

−

u

¯

|

dxdt

+

δ

4

Z Z

Q\Qτ

|

u

ˆ

−

u

¯

|

2

dxdt

−

cρ

k

u

ˆ

−

u

¯

k

_L22_(Q)

− |

r

_2,L

|

≥

τ

2

ZZ

Qτ

|

u

ˆ

−

u

¯

|

dxdt

+

δ

4

Z Z

Q\Qτ

|

u

ˆ

−

u

¯

|

2

dxdt

−

cρ

k

u

ˆ

−

u

¯

k

2_L2_(Q)

− |

r

_2,L

|

.

Since

k

u

ˆ

−

u

¯

k

L∞_(Q)

< ρ <

1

, we have

|

u

ˆ

−

u

¯

| ≥ |

u

ˆ

−

u

¯

|

2 almost everywhere in

Q

. Due to the

remainder estimate (3.14), we can conclude

J

(ˆ

y,

u

ˆ

)

−

J

(¯

y,

u

¯

)

≥

τ

2

ZZ

Qτ

|

u

ˆ

−

u

¯

|

2

dxdt

+

δ

4

ZZ

Q\Qτ

|

u

ˆ

−

u

¯

|

2

dxdt

−

cρ

k

u

ˆ

−

u

¯

k

_L22_(Q)

−

c

_L

k

u

ˆ

−

u

¯

k

_L∞_(Q)

k

u

ˆ

−

u

¯

k

2 L2_(Q)

≥

(

δ

′

−

cρ

)

k

u

ˆ

−

u

¯

k

2 L2_(Q)

≥

δ

′

2

k

u

ˆ

−

u

¯

k

2 L2_(Q)

for sufficiently small

ρ >

0

. Case II:

h

∈

C

(¯

u

)

\

C

β,τ

(¯

u

)

In this case we have to deduce the quadratic growth condition from the first order optimality conditions. Note that the linearized state

y

h

=

G

′

(¯

u

)

h

satisfies the following inequality

< y

h

,

µ >

¯

C(Q),M(Q)

> β

ZZ

Q\Qτ

|

h

(

x, t

)

|

dxdt

(4.12)

Now the first order Taylor expansion of the Lagrangian in (4.8) and the fact

G

(ˆ

u

)