()


CUBO A Mathematical Journal
Vol.13, No¯ 02, (73–84). June 2011

Closure of Pointed Cones and Maximum Principle in
Hilbert Spaces

Paolo d’Alessandro

Math. Department,

Third University of Rome,

email: dalex@mat.uniroma3.it

ABSTRACT

We prove, in a Hilbert space setting, that all targets of the minimum norm optimal

control problems reachable with inputs of minimum norm ρ are support points for the

the set reachable by inputs with norm bounded by ρ. This amount to say that the

Maximum Principle always holds in Hilbert Spaces.

RESUMEN

En este art́ıculo se demuestra que, para el problema de control óptimo a un nivel

mı́nimo en los espacios de Hilbert, todos los estados alcanzables con un nivel mı́nimo

de entrada de ρ son puntos de apoyo para el conjunto de estados alcanzables por la

norma de entrada inferior o igual a ρ. Esto es equivalente a decir que el Principio

Máximo siempre es válido en los espacios de Hilbert.

Keywords and phrases: Linear Control Systems in Hilbert Spaces, Norm Optimal Control,

Maximum Principle.

Mathematics Subject Classification: 93E20, 93E25.



74 Paolo d’Alessandro CUBO
13, 2 (2011)

1. Introduction

Arguing in a Hilbert space setting, suppose that the minimum norm of inputs, which steer

the origin to a state ζ in a finite interval of time [0, Γ ] is ρ. That the Maximum Principle holds,

seen through the convex analysis optics, means that the target ζ is a support vector for the set Rρ
of all vectors reachable under an input of norm less than or equal to ρ. In other words the normal

cone to Rρ at ζ is non-trivial. In the literature vectors in the normal cone, which live in the dual

space, are also called multipliers, a term that has a more general meaning.

To verify that Rρ has support at ζ, one might hope to apply to Rρ and {ζ} the celebrated

Separation Theorem for linear topological spaces, stating that, given two non-void convex sets A

and B, and assuming that A has interior, they can be can be separated by a continuous linear

functional if and only if B ∩ Ai = φ.

Unfortunately, this application is not possible in the infinite dimensional case because it is not

true in general that Rρ has interior.

It is very well known, and easy to show, that the set of support points S is contained and is

dense in the set R∧ρ of target points reachable with minimum input norm ρ. This might suggest

that some of the above targets are not support point. On the other hand the cited Separation

Theorem above is indeed a sufficient condition, in view of the presiding hypothesis (one of the sets

has interior), so that it leaves open the problem of determining if separation holds for all targets

or if, instead, the dense subset of support points is proper.

Much attention (see [1]) has been devoted to more sophisticated Banach space settings, ob-

taining a generalization of the Maximum principle, thanks to the definition of a larger linear space

of multipliers, which contains the dual space. Similarly, this leaves to determine which multipliers

are in the dual space, although sufficient conditions are known.

For details as well as for accurate historical remarks and proper credits, reference can be made

to the vast and outstanding work by Fattorini, which covers a variety of Banach space settings.

Recent work is, besides [1], [6] and [7].

The purpose here is to answer for Hilbert spaces the question connected to the aforementioned

density results: are there vectors of R∧ρ at which Rρ has no support?

The question remained open for quite a long time. And the answer is ”no”. The argument lean

on a (very general) result of the theory of cones and on strict convexity of Hilbert space norms.

Thus generalization are possible (although beyond the present purposes), but our technique cannot

pass the barrier of the requirement of strict convexity of norms.

More specifically, we will show that the tangent cone to Rρ at any ζ ∈ R
∧
ρ is pointed. That

some vector ζ ∈ R∧ρ is not a support point for Rρ is equivalent to say that the polar cone to the

tangent cone to Rρ at ζ is trivial. This is in turn equivalent to say that the closure of the tangent

cone is the whole space. That the closure of a pointed cone be the whole space is a rather counter-

intuitive proposition, and in fact we prove that this is not the case, in any linear topological space.



CUBO
13, 2 (2011)

Closure of Pointed Cones and Maximum Principle in Hilbert Spaces 75

From this fact it follows, as an immediate consequence, that all ζ ∈ R∧ρ are support points for Rρ.

No assumptions will be made on either separability or full control. More might be said in the

separable case, but this is not dealt with here.

With an apology, our exposition covers some well known basic facts and complements, in order

to enhance readability. A more succinct exposition would result in a choppy and difficult to follow

narration.

2. Setting

We refer to the Cauchy problem with u ∈ L2([0, Γ ], H1) as given in [2]. The abstract differential

equation has the form:
.
x(t) = Ax(t) + Bu(t)

where the spaces H1, H are real Hilbert spaces, B is an operator (bounded linear transformation)

H1 → H, A is the infinitesimal generator of a strongly continuous semigroup {T (t)} on H and
x(0) = x is given. The equation is intended in weak sense and its unique solution is expressed by

the formula of variation of constants:

x(t) = T (t)x +

∫

[0,t]

T (t − s)Bu(s)ds

which formally recurs also in other settings, like that of differential equations in Banach Spaces.

In our specific case the integral is a Pettis integral. The case H1 = H and B = I is referred to as

full control case, but here we do not make any assumption in this respect. For simplicity, dealing

with norm optimal control problems, we assume x = 0. The general case is a variant of this and

can be obtained along the lines of [1].

It is assumed that we can reach a certain target vector ζ ∈ H at time Γ > 0 or:

ζ =

∫

[0,Γ ]

T (Γ − s)Bu(s)ds = LΓ u

and we look for the minimum norm input that does the job of reaching ζ. The linear transformation

LΓ : L2([0, Γ ], H1) → H is well known to be continuous. This is all we will need in the sequel as to
the properties of this operator.

If the control steering the system from the origin at time 0 to ζ at time Γ is unique, because

the operator LΓ is one to one, then, according to Fattorini, who introduced this concept in sixties,

the system is called rigid. Fattorini constructed an example of rigid system showing that this

phenomenon can actually occur (see e.g. [1] and references therein). In this case, characterizing the

optimum control, becomes obviously pointless. We note that our arguments hold good even when

LΓ is one to one. In this case though, exploiting the fact that ζ is a support point, can only lead

to retrieving the unique control that solves the reachability problem.



76 Paolo d’Alessandro CUBO
13, 2 (2011)

It is immediate to show that the minimum norm control exists and is unique, since this is a

straightforward consequence of the Projection Theorem.

3. Lemmata and Definitions

The environment is, unless otherwise stated, a real Hilbert space H. The symbols L(C) and

L−(C) denote, respectively, the linear and closed linear hull of the non-void set C, whereas Co(C)

and Co−(C) denote, respectively, its conical and closed conical hull.

Given a convex cone C in H its polar cone is the (always closed) cone:

Cp = {y : (y, x) ≤ 0: ∀x ∈ C}

Definition 1. The tangent cone to C at y ∈ C is the cone Co(−y + C), The normal cone to C at y

is the polar of the tangent cone to C at y.

Consider a convex set C and ζ ∈ C. We say that ζ is a support point for C if there exists a

nonzero vector n (which can be taken with unit norm) such that:

(n, ζ) ≥ (n, z), ∀z ∈ C

This is equivalent to say that C is contained in the closed half-space defined by the continuous

linear functional (n, .):

{y : (n, y) ≤ (n, ζ)}

whose limiting (closed) hyperplane contains {ζ}. In still another equivalent terminology, this is

equivalent to say that the tangent cone to C at ζ has a non-trivial polar cone (because n is in such

cone).

Let n be an unit norm vector. The closed convex set:

{x : a ≤ (n, x) ≤ b}

(with b ≥ a) is called a sandwich. The number b − a is the thickness of the sandwich. If the

thickness is zero, the sandwich is a closed hyperplane. The sandwich is symmetrical if it has the

form

{x : −a ≤ (n, x) ≤ a}

for some a ≥ 0.

The next Lemma regards closed cones in Hilbert spaces

Lemma 2. A closed cone in a Hilbert space H is proper if and only if it is contained in a closed

half-space.

Demostración. Let C be a closed proper cone. Then there is a singleton {y} disjoint from C.

Singletons are convex and compact and therefore the Strong Separation Corollary 14.4 in [4]

applies. The rest is immediate.



CUBO
13, 2 (2011)

Closure of Pointed Cones and Maximum Principle in Hilbert Spaces 77

We recall briefly some other relevant notions about cones in infinite dimensional Hilbert Spaces.

Incidentally, the literature on this topic tend to be either finite dimensional or infinite dimensional,

but typically on the footsteps of the seminal work of Choquet, aiming at extending the Krein

Milman Theorem in a measure theory setting.

Definition 3. A (convex) cone C is pointed if C ∩ −C = {0}. A cone C is blunt if L−(C) = H

The following Lemma well known.

Lemma 4. If a closed C cone is pointed, its polar cone is blunt, that is, L−(Cp) = H.

Demostración. The proof is simple and based on elementary computations that, for brevity, are

taken for granted here, but, on the other hand, are rather intuitive, since polarization is the

analogous for cones of orthogonal complementation for subspaces. We can write:

{0} = C ∩ −C = [Cp + (−Cp)]p = L(Cp)p

and, taking polars of the first and last cone the desired conclusion follows.

If a pointed cone is not closed what can we say of its polar cone? Notice that according to

Lemma 2, if the closure of the cone is proper, then the cone has a nontrivial polar cone. For the

polar to be trivial instead, again in view of the same Lemma, the closure of the cone must be the

whole space, despite the fact that the cone is pointed. But we prove here that this cannot happen

even in general, as stated by the following:

Theorem 5. The closure of a pointed cone in a linear topological space is a proper cone.

Demostración. Suppose that it is not true, that is there is a pointed cone C in a linear topological

space E, such that C− = E. Consider a finite dimensional subspace F, which intersect C in a

non trivial, necessarily pointed, cone. Actually we can take instead of F, its subspace L(F ∩ C),

without restriction of generality. For simplicity we leave the symbol F unchanged, and equip F with

the relative topology. Next notice that, as is well known, because F is the finite dimensional, the

pointed convex cone Υ = F ∩ C has interior. Thus it can be separated by a continuous linear

functional from the origin and therefore it is contained in a closed semi-space. It follows that

the closure of Υ in F is contained in a closed half-space and therefore is a proper cone. But by

Theorem 1.16 in [5], such closure is C− ∩ F. By the initial assumption C− ∩ F = E ∩ F = F. This is

a contradiction and therefore the proof is finished.

4. Existence and uniqueness in L2(H1)

Assume that for some ζ ∈ H, ∃uζ such that x = ζ = LΓ (uζ). The set of all u satisfying

ζ = LΓ (u) is given by uζ + N (LΓ ). This is a closed affine space, because LΓ is continuous. If

N (LΓ ) is trivial, the unique uζ solving ζ = LΓ (uζ) is already optimum. Optimization in this case



78 Paolo d’Alessandro CUBO
13, 2 (2011)

is pointless, but the arguments below hold good anyway. To obtain the minimum norm solution we

can apply the projection theorem and project the origin on this closed convex set. Moreover, we

know, from the celebrated Projection Theorem for Hilbert spaces, that this projection exists and

is unique and hence the minimum norm solution always exists and is unique. We call this unique

minimum norm control uo.

We can put:

‖uo‖L2(H1) = ρ

which is the optimum value of the norm. In particular we can say that the optimum control belongs

to the closed sphere Sρ of radius ρ, around the origin in L2([0, Γ ], H1) (briefly L2(H1)):

uo ∈ Sρ

and so:

ζ ∈ LΓ (Sρ) ⊂ R(LΓ )

It is immediate to verify that:

LΓ (Sρ) = {z : mı́n{‖u‖ : LΓ (u) = z} ≤ ρ}}

For notational simplicity we put LΓ (Sρ) = Rρ.

5. Reachability

In this section we recast a few well known facts of reachability theory.

We noted that ζ ∈ LΓ (L2(H1)) = R(LΓ ) = RΓ . This is the reachable set at time Γ and is a

linear subspace of H, which is in general not closed. However, we can always argue in the Hilbert

space LΓ (L2(H1))
− and thus assume, without restriction of generality, that RΓ is a dense linear

subspace in H. This is equivalent to N (L
∗

Γ
) = {0}. To streamline the exposition this assumption

will be in force thoroughly.

Note that L∗Γ is the map x → B
∗T ∗(Γ − .)x. Thus x ∈ N (L

∗

Γ
) if and only if the continuous

function B∗T ∗(Γ − .)x is identically zero. Next notice that:

N (L
∗

Γ
) = N (L

Γ
L∗Γ )

for N (L
∗

Γ
) ⊂ N (L

Γ
L∗Γ ) is obvious and, on the other end:

LΓ L
∗

Γ x = 0 ⇒ (x, LΓ L
∗

Γ x) = ‖L
∗

Γ x‖
2 = 0

Therefore, under the present hypothesis that N (L
∗

Γ
)⊥ = R−

Γ
= H, the selfadjoint operator

GΓ = LΓ L
∗

Γ



CUBO
13, 2 (2011)

Closure of Pointed Cones and Maximum Principle in Hilbert Spaces 79

has a trivial kernel, and so it is one to one.

Moreover, obviously R(GΓ ) ⊂ R(LΓ ) = RΓ .

Indeed, R(GΓ )
− = R−

Γ
, for:

R(GΓ )
− = N (GΓ )

⊥ = N (L
∗

Γ
)⊥ = R−Γ

Thus both R(GΓ ) and R(LΓ ) are dense, so that for any x ∈ R(GΓ ) and ε > 0 there is an

y ∈ R(LΓ ) such that ‖x − y‖ ≤ ε and vice-versa.

Notice that, by a change of variables, ∀x ∈ H:

GΓ x =

Γ∫

0

T (σ)BB∗T ∗(σ)xdσ

We claim that we can define the integral as a Riemann integral. In fact we can prove that

the integrand is a continuous function. To this purpose first note that the function BB∗T ∗(σ)x is

continuous. Then, to show that the integrand function is continuous, apply the exponential growth

property of the semigroup and following well known:

Theorem 6. Consider a set T ⊂ LC(H,H) and suppose that it is bounded in the norm operator

topology. Then the evaluation map is jointly continuous in T ×H where T is equipped with the

(relativized) pointwise topology.

If the state y is reachable and y ∈ R(GΓ ) ,

y = GΓ x

has solution in x, so that

y =

Γ∫

0

T (σ)BB∗T ∗(σ)dσx =

Γ∫

0

T (Γ − τ)Bw(τ)dτ

where w(τ) = B∗T ∗(Γ − τ)x. In this case, there is a smooth control that solves the problem.

6. The Quasi-Topology of Rρ and Main Theorem

There are a number of interesting properties of Rρ = LT (Sρ), which depend both on the

environment (Hilbert space), on continuity of LT and on the fact that the closed sphere Sρ is

convex and weakly compact. In particular recall that s − s (strong-strong) continuity is equivalent

to w − w (weak-weak) continuity and, therefore, LT is w − w continuous.

The set Rρ is obviously:



80 Paolo d’Alessandro CUBO
13, 2 (2011)

Convex (as image under an operator of a convex set) with 0 ∈ Rρ, and symmetrical.

Weakly compact, as image of a weakly compact set under a w − w continuous linear trans-

formation

Weakly closed + convex and hence strongly closed

Bounded (as image under an operator of a bounded set)

Not a convex body in general.

We now describe what we mean with the quasi-topology of Rρ, adding to the above list the

fact that Rρ has a ”quasi-interior”, at whose points it is densely radial and circled, and a quasi-

boundary, which is the complement in Rρ of the quasi-interior. These terms are justified because

this quasi-topology can be realized as an actual topology, using the topology introduced by Fattorini

for Rρ, based on the norm:

p(z) = ı́nf{‖u‖ : LΓ u = z}

This concept is of primary importance in more general and complex Banach space settings, but it

will not be used here.

Definition 7. The set of all vectors x in Rρ, such that the minimum norm control to reach x has

norm ρ, is called quasi-boundary of Rρ and denoted by R
∧
ρ . The set of all vectors x in Rρ, such

that the minimum norm control to reach x has norm ρ′ < ρ, is called quasi-interior of Rρ and

denoted by R∨ρ .

It is well known that no point of the quasi-interior can be a support point for Rρ.

Consider the origin, which belongs to R∨ρ . The set Rρ is densely radial at the origin. In fact

consider any point z ∈ RΓ and let η be the minimum norm of the unique control u that reaches z.

If η ≤ ρ the whole segment [0 : z] ⊂ LΓ (Sρ). Otherwise take the point z
′ reached by the control

ρ
η
u, and [0 : z′] ⊂ LΓ (Sρ). Observe that ∀z ∈ RΓ is positively proportional to a vector in LΓ (Sρ).

In other words:

∪{αLΓ (Sρ) : α > 0} = RΓ

Moreover it is clear that αLΓ (Sρ) ⊂ LΓ (Sρ) for any positive α ≤ 1, so that LΓ (Sρ) is also circled.

On the other hand, if for ζ ∈ LΓ (Sρ) the minimum norm of the corresponding control to reach

ζ is ρ, by a very well known argument, see e.g. [1], (1 + ε)ζ /∈ LΓ (Sρ) for ∀ε > 0.

All the points of the quasi-interior have the same dense radiality property as the origin. In

fact, if ξ ∈ R∨ρ , ξ 6= 0, then the minimum norm of the control that steers the origin to the state to

ξ is some ρ′ with 0 < ρ′ < ρ. Let uξ be the corresponding minimum norm control. Radiality in

the direction of the origin or of ξ itself is obvious. Next consider any z ∈ RΓ with z 6= 0 and z not

proportional to ξ, and let uz be corresponding the minimum norm control. It will be ‖uz‖ = γ > 0.

Then it is immediate that ξ + αz ∈ Rρ for any α, such that: 0 < α ≤
ρ−ρ

′

γ
.

The role of the dense radiality is emphasized by the proof of the anticipated well known result:



CUBO
13, 2 (2011)

Closure of Pointed Cones and Maximum Principle in Hilbert Spaces 81

Theorem 8. No point of R∨ρ can be a support point of Rρ.

Demostración. Suppose that a point y ∈ R∨ρ is a support point. Then the projection of some vector

z /∈ Rρ on Rρ is y. By the Projection Theorem we have:

(z − y, x − y) ≤ 0, ∀x ∈ Rρ

If equality holds for all x ∈ Rρ then Rρ and hence also RΓ would be contained in a closed hyperplane,

contradicting that RΓ is dense in the space. So ∃x ∈ Rρ such that (z − y, x − y) < 0. Because of

the radiality property at y we can take in lieu of x, a state w = y + α(y − x) ∈ Rρ for some α > 0.

But then (z − y, w − y) = α(z − y, y − x) > 0 contradicting the Projection Theorem.

At this point we know that any support point is in R∧ρ .

It is well known that support points are dense in the quasi-boundary of Rρ (e.g.[2]).

To show this, one may use the following sequence of support points converging to an arbitrary

ζ ∈ R∧ρ . For any positive integer i, (1 +
1
i
)ζ /∈ Rρ. Because the projection PRρ on Rρ is continuous,

the sequence {PRρ ((1 +
1
i
)ζ)} converges strongly to ζ and obviously, because the points in the

sequence are projections, they are all support points (and hence also lie on the quasi-boundary of

Rρ).

However, more is true for arbitrary vectors of the quasi-boundary of Rρ in the present Hilbert

space setting, as we show in the next:

Theorem 9. All vectors in R∧ρ are extreme. No other point of Rρ can be extreme

Demostración. Suppose that for the quasi-boundary vector ζ it is true that ζ = ζ1+ζ2
2

with ζ1
and ζ2 in Rρ, and let ρ1 ≤ ρ be the norm of the minimum norm control u1 that steers the system

to ζ1 and ρ2 ≤ ρ be the norm of the minimum norm control u2 that steers the system to ζ2.

The control u1+u2
2

has norm strictly less than ρ, because the norm of a Hilbert space is strictly

convex, and steers the system to ζ. But this contradicts that ζ ∈ R∧ρ , and so the proof of the first

statement is finished. Points of the quasi-interior cannot be extreme in view of the specific dense

radiality property we have illustrated. Therefore we are done.

Corollary 10. The tangent cone to Rρ at any ζ ∈ R
∧
ρ is pointed

Demostración. Obviously if it were not so it would be contradicted that ζ is an extreme point.

At this point, putting together this Corollary with Lemmata 5 and 2 we have established the

following main

Theorem 11. Any point ζ ∈ R∧ρ is a support point of Rρ. In other words

∀ζ ∈ R∧ρ , ∃n ∈ H
∗ = H, ‖n‖ = 1



82 Paolo d’Alessandro CUBO
13, 2 (2011)

such that

(n, ζ) ≥ (n, z), ∀z ∈ C

Given this result we may expect all the more a large normal fan for Rρ. Indeed the normal

fan is the whole space, but this fact is independent on the main Theorem, as we illustrate in the

next section.

7. The Set of Support Points and the Normal Fan

In this Section we collect some, mostly well known, facts about the set of support points S

(which we now know to be the same as R∧ρ ) and the normal fan. At each support point ζ the

tangent cone has a non-trivial polar cone, also called the normal cone at ζ. The union of these

cones is the normal fan of Rρ. Naturally, it is often convenient to normalize vectors in the normal

fan.

Suppose ζ is a support point of Rρ. Using the pairing in H and the CBS inequality, we have

a well known expression for ζ. Let n a unit norm vector in the normal cone at ζ. Then it must be

for any y ∈ Rρ:

(n, y − ζ) = (n, LΓ uy − LΓ uζ) ≤ 0

or

(n, LΓ uy) ≤ (n, LΓ uζ)

or

(L∗Γ n, uy) ≤ (L
∗

Γ n, uζ)

so that, by the CBS inequality, the optimal control corresponding to ζ has the expression:

uζ = ρ
L∗Γ n

‖L∗
Γ
n‖

, a.e.

and:

ζ = LΓ
ρ

‖L∗
Γ
n‖

L∗Γ n =
ρ

‖L∗
Γ
n‖

GΓ n

On the other hand, for an arbitrary unit norm vector n consider the linear program:

máx{(n, z) : z ∈ Rρ}

The existence of solutions of this linear program derive from the fact that Rρ is convex and weakly

compact and the functional is continuous (and hence also weakly continuous). Let ζ be a solution,

then for any y ∈ Rρ:

(n, y − ζ) ≤ 0

so that ζ is a support point and has the above expression.



CUBO
13, 2 (2011)

Closure of Pointed Cones and Maximum Principle in Hilbert Spaces 83

Thus

S = {ζ : ζ =
ρ

‖L∗
Γ
n‖

GΓ n, ‖n‖ = 1}

Notice that because GΓ is one-to-one, to each support ζ there corresponds a unique normal n. In

other words, all normal cones are rays. We collect these facts and more in the following:

Theorem 12. The normal fan of Rρ, less the origin, is the whole space H\{0}. All normal cones at

support vectors ζ are rays, under the correspondence between unit normal vectors n and support

points ζ:

ζ =
ρ

‖L∗
Γ
n‖

GΓ n

The correspondence between unit vectors in the normal fan and the corresponding support points

is one-to-one. If ζ is a support point {ζ} is an exposed face of Rρ. Equivalently , for any pair

normal-support point it is true that:

(n, y − ζ) < 0, ∀y ∈ Rρ

Consequently (since 0 ∈ Rρ) the value of the functional (n, .) at ζ is always positive.

Demostración. It remains to be proved the last statement. Suppose a ζ ∈ Rρ is a support point, so

that it is necessarily a quasi-boundary and extreme point. Suppose that for some normal n there

is z such that:

(n, z − ζ) = 0

then z is a support point as well with normal n. In fact, ∀y ∈ Rρ:

(n, y − z) = (n, y − ζ) + (n, ζ − z) ≤ 0

Therefore z is a support point and hence it is an quasi-boundary point, so that the minimum norm

control that steers the system to z has norm ρ. Denote by uz the minimum norm control to reach

z and by uζ the minimum norm control to reach ζ. By the same argument above all points in

[ζ : z] are support points with normal n. In fact for 0 < α < 1, ∀y ∈ Rρ:

(n, y − (ζ + α(z − ζ))) ≤ 0

But the point ζ+z
2

can be reached by the control
uζ+uz

2
and ‖

uζ+uz
2

‖ < ρ. This means that ζ+z
2

is a quasi-interior point, which is a contradiction and the proof is finished.

Returning to the linear program:

máx{(n, z) : z ∈ Rρ}

Call the maximum m, then

m =
ρ

‖L∗
Γ
n‖

(n, GΓ n) = ρ(n, GΓ n)
1/2 = ρ‖L∗Γ n‖

1/2



84 Paolo d’Alessandro CUBO
13, 2 (2011)

Recall that m > 0. Because Rρ is symmetric, it is contained in the symmetrical sandwich:

Rρ ⊂ {y : −m ≤ (n, y) ≤ m}

and the limiting hyperplanes of the sandwich meet Rρ only in the two points ζ and −ζ. Also the

sandwich cannot degenerate to a hyperplane, because its thickness is 2m > 0.

Received: October 2009. Revised: January 2010.

Referencias

[1] H.O. Fattorini, Infinite Dimensional Linear Control Systems, Elsevier, Amsterdam 2005.

[2] A.V. Balakrishnan, Applied Functional Analysis,Springer-Verlag Berlin-Heidelberg-New

York 1976.

[3] P.R. Halmos, A Hilbert Space Problem Book Van Nostrand, New York, 1967.

[4] J.L. Kelley and I. Namioka, Linear Topological Spaces, Springer, New York, 1963

[5] J.L. Kelley, General Topology, Springer, New York, 1955.

[6] H.O. Fattorini Smoothness of the Costate and the Target in the Time and Norm Optimal

Problems Optimization, Vol.55, No.2, 2006, 19-36

[7] H.O. Fattorini Regular and Strongly Regular Time and Norm Optimal Controls to appear


	Introduction
	Setting
	Lemmata and Definitions
	Existence  and uniqueness in L2(H1)
	Reachability
	The Quasi-Topology of R and Main Theorem
	The Set of Support Points and the Normal Fan