CUBO A Mathematical Journal
Vol.14, No¯ 02, (81–90). June 2012

An Immediate Derivation of Maximum Principle in Banach
spaces, Assuming Reflexive Input and State Spaces.

Paolo d’Alessandro

Department of Mathematics,

Third university of Rome.

email: dalex@mat.uniroma3.it

ABSTRACT

We consider a standard setting for the norm optimal problem in Banach spaces and

show that with a simple argument which invokes some appropriately selected powerful

general Theorems for Banach spaces a straightforward derivation of the Maximum

Principle is obtained.

RESUMEN

Consideramos una formulación estándar para el problema de norma optimal en espacios

de Banach y mostramos que con un argumento simple que invoca algunos fuertes teo-

remas generales de la teoŕıa de espacios de Banach elegidos apropiadamente se deriva

directamente el Principio del Máximo.

Keywords and Phrases: Linear Control Systems in Banach Spaces, Norm Optimal Control,

Support of Closed Convex Sets

2010 AMS Mathematics Subject Classification: 49K20.



82 Paolo d’Alessandro CUBO
14, 2 (2012)

1 Introduction

We consider an optimal control setting proposed by Fattorini [1], where the control functions are

in the space L
∞
([0, Γ], Eu), and Eu is a real Banach space. To simplify the matter we make the

assumption that Eu and state space E (another real Banach space) be a reflexive Banach spaces

(but, anyway, L
∞
([0, Γ], Eu) is not reflexive). The spaces Eu and E are not assumed to be separable.

As explained later on, the hypothesis on Eu has a justification, in that it is a possible way to make

the setting to work, which requires L
∞
([0, Γ], Eu) to be the adjoint of another space. The hypothesis

on E make it easier on the semigroup front. Essentially this is one out of the many settings covered

in [1], but handled with a different technique.

Targets states for which the Maximum Principle may hold must be support points of the set of

states reachable under bounded norm. Applying the results in [8] we show that a weath of support

points exists. Then, suitably blending certain ingredients and, more precisely: the properties of the

setting, the celebrated Bishop Phelps Support Theorem, the determination and characterization of

the radial kernel of the bounded norm reachable set, and, finally, an important technical Lemma by

Fattorini, we obtain a very simple derivation of the Maximum Principle for a dense set of targets

and functionals in E∗. In addition, we geometrically characterize the set of targets for which the

Principle holds, along with the conditions, which decide whether the principle is only necessary or

necessary and sufficient.

Naturally, all this simplicity is made possible by the profound and powerful results we invoke

along the way. But, on the other hand, this is what powerful theorems are mainly useful for: make

life easier.

There is clearly the need to investigate the connections of our analysis with recent research

work on Maximum principle, mainly by Fattorini. We comment briefly on this in the conclusions,

touching upon the open problems it poses.

2 Unconstrained and Constrained Reachable Sets

Consider the variation of constants formula:

x(t) = T(t)x +

∫

[0,t]

T(t − σ)Bu(σ)ds

where t ∈ [0, Γ], {T(t)}, is a C0 semigroup on a real Banach space E, B: Eu → E is an operator

on the real Banach space Eu, u ∈ L∞([0, Γ], Eu), and the integral is a Bochner integral. The case

Eu = E and B = I is referred to as full control case, but here we do not make the full control

assumption. We assume both E and Eu reflexive. These hypotheses will be in force throughout

the paper.

For simplicity, dealing with norm optimal control problems, we assume x = 0. The general



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An Immediate Derivation of Maximum Principle in Banach spaces ... 83

case where the system starts from a non-zero state x is dealt with considering the target ζ − T(Γ)x

in place of ζ (see [1]).

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

ζ =

∫

[0,Γ]

T(Γ − σ)Bu(σ)ds = LΓ u

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

transformation LΓ : L∞([0, Γ], Eu) → E is well known to be continuous.

On the other hand, in our setting (see [1]):

L
∞
([0, Γ], Eu) = (L1([0, Γ], E

∗

u))
∗

Remark 1. As stated by [1], if we had put Eu = F, and had taken X such that F = X
∗ then

for L
∞
([0, Γ], F) = (L1([0, Γ], X))

∗ to hold, barring separability assumptions, it remains to assume

reflexivity of X. For simplicity, we have taken directly Eu reflexive which implies E
∗

u reflexive.

Thus our hypothesis on value space Eu is justified, as long as we are interested in a setting where

the above duality relation on time function spaces holds. Reflexivity of E provides instead a

simplification in terms of semigroup theory: in fact, in this case, the adjoint semigroup is C0, and

we are dispensed to invoke the Phillips dual, as in the general case (again [1]).

This setting has the advantage that we find ourselves on the dual side, where the situation is

much more favorable. While in general, in view of the celebrated James’ Theorem, the unit ball of

a Banach space is not weakly compact, in the dual, thanks to the Banach-Alaoglu Theorem ([6]),

the unit ball is weak∗ compact.

This circumstance is obviously useful if we can prove that the operator LΓ is weak
∗ to weak

continuous.

We begin recalling that the assumed strong measurability of u(.) implies weak measurability

([7]). Henceforth we use the symbol < ., . > to denote the canonical pairing functionals. If different

pair of spaces are involved we either use a suffixes, or leave distinctions to the context when we

feel it is safe to do so.

Proposition 2. The operator LΓ is weak
∗ to weak continuous.

Proof. Consider any y ∈ E∗. Recall that under our hypothesis that E be reflexive {T∗(.)} is also a

C0 semigroup. Thus

g(.) = B∗T∗(Γ − .)y ∈ C([0, Γ], E∗u) ⊂ L1([0, Γ].E
∗

u)

In particular it is weakly measurable. Next recall that continuous linear functionals and transfor-

mation can go in and out the Bochner integral ([7]), and that, since E is reflexive {T∗(.)} is a C0.

semigroup. So, consider a weak∗ convergent net {uα} → u in L∞([0, Γ], Eu) and write:

< y,

∫

[0,Γ]

T(Γ − σ)Buα(σ)ds >=



84 Paolo d’Alessandro CUBO
14, 2 (2012)

∫

[0,Γ]

< B∗T∗(Γ − σ)y, uα(σ) > ds =

< g(.), uα(.) >L1L∞

Thus the net {< g(.), uα(.) >L1L∞} converges and the proof is finished.

This result implies that Rρ = LΓ (Bρ), the image under LΓ of the unit ball Bρ of L∞([0, Γ], Eu),

is weakly compact and hence weakly closed. Moreover, since Rρ is convex, it is also strongly closed.

3 Properties of Bounded Norm Reachable Set.

Let RΓ = R(LΓ ), which is the reachable set in the interval [0, Γ]. We may assume, without

restriction of generality that R−
Γ
= E. If this where not the case it suffices to consider R−

Γ
in lieu of

E. Generality is not restricted because a closed subspace of a reflexive Banach space is reflexive.

For convenience we summarize the relevant properties of the constrained reachable set LΓ (Bρ) =

Rρ.

• Convex and circled (and hence also symmetric). In particular 0 ∈ Rρ.

• L(Rρ) = RΓ

• Weakly compact and both weakly and strongly closed.

• In general it has no interior.

In special cases Rρ might well have interior, but in what follows we assume R
i
ρ = φ.

We add to this list a further important property.

First we state the following:

Definition 3. We define:

R∨ρ = {z : z ∈ Rρ, inf{‖u‖, LΓ (u) = z} < ρ}

and put:

R∧ρ = Rρ\R
∨

ρ

Proposition 4. R∨ρ is the radial kernel of Rρ in RΓ. If ζ ∈ R
∧
ρ then ∃ũ such that LΓ (ũ) = ζ,

‖ũ‖ = ρ = min{‖u‖ : LΓ (u) = ζ}. A necessary condition for a state ζ to be a support point of Rρ

is that ζ ∈ R∧ρ .



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An Immediate Derivation of Maximum Principle in Banach spaces ... 85

Proof. Clearly 0 ∈ R∨ρ . If w 6= 0 and w ∈ RΓ , ∃u 6= 0 s.t. LΓ (u) = w. If 0 ≤ ε < ρ, then the

non-zero state:

LΓ (ε
u

‖u‖
) ∈ Rρ

thus Rρ is radial at zero in Rρ. Next, if ξ 6= 0, ξ ∈ R
∨
ρ , then ∃u s.t.‖u‖ = ρ

′, 0 < ρ′ < ρ and

LΓ (u) = ξ . Then for an arbitrary z 6= ξ, 0 6= z ∈ RΓ , let LΓ (uz) = z − ξ and ‖uz‖ = γ 6= 0. The

state ξ+ α(z − ξ), with α ≤ ρ−ρ
′

γ
, is reachable by the control u + αuz, whose norm is less or equal

to ρ, by the triangle inequality. Thus ξ+ β(z − ξ) ∈ Rρ for 0 ≤ β ≤ α and Rρ is radial at ξ. Next

ζ ∈ R∧ρ implies

inf{‖u‖, LΓ (u) = ζ} ≥ ρ

But, also, ζ ∈ Rρ implies ∃u s.t.‖u‖ ≤ ρ and so the second statement is proved. The just proved

radiality property prevents any state in R∨ρ to be a support point of Rρ. In fact suppose that for

some ζ ∈ R∨ρ there exists a continuous linear functional f, such that < f, Rρ >≤< f, ζ >. If it

were < f, ζ >= 0 then < f, Rρ >= {0} =< f, L(Rρ) >=< f, RΓ >. But the fact that RΓ is contained

in a closed hyperplane contradicts the fact that RΓ is dense. If < f, ζ >= α > 0, then because

by radiality, for some ε > 0, ξ = (1 + ε)ζ ∈ Rρ, we can write < f, ξ >>< f, ζ >, contradicting

separation. This concludes the proof.

Remark 5. If Rρ had interior then R
∨
ρ = R

i
ρ and B(Rρ) = R

∧
ρ , and all points of R

∧
ρ would be

support points. If not then, since Rρ is closed, B(Rρ) = Rρ, but this proposition tell us that

we can find support points only in the proper subset R∧ρ . In this case we may view the above

partition as a quasi-topological decomposition in which R∨ρ plays the role of quasi-interior and R
∧
ρ

as a quasi-boundary.

The results on the support problem given in [8] hold good here because they are surely true

for Hausdorff complete locally convex spaces. We summarize the argument here. First notice that

the tangent cone to a convex set at an extreme point is always pointed. Theorem 5 in [8] states

that the closure of a pointed cone in a linear topological space is a proper cone. In the same paper

Lemma 2 states that a closed proper cone is contained in a closed semispace; the statement is

made for Hilbert spaces but it is obvious from it simple proof that it is indeed valid for Hausdorff

complete locally convex spaces. It follows that all extreme points of a convex set are support

points. Next, by the Krein-Milman Theorem, the set ex(Rρ) of extreme points of Rρ is non-void

(and generates Rρ by closed convex extension). Clearly ex(Rρ) ⊂ R
∧
ρ , because there cannot be

radiality in an extreme point. As recalled, all points of ex(Rρ) are support points. Let us call the

set of all support points Sρ. We collect this remarks in the following:

Proposition 6. ex(Rρ) ⊂ Sρ ⊂ R
∧
ρ . In particular Sρ 6= φ.

4 Support for Void Interior Convex Sets

In a nutshell the maximum principle is about showing that a target is a support point for Rρ and

then translating the supporting condition in a pointwise in time condition for the optimal control.



86 Paolo d’Alessandro CUBO
14, 2 (2012)

The support condition is a special separation condition (separation between a singleton in

a set and the set itself). One classical way to provide support points for a set is to invoke a

topological separation theorem, which states that, in a linear topological space, given two convex

sets A and B and assuming that A has interior, there is a continuous linear functional separating

A and B iff B ∩ Ai = φ. One obviously uses this Theorem taking B = {ζ} with ζ ∈ B(A) thereby

obtaining a support point for A−. The derivation of this Separation Theorem in [6] is essentially

pre-topological and based on the theory of cones.

Despite the ”iff” we are in presence of a masked sufficient condition, because of the presiding

hypothesis that Ai 6= φ. Thus we cannot exclude the possibility of finding support points for void

interior sets.

If the dimension is finite, then every convex set has (relative) interior, so that the application

of this separation theorem to find support targets is direct and general.

In infinite dimension, as already mentioned, Rρ has no interior in general.

We see various possible techniques to overcome this hurdle.

The first consists in re-topologizing Rρ in such a way that it has interior in the new topology.

This is the technique introduced by Fattorini (see [1]), who has developed a very complete and

advanced theory for both norm and time optimality. Because the new topology is stronger, in the

larger dual space, singular functionals appear (non-zero functionals that are zero on the domain of

the infinitesimal generator).

The second possibility (developed in [8]) consisted in introducing and applying a Support

Theorem for extreme points of convex sets (like all other separation/support theorems, this too is

based on theory of cones). This technique works well in a Hilbert space setting. In the present

setting it allows us to exhibit an already large set of support points via the Krein-Milman Theorem.

But it would be nice to tell more about the structure of the set of support points of Rρ.

To this effect we apply a further tool: the celebrated Bishop-Phelps Theorem for closed sets

in Banach spaces (see [9]).

We recall the relevant part of the Bishop-Phelps theorem:

Theorem 7. A closed convex set of a real Banach space has a non-void set of support points which

is dense in its boundary.

Remark 8. Some authors have shown that the corresponding statement for complex Banach spaces

does not hold (see e.g. [10]). This is completely irrelevant here.

Basically the idea of the proof of the Bishop Phelps Theorem is to observe that the cone

generated by a translated ball not containing the origin is a pointed cone with interior. If we can

place, by translation, the apex of this cone on a point of the convex set in such a way that the

apex is the only point in their intersection, then the cone is separated from the convex set. Note

the ingenuous swap of roles: here it is the cone in charge of insuring that at least one of the convex

sets has non-void interior, so that the convex set, for which support is sought, is allowed to have



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An Immediate Derivation of Maximum Principle in Banach spaces ... 87

void interior. But such separation and the fact that the second set is a cone, imply that the point

in question is a support point for the convex set.

5 Norm Optimality

Putting together the Bishop-Phelps Theorem and Proposition 4, we obtain the following result:

Theorem 9. The set Sρ of all support points for Rρ, which we proved to be non-void and contained

in R∧ρ , is dense in R
∧
ρ . For all points ζ ∈ Sρ, the minimum norm of controls that steer the system

from 0 (at t = 0) to ζ (at t = Γ) exists and is ρ.

The proof of this Theorem is contained in the previous analysis and can be omitted. Notice

that if the interior of Rρ is void (as we are assuming) then Sρ is dense in the whole Rρ and hence

also in the set R∧ρ into which is contained. If the interior is non-void, then Sρ is dense in B(Rρ),

which, in such case, coincides with R∧ρ .

Let’s write down the support condition for ζ ∈ Sρ. There exists a continuous linear functional

f ∈ E∗ such that:

< f, ζ >≥< f, z > , ∀z ∈ Rρ

The same condition holds, however, for all ξ ∈ Rρ such that:

< f, ξ >=< f, ζ >

Define

Ω = {ξ : ξ ∈ Rρ, < f, ξ >=< f, ζ >}

Note that Ω is a closed convex set, as a matter of facts it is a closed exposed face of Rρ and

obviously:

Ω ⊂ Sρ

It is well possible that Ω = {ζ}, in which case ζ is an exposed extreme point. But it may happen,

as well, that Ω is a proper superset of {ζ}.

Next, to move back to the control space, we begin recalling the following well known propo-

sition ([6]):

Proposition 10. Let T be a linear transformation E → G and C be a convex subset of E. If A is a

face of T(C) then T−1(A) ∩ C is a face of C.

It follows that L−1
Γ

(Ω) ∩ Bρ is a closed face of Bρ. It is also an exposed face because we can

retrieve it as a face of Bρ, generated by the support functional

g = L∗Γ f = B
∗T∗(Γ − .)f ∈ C([0, Γ], E∗u) ⊂ L1([0, Γ], E

∗

u)

that ”pushes back” the support functional f ∈ E∗.



88 Paolo d’Alessandro CUBO
14, 2 (2012)

In fact for each ξ ∈ Ω, for all the corresponding uξ ∈ L
−1
Γ

(Ω) ∩ Bρ it must be (in the next

formula there are two different pairing functionals, but we leave unchanged the symbol):

< f, ξ >=< f, LΓ uξ >=< L
∗

Γ f, uξ >≥< f, LΓ u >=< L
∗

Γ f, u > , ∀u ∈ Bρ

If instead u ∈ Bρ\L
−1
Γ

(Ω) ∩ Bρ:

< g, uξ >>< g, u >

Thus we have proved that the functional g is a support functional for Bρ at all points of L
−1
Γ

(Ω)∩Bρ

and this set is a closed and exposed face of Bρ.

Remark 11. A consequence of James’ Theorem insures that, for a non-reflexive Banach spaces there

exist some continuous linear functionals, that do not attain their supremum on the unit ball (but,

of course, by the Separation Theorem, there is also a profusion of continuous linear functionals

that do attain their supremum on the unit ball). Clearly g is is not one of those pathological

continuous linear functionals, because we have proved that it attains its supremum on Bρ.

If we use the above condition to characterize uξ, we have a necessary condition for the optimum

controls corresponding to the target ζ. The condition is not sifficient if Ω\ {ζ} 6= φ. If, by the

contrary, no other target is involved, or Ω = {ζ}, the condition becomes sufficient (independently

of the fact that the optimum control be unique). Reall that Ω = {ζ}means that ζ is an exposed

extreme point. The proof is implicit in the above discussion illustrating the correspondence between

the two exposed faces of Rρ and Bρ. We register this fact in the following:

Theorem 12. The condition on u:

< L∗Γ f, u >= max {< L
∗

Γ f, u > : u ∈ Bρ}

is necessary for a control u to be the norm optimal for the target ζ The condition becomes also

sufficient if Ω = {ζ}, or, equivalently, if ζ is an exposed extreme point.

6 Maximum Principle

To state the Maximum Principle, we need to recast the support condition in an equivalent condition

on the optimal input, which characterizes the input pointwise in time.

Given the reflexivity assumptions in force, the function

L∗Γ f(.) = B
∗T∗(Γ − .)f

is continuous thus Borel and weakly measurable. Also continuity of the norm implies that

‖L∗
Γ
f(.)‖E∗ is a continuous and thus measurable function. The support condition for the opti-

mum control u yields:

< L∗Γ f, u >L
1
L∞

=

∫

[0,Γ]

< L∗Γ f, u >E∗uEu (σ)dσ =



CUBO
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An Immediate Derivation of Maximum Principle in Banach spaces ... 89

= max






∫

[0,Γ]

< L∗Γ f, u > (σ)dσ: u ∈ Bρ





≤ ρ

∫

[0,Γ]

‖L∗Γ f‖E∗u(σ)dσ =

=

∫

[0,Γ]

max{< L∗Γ f, v >: ‖v‖Eu ≤ ρ}(σ)dσ < ∞

On the other hand, by Lemma 2.2.10 in [1], the inequality can be substituted by equality (indeed

Fattorini proved this for arbitrary Banach spaces) and so the support condition implies (and clearly

is implied by):

< B∗T∗(Γ − σ)f, u(σ) >= max{< B∗T∗(Γ − σ)f, v >: ‖v‖Eu ≤ ρ}

a.e. for σ ∈ [0, Γ]. This characterization of the optimal control is the Maximum Principle. It yields a

set of optimal controls. In view of the equivalence with the support condition, our consideration on

whether necessity or necessity and sufficiency prevail hold good as well for the Maximum Principle.

Thus if ζ is a an exposed extreme points all controls defined by the principle are optimal (or the

condition is necessary and sufficient). Otherwise we can only say that the optimal controls are

among those functions satysfying this condition (the condition is necessary).

7 Conclusions

The following considerations are inspired by the cited work by Fattorini, including a private com-

munication.

One open problem is the relationship between the dense subset of targets for which the Max-

imum Principle holds for functionals in E∗, that we have shown to exists, and the domain of

A.

In [1] the Maximum Principle is proved for targets in D(A) using functionals in a linear space

Z larger than E∗.

On the other hand [3] shows that, in general, if the target is in D(A), then the functional in

E∗does not always exist.

He also conjectures that this implication may fail even under the assumption that the semi-

group is selfadjoint and the state space is a Hilbert space, albeit the implication has been found

to hold for the left translation semigroup and E = L2([0, ∞)) ([4]).

Thus he puts the question of finally determining a condition, stronger than target ζ ∈ D(A),

ensuring that the functionals appearing in the Maximum Principle exists in E∗.

This question is of course crucial but it is open at the moment.

In some cases this whole issue is known to be connected the other important aspect of regularity

of optimal control (e.g. [2]). A further motivation for orientating research toward these open

problems.



90 Paolo d’Alessandro CUBO
14, 2 (2012)

Received: October 2010. Revised: October 2011.

References

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

[2] H.O. Fattorini, ”Strong Regularity of Time and Norm Optimal Controls”, Submitted

[3] H.O. Fattorini, ”Linear Control Systems In Sequence Spaces” Functional Analysis and evolu-

tion equations, The Gunter Lumer Volume 2007, pp 273-290.

[4] H.O. Fattorini, ”Regular and Strongly Regular Time and Norm Optimal Controls” Submitted.

[5] H.O. Fattorini, Private Communication.

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

[7] K. Yosida, ”Functional Analysis”, Springer-Verlag, New York, 1974.

[8] P. d’Alessandro, ”Closure of Pointed Cones and Maximum Principle in Hilbert Spaces”, CUBO

a Mathematical Journal, Vol.13, No.2 (73-84), June 2011.

[9] R.R. Phelps, ”Support Cones in Banach Spaces and Their Application”, Adv. in Math. 13

(1974), 1-19.

[10] W.B. Johnson, J. Lindenstrauss Eds, ”Handbook of The Geometry of Banach Spaces”, Vol.

1, Elsevier, Amsterdam, 2001.


	 Introduction
	Unconstrained and Constrained Reachable Sets
	Properties of Bounded Norm Reachable Set.
	Support for Void Interior Convex Sets
	Norm Optimality
	Maximum Principle
	Conclusions