CUBO A Mathematical Journal

Vol.10, N
o
¯ 01, (77–92). March 2008

Regular and Strongly Regular

Time and Norm Optimal Controls

H. O. Fattorini

Department of Mathematics, University of California

Los Angeles, California 90095-1555

email: hof@math.ucla.edu

ABSTRACT

Pontryagin’s maximum principle in its infinite dimensional version provides (sep-

arate) necessary and sufficient conditions for both time and norm optimality for

the system y
′
= Ay + u (A the infinitesimal generator of a strongly continuous

semigroup). Among controls that satisfy the maximum principle, a smoothness

distinction can be defined in terms of smoothness of the final value of the costate.

This paper addresses some issues related to this distinction.

RESUMEN

El principio del máximo de Pontryagin, en su version de dimension infinita, pro-

porciona condiciones necesarias y suficientes (separadamente) para optimalidad



78 H. O. Fattorini CUBO
10, 1 (2008)

en el tiempo y en la norma para el sistema y
′
= Ay + u (A el generador infinites-

imal de un semigrupo fuertemente cont́ınuo). Entre los controles que satisfacen

el principio del máximo se puede establecer una jerarqúıa de regularidad en

términos de la regularidad del valor final del co-estado. Este art́ıculo considera

algunas cuestiones relacionadas con ésta jerarqúıa.

Key words and phrases: linear control systems in Banach spaces, time optimal problem,

norm optimal problem

Math. Subj. Class.: 93E20, 93E25.

1 Introduction.

We consider the control system

y
′
(t) = Ay(t) + u(t) , y(0) = ζ (1.1)

with controls u(·) ∈ L∞(0, T ; E), where A is the infinitesimal generator of a strongly contin-

uous semigroup S(t) in a Banach space E. In the norm optimal problem we drive the initial

point ζ to a point target,

y(T ) = ȳ

in a fixed time interval 0 ≤ t ≤ T minimizing ‖u(·)‖L∞(0,T ;E), while in the time optimal prob-

lem we drive to the target with a bound on the norm of the control (say ‖u(·)‖L∞(0,T ;E) ≤ 1)

in optimal time T. Solutions or trajectories

y(t) = S(t)ζ +

∫ t

0

S(t − σ)u(σ)dσ

of the initial value problem (1.1) are continuous and denoted by y(t) = y(t, ζ, u). For the

time optimal problem, controls in L
∞

(0, T ; E) with norm ‖u(·)‖L∞(0,T ;E) ≤ 1 are named

admissible.

Separate necessary and sufficient conditions for both norm and time optimality can

be given in terms of the maximum principle, which requires the construction of spaces of

multipliers (final values of the costates). We summarize [5] or [7, 2.3]. When the infinitesimal

generator A has a bounded inverse, we define the space E
∗
−1 as the completion of E

∗
in the

norm

‖y∗‖E∗
−1

= ‖(A−1)∗y∗‖E∗ .

Each S(t)
∗

can be extended to an (equally named) operator S(t)
∗

: E
∗
−1 → E

∗
−1, and the



CUBO
10, 1 (2008)

Regular and strongly regular ... 79

space Z
1
(T ) consists of all z ∈ E∗−1 such that S(t)

∗
z ∈ E∗ and1

‖z‖Z1(T ) =

∫ T

0

‖S(t)∗z‖dt < ∞ . (1.2)

Equipped with ‖ ·‖Z1(T ), Z
1
(T ) is a Banach space. All spaces Z

1
(T ) coincide and all norms

‖ · ‖Z1(T ) are equivalent for T > 0. Z
1
(T ) is an example of a multiplier space, defined

as an arbitrary linear space Z ⊇ E∗ to which S(t)∗ can be extended in such a way that

S(t)
∗Z ⊆ E∗. When A does not have a bounded inverse, the construction of the spaces

above is modified as follows. Since A is a semigroup generator, (λI − A)−1 exists for λ > ω

and E
∗
−1 is the completion of E

∗
in any of the equivalent norms

‖y∗‖E∗
−1

,λ = ‖((λI − A)
−1

)
∗
y
∗‖E∗ , (λ > ω) .

The definition of Z
1
(T ) (and of multiplier spaces) is the same. See [8, 2.3] for details.

A control u(·) ∈ L∞(0, T ; E) satisfies Pontryagin’s maximum principle if

〈S(T − t)∗z, ū(t)〉 = max
‖u‖≤ρ

〈S(T − t)∗z, u〉 a. e. in 0 ≤ t < T , (1.3)

〈· , ·〉 the duality of the space E and the dual E∗, with ρ = ‖u(·)‖L∞(0,T ;E) and z in some

multiplier space Z. We call z the multiplier and S(T − t)∗z the costate corresponding to the

control ū(t). We work under the standing assumption that (1.3) is nonempty; this means

S(T −t)∗z is not identically zero in the interval 0 ≤ t < T, although we don’t mind S(T −t)∗

vanishing in part of the interval (in which part (1.3) provides no information on ū(t)). The

assumption that (1.3) is nonempty implies in particular that z 6= 0. The maximum principle

takes a simple form when E is a Hilbert space; if fact, it reduces to

ū(t) = ρ
S(T − t)∗z

‖S(T − t)∗z‖
a. e. in 0 ≤ t < T , (1.4)

where S(T − t)∗z 6= 0 in 0 ≤ t < T.

A large part of the theory of optimal controls for the system (1.1) deals with the relation

between optimality and the maximum principle (1.3), a relation which is elementary in

finite dimension but becomes rather involved in an infinite dimensional space E. All one

has (at present) are separate necessary and sufficient conditions for optimality based on

the maximum principle (Theorem 1.1 below). We call an optimal control ū(t) regular if it

satisfies (1.3) with z ∈ Z1(T ).

1At this level of generality, the semigroup S(t)∗ may not be strongly continuous, or even strongly mea-
surable (consider, for instance, the translation semigroup S(t)y(x) = y(x − t) in E = L1(∞, ∞)). However,
S(t)∗ is always E-weakly continuous, which guarantees that ‖S(t)∗‖ is lower semicontinuous, hence measur-
able. This gives sense to the integral (1.2). Note also that in existing literature (for instance, [7]) Z1(T ) is
called Z(T ) (sometimes Zw(T ) for “weak” to emphasize that S(t)

∗ may not be strongly continuous). We
use the superindex 1 since spaces Zp(T ) (p 6= 1) will be introduced later.



80 H. O. Fattorini CUBO
10, 1 (2008)

Theorem 1.1. Assume ū(t) drives ζ ∈ E to ȳ = y(T, ζ, ū) time or norm optimally in the

interval 0 ≤ t ≤ T and that

ȳ − S(T )ζ ∈ D(A) . (1.5)

Then u(t) is regular. Conversely, let ū(t) be a regular control. Then ū(t) drives ζ ∈ E to

ȳ = y(T, ζ, ū) norm optimally in the interval 0 ≤ t ≤ T ; if ρ = 1 the drive is time optimal.

For the proof see [5, Theorem 5.1], [7, Theorem 2.5.1]; we note that in the sufficiency

half of Theorem 1.1 no conditions of the type of (1.5) are put on the initial value ζ or the

target ȳ.
2

A control u(·) is called strongly regular if it satisfies (1.3) with z ∈ E∗. The notion

of strongly regular control adds nothing to the two implications in Theorem 1.1, but it is

of interest in applications. In fact, if E
∗

is a Hilbert space then (1.4) shows that a strongly

regular control is (at least) continuous in 0 ≤ t ≤ T, whereas a merely regular control may

“oscillate” at the endpoint T of the control interval. This makes a difference, for instance,

in numerical approximations of the optimal control.
3

The question addressed in this paper is, characterize the control systems (1.1) for which

all (time, norm) optimal controls are strongly regular. Part of the answer to this question is

known; a sufficient condition for all optimal controls being strongly regular is

S(t)E = E (t > 0) . (1.6)

This condition is valid in any Banach space (Theorem 2.1 below). The main contribution of

this paper is the opposite implication, which we only prove under special assumptions on E

(Corollary 4.8). We also show (Remark 4.9) that if these special assumptions are dropped,

the implication ceases to be true.

2 Reversible semigroups.

Semigroups satisfying (1.6) we call reversible. In this section, no restrictions are placed upon

the Banach space E.

Theorem 2.1. Let S(t) be a reversible semigroup. Then all optimal controls for (1.1) are

strongly regular, that is, they satisfy (1.3) with z ∈ E∗.

2The statement on time optimality, however, needs additional assumptions on the initial condition ζ and
the target ȳ. These conditions are satisfied if either ζ = 0 or ȳ = 0 [6], [7, Theorem 2.5.7]. We point out that
the conditions are on the “size” of ζ ȳ, not on their smoothness like (1.5); for instance, for ζ = 0, ȳ may be
an arbitrary element of E. We also need to assume that S(t)∗z 6= 0 in the entire interval 0 ≤ t ≤ T.

3Piermarco Cannarsa has pointed out situations involving optimal controls for semilinear equations, where
strong regularity of (linear) optimal controls is actually needed; plain regularity is not enough.



CUBO
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Regular and strongly regular ... 81

The proof of Theorem 2.1 requires some auxiliary results.

Lemma 2.2. Let the E∗-valued, E-weakly continuous function f (t) satisfy

∫ T

0

〈f (t), u(t)〉dt ≤ C

(
∫ T

0

‖u(t)‖p
)1/p

dt (u(·) ∈ L∞(0, T ; E)) (2.1)

for some p, 1 ≤ p < ∞. Then, if p > 1 and 1/p + 1/q = 1 we have

(
∫ T

0

‖f (t)‖q
)1/q

≤ C , (2.2)

with equality in (2.2) if C is the smallest constant satisfying (2.1). If p = 1,

‖f (t)‖ ≤ C (0 ≤ t ≤ T ) , (2.3)

with equality in (2.3) if C is the smallest constant satisfying (2.1).

For p > 1 the proof of Lemma 2.2 is essentially similar to that of [7, Lemma 2.2.1 and

Lemma 2.2.10] thus we omit it. For p = ∞, assume (2.3) fails. Then there exists y ∈ E and

a nontrivial interval e such that 〈f (t), y〉 ≥ (C + ǫ)‖y‖. Setting

u(t) =

{

y t ∈ e
0 t /∈ e

we obtain

∫ T

0

〈f (t), u(t)〉dt =

∫

e

〈f (t), y〉dt ≥ (C + ǫ)|e|‖y‖ = (C + ǫ)

∫

e

‖u(t)‖dt ,

contradicting (2.1). This completes the proof.

Given T > 0 and 1 ≤ p ≤ ∞, the reachable space Rp(T ) (at time T ) of the system (1.1)

consists of all

y = y(t, 0, u) =

∫ T

0

S(T − σ)u(σ)dσ u(·) ∈ Lp(0, T ; E) ,

and is equipped with the norm

‖y‖Rp(T ) = inf

{

‖u‖Lp(0,T ;E);

∫ T

0

S(T − σ)u(σ)dσ = y

}

,

which makes R
p
(T ) a Banach space, isometrically isomorphic to the quotient space

L
p
(0, T ; E)/N p



82 H. O. Fattorini CUBO
10, 1 (2008)

where N p is the closed subspace of Lp(0, T ; E) of all u(·) with

∫ T

0

S(T − σ)u(σ)dσ = 0 .

We note in passing that all spaces R
∞

(T ) coincide (with equivalent norms) for T > 0. This

is proved in [2], [7, 2.1] and can be extended to p < ∞, but is not particularly relevant here.

If r > p Hölder’s inequality gives

∫ T

0

‖u(σ)‖pdσ =

∫ T

0

1 · (‖u(σ)‖r)p/rdt ≤ T (r−p)/r
(

∫ T

0

‖u(σ)‖rdr

)p/r

,

thus

‖y‖Rp(T ) ≤ T
(r−p)/pr‖y‖Rr(T ) ,

and it follows that R
r
(T ) →֒ Rp(T ). (the symbol →֒ means “is imbedded in”. Another

application of Hölder’s inequality produces

‖y‖ ≤ ‖S(T − ·)‖Lp/(p−1)(0,T )‖y‖Rp(T ) ≤ T
(p−1)/p‖S(T − σ)‖L∞(0,T )‖y‖Rp(T ) ,

so that R
p
(T ) →֒ E. Finally, if y ∈ D(A), integration by parts gives

y =

∫ T

0

S(T − σ)
y − σAy

T
dσ

thus, if we equip D(A) with its customary graph norm, we have D(A) →֒ R∞(T ). Putting

all the imbeddings together,

D(A) →֒ R∞(T ) →֒ Rr(T ) →֒ Rp(T ) →֒ E (p < r) . (2.4)

All imbeddings except the first are dense in the norm of the bigger space. For the imbeddings

R
∞

(T ) →֒ Rr(T ) →֒ Rp(T ) this follows from denseness of L∞(0, T ; E) (thus of Lr(0, T ; E))

in L
p
(0, T ; E), and for R

p
(T ) →֒ E from denseness of D(A) in E (for these results and more

details about function spaces of E-valued functions see [1, Chapter III] or [9, Chapter III].

Whether or not D(A) is dense in R
∞

(T ) in the norm of the latter space is one of the main

themes of [7, Chapters 2 and 3]. The following result follows immediately from denseness of

R
∞

(T ) in E and from the open mapping principle.

Lemma 2.3. We have R∞(T ) = E (with equivalent norms ) if and only if

‖y‖R∞(T ) ≤ C‖y‖ (y ∈ R
∞

(T )) .

Lemma 2.4 below is proved in [7, Theorem 2.2.3 and beginning of 2.2]:

Lemma 2.4. R∞(T ) = E (with equivalent norms) if and only if S(t) is reversible.



CUBO
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Regular and strongly regular ... 83

Theorem 2.5. Assume S(t) is reversible. Then

Z
1
(T ) = E

∗
(T > 0) (2.5)

with equivalent norms.

Proof. Assume S(t) is reversible. Then, by Lemma 2.4, R∞(T ) = E with equivalent norms.

It follows that R
∞

(T )
∗

= E
∗

with equivalent norms as well.

Let z ∈ Z1(T ). We can define a bounded linear functional ξz on R
∞

(T ) by

〈ξz , y〉 =

〈

ξz ,

∫ T

0

S(T − σ)u(σ)dσ

〉

=

∫ T

0

〈S(T − σ)∗z, u(σ)〉dσ . (2.6)

It can be easily seen that (2.6) pays heed to the equivalence relation in R
∞

(T ) = L
∞

(0, T ; E)

/N ∞ [5], [7, Lemma 2.3.5] and it follows from (1.2) that ξz is bounded in the norm of R
∞

(T ),

precisely

‖ξz‖ =

∫ T

0

‖S(T − σ)∗z‖dσ =

∫ T

0

‖S(σ)∗z‖dσ . (2.7)

The inequality ≤ in (2.7) is obvious; for the equality, see [5] or [7, 2.3]. By Lemma 2.4, ξz

is as well bounded in the norm of E. Accordingly,

∫ T

0

〈S(T − σ)∗z, u(σ)〉dσ ≤ C

∥

∥

∥

∥

∫ T

0

S(T − σ)u(σ)dσ

∥

∥

∥

∥

.

This implies

∫ T

0

〈S(T − σ)∗z, u(σ)〉dσ ≤ C

∫ T

0

‖u(σ)‖dσ (u(·) ∈ L∞(0, T ; E))

and Lemma 2.2 shows that

‖S(t)∗z‖ ≤ C (0 < t ≤ T ) . (2.8)

We show that (2.8) implies z ∈ E∗. Let {tn} be a positive decreasing sequence with tn → 0.

Then

|〈S(tn)
∗
z − S(tm)z, y〉| = |〈S(tm)

∗
z, S(tn − tm)y − y〉| ≤ C‖S(tn − tm)y − y‖

for n < m, so that {S(tn)z} is Cauchy in the E-weak topology of E
∗
. Accordingly, it

converges E-weakly to y
∗ ∈ E∗, and we have

〈S(t)∗z, y〉 = 〈S(t − tn)
∗
S(tn)

∗
z, y〉 = 〈S(tn)

∗
z, S(t − tn)y〉 → 〈S(t)

∗
y
∗
, y〉 ,

thus

S(t)
∗
z = S(t)

∗
y
∗

(t > 0) . (2.9)



84 H. O. Fattorini CUBO
10, 1 (2008)

The operator (A
−1

)
∗

: E
∗
−1 → E

∗
is 1-1 (and onto). Applying (A

−1
)
∗

to both sides of (2.9)

and applying the functionals on both sides to an element y ∈ E we obtain

〈(A−1)∗z, S(t)y〉 = 〈(A−1)∗y∗, S(t)y〉 (t > 0) .

Letting t → 0 we obtain (A−1)∗z = (A−1)∗y∗, thus z = y∗ as claimed. This ends the proof.

We note the following interesting byproduct of the proof of Theorem 2.5 (in particular,

of the lines following (2.8)). Define Z
∞

(T ) as the space of all z ∈ E∗−1 such that S(t)
∗
z is

bounded in 0 ≤ t ≤ T equipped with the norm

‖z‖Z∞(T ) = max
0≤t≤T

‖S(t)∗z‖ .

Then (with no conditions on the space E or the semigroup S(t)),

Lemma 2.6. We have

Z
∞

(T ) = E
∗

with equivalent norms.

3 Regular implies strongly regular, I.

The first question is this. Assume that (2.5) fails, that is, that the inclusion

Z
1
(T ) ⊃ E∗ (3.1)

is strict. Does this mean that there are regular controls which are not strongly regular? To

attempt to answer this question is complicated by lack of uniqueness of z in the maximum

principle (1.3) as in the following example, which is taken from [8].

Example 3.1. Consider the space E = ℓ0 consisting of all numerical sequences y =

{yn} = {y1, y2, . . . } such that limn→∞ yn = 0 , equipped with the norm ‖y‖0 = maxn≥1 |yn|.

The dual is E
∗

= ℓ
1
, the space of all numerical sequences y

∗
= {y∗n} such that ‖y

∗‖1 =
∑∞

n=1 |y
∗
n| < ∞, the duality of both spaces given by 〈y

∗
, y〉 =

∑∞
n=1 y

∗
nyn. The semigroup

and generator are

S(t){yn} = {e
−nt

yn} , A{yn} = −{nyn} , (3.2)

A with maximal domain (limn→∞ n|yn| = 0). The space E
∗
−1 consists of all sequences {yn}

with

‖(A−1)∗{y∗n}‖ =
∞
∑

n=1

|y∗n|

n
< ∞ . (3.3)



CUBO
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Regular and strongly regular ... 85

If {y∗n} ∈ E
∗
−1 we have

∫ T

0

‖S(t)∗z‖dt =

∥

∥

∥

∥

{
∫ T

0

e
−nt

y
∗
n

}
∥

∥

∥

∥

=

∞
∑

n=1

|y∗|
1 − e−nT

n
≤ ‖(A−1)∗{y∗n}‖ ,

thus E
∗
−1 = Z

1
(T ) and the inclusion (3.1) is strict. Due to existence requirements for

optimal controls for (1.1) with this choice of space and generator, controls are taken in

L
∞
w (0, T ; ℓ

∞
) rather than in L

∞
(0, T ; ℓ

0
), where ℓ

∞
is the space of all bounded numerical

sequences y = {yn} equipped with the norm ‖y‖∞ = maxn≥1 |yn|. This means the u in the

maximum principle (1.3) belongs to ℓ
∞

rather than in ℓ
0
. See [8] for additional details. We

also take the following result from [8].

Theorem 3.2. An admissible control ū(t) = {ūn(t)} satisfies the maximum principle (1.3)

with z = {zn} in any multiplier space if and only if ūm(t) = 1 (0 ≤ t ≤ T ) or um(t) = −1

(0 ≤ t ≤ T ) for at least one m ≥ 1.

Proof. We take ρ = 1. The maximum principle for this space and generator is

〈S(T − t)∗{zn}, {ūn(t)}〉 =

∞
∑

n=1

e
−n(T −t)

znūn(t)

= max
‖{un}‖ℓ∞ ≤1

〈S(T − t)∗{zn}, {un}〉

= max
|un|≤1

∞
∑

n=1

e
−n(T −t)

znun

=

∞
∑

n=1

e
−n(T −t)|zn| ,

so that we must have ūm(t) = sign zm whenever zm 6= 0. Conversely, if the assumptions of

Theorem 3.2 are satisfied for {ūn(t)} we obtain the maximum principle (1.3) with {zn} = δmn

(δmn the Kronecker delta). This ends the proof.

Strictness of the inclusion (3.1) and uniqueness of z in the maximum principle (1.3)4 do

imply the existence of optimal controls that are regular but nor strongly regular. We just

take z ∈ Z1(T ) \ E∗ and use the sufficiency statement in Theorem 1.1.

Uniqueness of z holds (for instance) in Hilbert spaces. If ū(t) satisfies the maximum

principle with two different z, ζ ∈ Z1(T ) then, assuming (as always) that the maximum

principle is nonempty and taking ρ = 1 for simplicity,

ū(t) =
S(T − t)∗z

‖S(T − t)∗z‖
=

S(T − t)∗ζ

‖S(T − t)∗ζ‖

4“Uniqueness of z” obviously means “uniqueness up to multiplication by a constant”; if ū(t) satisfies the
maximum principle (1.3) with two different z, ζ then ζ = αz, α 6= 0. The condition that α 6= 0 is required
by the assumed nontriviality of (1.3).



86 H. O. Fattorini CUBO
10, 1 (2008)

in some interval T −ǫ ≤ t ≤ T. Multiplying by the product of the denominators and applying

(A
−1

)
∗

to both sides we obtain

‖S(T − t)∗ζ‖S(T − t)∗(A−1)∗z = ‖S(T − t)∗z‖S(T − t)∗(A−1)∗ζ

where (A
−1

)
∗
z, (A

−1
)
∗
ζ ∈ E∗, thus, if {tn} is a decreasing sequence with tn → 0 we have

S(T − tn)
∗
(A

−1
)
∗
ζ = αnS(T − tn)

∗
(A

−1
)
∗
z . (3.4)

Now, if y ∈ E is such that 〈y, (A−1)∗z〉 6= 0 we apply the functionals on both sides of (3.4)

to y and obtain

〈(A−1)∗ζ , S(T − tn)y〉 = αn〈(A
−1

)
∗
z , S(T − tn)y〉 , (3.5)

which shows that αn → α, thus we can take limits in (3.5), now written for arbitrary y ∈ E,

obtaining

〈(A−1)∗ζ , y〉 = α〈(A−1)∗z , y〉 ,

thus

ζ = αz (3.6)

where α 6= 0 due to the requirement that (1.3) be nonempty (see comments after (1.3)).

4 Regular implies strongly regular, II.

We show in this section the converse of Theorem 2.5. The first result is on one of the

imbeddings in (2.4),

D(A) →֒ Rp(T ) . (4.1)

Lemma 4.1. If 1 ≤ p < ∞ the imbedding (4.1) is dense.

Proof. Let {λn} be an increasing sequence with λn → ∞. It follows from the dominated

convergence theorem that if u(·) ∈ Lp(0, T ; E) then λnR(λn; A)u(·) → u(·) in the norm of

L
p
(0, T ; E) thus

λnR(λn; A)

∫ T

0

S(T − σ)u(σ)dσ →

∫ T

0

S(T − σ)u(σ)dσ

in the norm of R
p
(T ). This ends the proof.

Given 1 ≤ q < ∞, the space Zq(T ) ⊆ Z1(T ) consists of all z ∈ Z1(T ) such that

∫ T

0

‖S(t)∗z‖qdt < ∞ (4.2)



CUBO
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Regular and strongly regular ... 87

equipped with the norm ‖S(·)∗z‖Lq(0,T ). For q = ∞, the space was defined at the end of 2

(and shown to coincide with E
∗
).

Theorem 4.2. The dual space Rp(T )∗, 1 ≤ p < ∞ is algebraically and metrically isomorphic

to Zq(T ), 1/q + 1/p = 1.

The proof is based on the calculation of the dual for p = ∞, which we outline below.

Bounded functionals ξz on R
∞

(T ) of the form (2.6) are called regular, and R(T ) ⊆ R∞(T )∗

is the subspace of all regular functionals. Bounded functionals ξs on R
∞

(T ) that vanish in

D(A) ⊆ R∞(T ) are called singular; the space of all such functionals is S(T ) ⊆ R∞(T )∗.

Application of the Hahn - Banach theorem gives

S(T ) = {0} ⇐⇒ D(A) is dense in R∞(T ) (in the norm of R∞(T )) .

Theorem 4.3. [7, Theorem 2.4.1]. We have5

R
∞

(T )
∗

= R(T ) ⊕ S(T ) (Banach direct sum ) .

Proof of Theorem 4.2. Let ξ be a bounded linear functional in Rp(T ). Then (due to the

second imbedding (2.4)) ξ is a bounded linear functional in R
∞

(T ) as well, hence, due to

Theorem 4.3. we have ξ = ξz + ξs with ξz regular and ξs singular. If u(·) ∈ L
∞

(0, T ; E) and

∫ T

0

S(T − σ)u(σ)dσ ∈ D(A) (4.3)

we have
〈

ξ ,

∫ T

0

S(T − σ)u(σ)dσ

〉

=

〈

ξz ,

∫ T

0

S(T − σ)u(σ)dσ

〉

=

∫ T

0

〈S(T − σ)∗z, u(σ)〉dσ. (4.4)

Now, D(A) is dense in R
p
(T ) and R

∞
(T ) is dense in R

p
(T ), thus (4.4) can be extended

to all elements (4.3) of R
p
(T ) whether or not they belong to D(A). Since ξ is bounded in

R
p
(T ) we have

∫ T

0

〈S(T − σ)∗z, u(σ)〉dσ ≤ ‖ξ‖Rp(T )∗

∥

∥

∥

∥

∫ T

0

S(T − σ)u(σ)dσ

∥

∥

∥

∥

Rp(T )

. (4.5)

For the case p > 1 this implies

∫ T

0

〈S(T − σ)∗z, u(σ)〉dσ ≤ ‖ξ‖Rp(T )∗

(
∫ T

0

‖u(σ)‖pdσ

)1/p

(u(·) ∈ Lp(0, T ; E))

5“Banach direct sum” means algebraic direct sum plus bounded projections from the space into each of
the two subspaces.



88 H. O. Fattorini CUBO
10, 1 (2008)

and it follows from Lemma 2.2 that

S(·)∗z ∈ Zq(T ) , ‖z‖Zq(T ) =

(
∫ T

0

‖S(T − σ)‖qdσ

)1/q

= ‖ξ‖Rp(T )∗ ,

equality coming from the fact that ‖ξ‖Rp(T )∗ is the least constant that does the job in (4.5).

That an element of Z
q
(T ) produces a functional in R

p
(T ) through (2.6) is a consequence of

Hölder’s inequality. In the case p = 1, (4.5) implies

S(·)∗z ∈ Z∞(T ) , ‖z‖Z∞(T ) = sup
0≤t≤T

‖S(t)∗z‖ = ‖ξ‖Rp(T )∗ .

This ends the proof of Theorem 4.2.

We note that a proof of Theorem 4.2 which is independent of the (rather involved)

identification of R
∞

(T )
∗

can be given using the equality

L
p
w(0, T ; E)

∗
= L

q
(0, T ; E

∗
) (4.6)

for 1/p + 1/q = 1 ([10], [4]) where the subindex w means “E-weakly measurable”. However,

the description of the spaces L
p
w(0, T ; E)

∗
(and the definition of the norms) is rather involved

as well.

Corollary 4.4. We have

R
1
(T ) = E (4.7)

with equivalent norms.

Proof. We have R1(T ) →֒ E and R1(T ) is dense in E. On the other hand, by Lemma 2.6 we

have R
1
(T )

∗
= Z

∞
(T ) = E

∗
with equivalent norms. This is easily seen to imply equivalence

of the norms of R
1
(T ) and E. In fact, it suffices to note that, as a consequence of the Hahn

- Banach theorem we have

‖y‖ = sup
‖y∗‖≤1

〈y∗, y〉 . (4.8)

for any Banach space E and its dual E
∗
. Equivalence of the norms and the fact that R

1
(T )

is dense in E implies (4.7).

So far, all results in this section have been proved for an arbitrary Banach space E

and strongly continuous semigroup S(t). Our objective below is the proof of the converse of

Theorem 2.5, thus we assume Z
1
(T ) = E

∗
. By intercession of Lemma 2.6, this is the same

as Z
1
(T ) = Z

∞
(T ), which in turn is equivalent to

∫ T

0

‖S(t)∗z‖dt < ∞ =⇒ ‖S(t)∗z‖ is bounded in 0 ≤ t ≤ T . (4.9)



CUBO
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Regular and strongly regular ... 89

It follows that all Z
p
(T ) coincide, 1 ≤ p ≤ ∞.

Lemma 4.5. Under (4.9) all Zp(T ) norms are equivalent to the norm in Z1(T ), with

constants that do not depend on p.

Proof. Independently of (4.9) we have

‖z‖Zp(T ) =

(
∫ T

0

‖S(t)∗z‖pdt

)1/p

≤ T 1/p‖S(·)z‖L∞(0,T ) = T
1/p‖z‖Z∞(T ) . (4.10)

If (4.9) holds then, since Z
1
(T ) = Z

∞
(T ) is a Banach space under the two norms, by the

open mapping principle these norms have to be equivalent: hence

‖z‖Z∞(T ) ≤ C‖z‖Z1(T )

which, combined with (4.10) gives

‖z‖Zp(T ) ≤ CT
1/p‖z‖1(T ) .

On the other hand, and independently of (4.9),

‖z‖Z1(T ) =
∫ T

0
‖S(t)∗z‖dt

≤ T (p−1)/p
(

∫ T

0
‖S(t)∗z‖pdt

)1/p

= CT
(p−1)/p‖z‖Zp(T ) .

Theorem 4.6. Assume (4.9) holds. Then

R
p
(T ) = E (1 ≤ p < ∞) ,

all norms equivalent to the norm of E with constants that do not depend on p.

Proof. Since Rp(T )∗ = Zp/(p−1)(T ) algebraically and metrically, Lemma 4.5 says that

R
p
(T )

∗
= Z

∞
(T ) = E

∗
, all norms equivalent to the norm of E

∗
with constants than do not

depend of p. The corresponding statement for R
p
(T ), E is a consequence of denseness of

R
p
(T ) in E and (4.8). This completes the proof.

Theorem 4.7. Let E be reflexive and separable. If

Z
1
(T ) = E

∗
.

then S(t) is reversible, that is, (1.6) holds.

Proof. We shall show that S(t) is reversible by proving that R∞(t) = E and using Lemma

2.4. If E is reflexive and separable then X = E
∗

is reflexive and separable as well and

L
q
(0, T ; E) = L

p
(0, T ; X)

∗



90 H. O. Fattorini CUBO
10, 1 (2008)

(1/p + 1/q = 1) where, due to the assumptions (and unlike in the generality of (4.6)) the

space on the left is described exactly in the same form as the space on the right. Since X

is separable, the space L
p
(0, T ; X) is separable as well for 1 ≤ p < ∞. This implies that the

L
p
(0, T ; X)-weak topology in any bounded subset of L

q
(0, T ; E) is defined by a metric ([1,

Theorem 3, p. 434]), which justifies the “passing to a subsequence” arguments below.

Under the assumptions, given y ∈ E we may avail ourselves of Theorem 4.6, and

construct a sequence {un(·)}, un(·) ∈ L
n
(0, T ; E) such that

∫ T

0

S(T − σ)un(σ)dσ = y , (‖un(·)‖Ln(0,T ;E) ≤ C‖y‖ , n = 2, 3, 4, . . . ) (4.11)

where C does not depend on n. Since

L
2
(0, T ; E) = L

2
(0, T ; X)

∗

we can select a subsequence of {un(·)} L
2
(0, T ; X)-weakly convergent in L

2
(0, T ; E); since

L
3
(0, T ; X) = L

3/2
(0, T ; X)

∗

we can select a subsequence of the previous subsequence that is L
3/2

(0, T, X)-weakly con-

vergent in L
3
(0, T, X) (thus L

2
(0, T ; X)-weakly convergent in L

2
(0, T ; E)); since

L
4
(0, T ; X) = L

4/3
(0, T ; X)

∗

we can select a subsequence of the previous subsequence that is L
4/3

(0, T ; X)-weakly con-

vergent in L
4
(0, T ; X) (thus L

3/2
(0, T ; X)-weakly convergent in L

3
(0, T ; E)), L

2
(0, T ; X)-

weakly convergent in L
2
(0, T ; E)), . . . and so on. Picking the diagonal sequence, we finally

obtain a sequence {un(·)} such that, eventually, it belongs to every L
m

(0, T ; E) and such

that

un(·) → ū(·) ∈ L
m

(0, T ; E) L
m/(m−1)

(0, T ; X)-weakly in L
m

(0, T ; E) .

(the fact that the limit ū(·) is the same in all spaces is elementary). Now, it follows from

the norm estimation in (4.11) that

‖ū(·)‖Lm(0,T ;E) ≤ C (2, 3, . . . )

with C independent of m, hence ū(·) ∈ L∞(0, T ; E). The first relation (4.11) implies

〈

y
∗
,
∫ T

0
S(T − σ)un(σ)dσ

〉

=
∫ T

0
〈S(T − σ)∗y∗, un(σ)〉dσ

→
∫ T

0
〈S(T − σ)∗y∗, ū(σ)〉dσ =

〈

y
∗
,
∫ T

0
S(T − σ)ū(σ)dσ

〉

(y ∈ E∗)



CUBO
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Regular and strongly regular ... 91

so that

y =

∫ T

0

S(T − σ)ū(σ)dσ

and the proof of Theorem 4.7 is finished.

Corollary 4.8. Let E be reflexive and separable. Assume all regular controls for (1.1) are

strongly regular and that z in the maximum principle (1.3) depends uniquely on ū(t) (as in

the comments preceding (3.6)). Then S(t) is reversible.

Proof. If the inclusion Z1(T ) ⊃ E∗ is strict, taking z ∈ Z1(T ) \ E∗ and using the sufficiency

statement in Theorem 1.1 we can construct a regular ū(t) which is not strongly regular.

Accordingly, we must have Z
1
(T ) = E

∗
and Theorem 4.7 applies.

Remark 4.9. Theorem 3.2. shows that the conclusion of Corollary 4.8 collapses if we drop

the assumptions that E be reflexive and that z be unique. In fact, if E = ℓ
1

and A is given

by (3.2) every control ū(t) that satisfies the maximum principle (1.3) with any multiplier

z = {zn} 6= 0 satisfies (1.3) as well with {zn} = {δmn} ∈ ℓ
1

= E
∗
, thus it qualifies as

strongly regular. However, the semigroup (3.1) is far from reversible.

Received: January 2007. Revised: September 2007.

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