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228 R. Farwig, H. Kozono and H. Sohr
Since g ∈ W
1/2,2
(∂Ω) ⊂ W
−1/2,2
(∂Ω), the very weak solutions u
j
, u
0
, u
A
constructed so far are also weak solutions, and, in particular, u
j
∈ W
1,2
(Ω
j
)and
u
j
W
1,2
(Ω
j
)
≤ c g
W
1/2,2
(∂Ω
j
)
; for this regularity argument see [4, Remarks 2(1)].
Next we define u on R
3
by u = u
j
in Ω
j
,1≤ j ≤ L, u = u
0
in Ω, u = u
A
in
A and u =0inR
3
\A. Obviously, u ∈ W
1,2
(R
3
), div u =0inR
3
,andu satisfies
the estimates
∇u
2
≤ c g
W
1/2,2
(∂Ω)
,
u
2
≤ c g
W
−1/2,2
(∂Ω)
.
(5.14)
Using L
2
-Fourier analysis we find a vector potential ψ ∈ W
2,2
(R
3
) such that
u =rotψ, ∇ψ, ∇
2
ψ
2
≤ c u
W
1,2
(R
3
)
. (5.15)
Indeed, since div u = 0, the equation rot ψ = u has a unique solenoidal solution ψ
defined in Fourier space via |ξ|
2
ˆ
ψ = iξ× ˆu. Obviously, ψ satisfies (5.15). To prove
that ψ ∈ L
2
(R
3
) we introduce the space
ˆ
W
1,q
(R
n
), 1 <q<∞, as the closure
of C
∞
0
(R
n
) with respect to the norm ∇(·)
q
, and its dual space
ˆ
W
−1,q
(R
n
) ≡
ˆ
W
1,q
(R
n
)
∗
.Evidently,ψ ∈ L
2
(R
n
)andψ
2
≤ c u
ˆ
W
−1,2
provided that u ∈
ˆ
W
−1,2
(R
3
). To see the latter assertion we exploit the fact that u has compact
support in B
R
and refer to the following Lemma 5.5 the proof of which will be
postponed to the end of the paper.
Lemma 5.5. Let D be a bounded domain in R
n
and let 1 <q<∞.Ifu ∈ L
q
(R
n
)
with supp u ⊂ D,thenu ∈
ˆ
W
−1,q
(R
n
) and satisfies the estimate
u
ˆ
W
−1,q
(R
n
)
≤ cu
L
q
(D),
(5.16)
where c = c(D, n, q) is a constant independent of u.
Summarizing (5.15), the estimate ψ
2
≤ c u
ˆ
W
−1,2
and (5.16) with q =2
we get that u =rotψ,
∇ψ
2
+ ∇
2
ψ
2
≤ c g
W
1/2,2
(∂Ω)
ψ
2
≤ c g
W
−1/2,2
(∂Ω)
(5.17)
with a constant c = c(Ω) > 0. Moreover, the map g → ψ is linear.
In the next step we define the vector field E = E
ε
by E =rot(θ
ε
ψ)where
θ
ε
∈ W
1,∞
is a cut-off function with support in an ε-neighborhood of ∂Ω. Following
[11, pp. 288–290] or [22, Ch. II, §1, Lemma 1.9, Lemma 1.10] for pointwise estimates
of θ
ε
and E we get for all w
1
,w
2
∈ W
1,2
0,σ
(Ω) the estimates
|w
1
E,∇w
2
Ω
|≤w
1
E
2
∇w
2
2
Global Leray-Hopf Weak Solutions 229
and with χ
ε
= χ
supp θ
ε
and d(x)=dist(x, ∂Ω)
w
1
E
2
2
≤ c
Ω
|w
1
|
2
ε
d(x)
|ψ(x)|+ |∇ψ(x)|χ
ε
2
dx
≤ cε
2
Ω
"
"
"
w
1
d(·)
"
"
"
2
dx
ψ
2
∞
+ c w
1
2
6
∇ψ
2
6
χ
ε
2
6
≤ c ∇w
1
2
2
ψ
2
H
2
(R
3
)
(ε
2
+ χ
ε
2
6
);
here ε>0 may be chosen arbitrarily small and is related to the size of supp θ
ε
which shrinks when ε → 0. Hence (1.15)
2
can be fulfilled for a.a. fixed t>0inthe
sense that
|w
1
E,∇w
2
Ω
|≤
1
4
∇w
1
2
∇w
2
2
,w
1
,w
2
∈ W
1,2
0,σ
(Ω) . (5.18)
Furthermore, since E,ϕ
Ω
= θψ, rot ϕ)
Ω
for all ϕ ∈ W
1,2
0
(Ω) we get by (5.17)
2
the estimate
E
W
−1,2
(Ω)
≤ c g
W
−
1
2
,2
(∂Ω)
. (5.19)
In the final step we define E(·) as in (5.11) satisfying (1.15)
2
.Giveng ∈
L
∞
(0, ∞; W
1
2
,2
(∂Ω)) fulfilling (5.9) for a.a. t>0 we find by the previous argu-
ments for a.a. t>0avectorfieldE(t)=rot(θψ(t)) satisfying (5.18) and, due to
(5.17),
E(t)
W
1,2
(Ω)
≤ c g(t)
W
1
2
,2
(∂Ω)
for a.a. t ∈ (0,T).
Hence E ∈ L
∞
(0, ∞; W
1,2
(Ω)), and (5.12)
1
, (1.15)
2
are easy consequences. Since
the map g → E is linear and g
t
∈ L
∞
(0, ∞; W
−1/2,2
(∂Ω)), the previous ar-
guments, the method of difference quotients and (5.19) also imply that E
t
∈
L
∞
(0, ∞; W
−1/2,2
(∂Ω)) and that (5.12)
2
holds.
Now Proposition 5.4 is completely proved.
To apply Proposition 5.4 to the Navier-Stokes system (1.1) via Theorem 1.3
we have to consider the Stokes system (1.4) for E more closely. In this setting
where E has already been defined by the boundary data g we have to determine
f
0
and E
0
in (1.4). Let h ≡ 0 so that by the construction in the proof of Proposition
5.4
f
0
= E
t
− ΔE, E =rot(θψ),
whichmaybewrittenalsointheformf
0
=divF
0
. By (5.12) we easily get that
F
0
∈ L
∞
(0, ∞; L
2
(Ω)) and that
F
0
2,∞;∞
≤ c
E
t
L
∞
(0,∞;W
−1,2
(Ω))
+ E
L
∞
(0,∞;W
1,2
(Ω))
≤ c
g
t
L
∞
(0,∞;W
−
1
2
,2
(∂Ω))
+ g
L
∞
(0,∞;W
1
2
,2
(∂Ω))
.
(5.20)
Moreover, since E ∈ L
2
loc
([0, ∞); W
1,2
(Ω)) and E
t
∈ L
2
loc
([0, ∞); W
−1,2
(Ω)), a
classical interpolation result states that E ∈ C
0
([0, ∞); L
2
(Ω)), the initial value
230 R. Farwig, H. Kozono and H. Sohr
E
0
= E(0) ∈ L
2
(Ω) is well defined and there exists a constant c>0 such that
E
0
2
≤ c
E
L
∞
(0,∞;W
1,2
(Ω))
+ E
t
L
∞
(0,∞;W
−1,2
(Ω))
≤ c
g
L
∞
(0,∞;W
1
2
,2
(∂Ω))
+ g
t
L
∞
(0,∞;W
−
1
2
,2
(∂Ω))
.
(5.21)
Furthermore, div E
0
=0andE
0
|
∂Ω
= g(0) where g(0) is well defined in L
2
(∂Ω).
Now we are ready to state our final result.
Corollary 5.6. Let Ω ⊂ R
3
be a bounded domain with boundary ∂Ω ∈ C
1,1
and
boundary components Γ
j
, 0 ≤ j ≤ L. Assume that f =divF , F ∈ L
∞
(0, ∞; L
2
(Ω)),
u
0
∈ L
2
(Ω) and that g satisfies (5.10) and the restricted flux condition (5.9).
Then the Navier-Stokes system (1.1) has a global weak solution u = v + E
where E satisfies (5.11), (5.12) and
v(t)
2
2
≤ e
−μ
0
t
v
0
2
2
+2
t
0
e
μ
0
τ
(F
1
2
2
+ E
4
4
) dτ
;
here F
1
= F + F
0
satisfies (5.20), v
0
= u
0
−E
0
where E
0
is subject to (5.21),and
with μ
0
from (4.8).
In particular, v and also u are bounded globally in time in L
2
(Ω).There
exists a bound κ
∗
independent of u
0
such that for all t>T
0
= T
0
(u
0
,g,g
t
) we
have v(t)
2
≤ κ
∗
.
Proof of Lemma 5.5. Let us first consider the case n
=
n
n−1
<q<∞.Forsuchq,
it holds 1 <q
=
q
q−1
<n, and the Sobolev inequality states that ϕ
L
nq
n−q
(R
n
)
≤
C∇ϕ
L
q
(R
n
)
for all ϕ ∈ C
∞
0
(R
n
). Hence taking η ∈ C
∞
0
(R
n
) satisfying η(x) ≡ 1
for x ∈ D,wehave
|u, ϕ| = |u, ηϕ| ≤ u
L
q
(D)
ηϕ
L
q
(D)
≤ cu
L
q
(D)
ϕ
L
nq
n−q
(R
n
)
≤ cu
L
q
(D)
∇ϕ
L
q
(R
n
)
for all ϕ ∈ C
∞
0
(R
n
). Hence u ∈
ˆ
W
−1,q
(R
n
), and u satisfies the estimate (5.16).
We next consider the case 1 <q≤ n
, i.e., n ≤ q
< ∞.Forsuchq,wesee
that the subspace S
D
≡{ϕ ∈ C
∞
0
(R
n
);
D
ϕ(x)dx =0} is dense in
ˆ
W
1,q
(R
n
). For
a moment, let us assume this density. Then it follows from the Poincar´e-Friedrichs
inequality ϕ
L
q
(D)
≤ c∇ϕ
L
q
(D)
for ϕ ∈ S
D
that
|u, ϕ| ≤ u
L
q
(D)
ηϕ
L
q
(D)
≤ cu
L
q
(D)
ϕ
L
q
(D)
≤ cu
L
q
(D)
∇ϕ
L
q
(R
n
)
for all ϕ ∈ S
D
.SinceS
D
is dense in
ˆ
W
1,q
(R
n
), the above estimate implies that
u ∈
ˆ
W
−1,q
(R
n
) with (5.16).
It remains to prove the density of the space S
D
in
ˆ
W
1,q
(R
n
). Take a function
ζ ∈ C
∞
0
(R
n
) such that ζ(x)=1for|x|≤1andζ(x)=0for|x| > 2, and define
ζ
k
(x) ≡ ζ(x/k)fork ∈ N. For every ϕ ∈ C
∞
0
(R
n
), we choose a sequence {ϕ
k
}
∞
k=1
by
ϕ
k
(x)=ϕ(x) −
1
|D|
D
ϕ(y)dy
ζ
k
(x),x∈ R
n
,k∈ N.
Global Leray-Hopf Weak Solutions 231
For sufficiently large k we have ζ
k
(x) ≡ 1 for all x ∈ D, and hence we may assume
that {ϕ
k
}
∞
k=1
⊂ S
D
.For1<q<n
, i.e., n<q
< ∞, it holds that
∇ϕ
k
−∇ϕ
L
q
(R
n
)
≤ c∇ζ
k
L
q
(R
n
)
≤ ck
−1+
n
q
→ 0ask →∞,
from which we conclude that S
D
is dense in
ˆ
W
1,q
(R
n
)provided1<q<n
.
For q = n
, i.e., q
= n,wehavesup
k∈N
∇ϕ −∇ϕ
k
L
n
(R
n
)
< ∞,andwe
easily conclude that
∇ϕ
k
∇ϕ weakly in L
n
(R
n
)ask →∞.
Applying Mazur’s lemma to the sequence {ϕ
k
}
∞
k=1
, we may select a sequence
{¯ϕ
k
}
∞
k=1
of convex combinations of {ϕ
k
}
∞
k=1
so that
∇¯ϕ
k
→∇ϕ strongly in L
n
(R
n
)ask →∞,
from which we also deduce the density of S
D
in
ˆ
W
1,n
(R
n
). This proves the lemma.
References
[1] H. Amann: On the strong solvability of the Navier-Stokes equations. J. Math. Fluid
Mech. 2 (2000), 16–98
[2] H. Amann: Navier-Stokes equations with nonhomogeneous Dirichlet data, J. Non-
linear Math. Phys. 10 Suppl. 1 (2003), 1–11
[3] R. Farwig, G.P. Galdi and H. Sohr: Very weak solutions of stationary and insta-
tionary Navier–Stokes equations with nonhomogeneous data. Progress in Nonlinear
Differential Equations and Their Applications. Vol. 64, 113–136, Birkh¨auser Verlag
Basel (2005)
[4] R. Farwig, G.P. Galdi and H. Sohr: A new class of weak solutions of the Navier-Stokes
equations with nonhomogeneous data. J. Math. Fluid Mech. 8 (2006), 423–444
[5] R. Farwig, H. Kozono and H. Sohr: Very weak, weak and strong solutions to the
instationary Navier-Stokes system. Topics on partial differential equations, ed. by P.
Kaplick´y,
ˇ
S. Neˇcasov´a. J. Neˇcas Center Math. Model., Lecture Notes, Vol. 2, 1–54,
Praha 2007
[6] R. Farwig, H. Kozono, H. Sohr: Global weak solutions of the Navier-Stokes equations
with nonhomogeneous boundary data and divergence. Rend. Sem. Mat. Univ. Padova
(to appear)
[7] A.V. Fursikov, M.D. Gunzburger, L.S. Hou: Inhomogeneous boundary value prob-
lems for the three-dimensional evolutionary Navier-Stokes equations. J. Math. Fluid
Mech. 4 (2002), 45–75
[8] G.P. Galdi: An Introduction to the Mathematical Theory of the Navier-Stokes Equa-
tions. Vol. II, Springer Verlag, New York, 1994
[9] Y. Giga: Analyticity of the semigroup generated by the Stokes operator in L
r
-spaces.
Math. Z. 178 (1981), 297–329
[10] Y. Giga and H. Sohr: Abstract L
p
estimates for the Cauchy problem with applica-
tions to the Navier-Stokes equations in exterior domains. J. Funct. Anal. 102 (1991),
72–94
232 R. Farwig, H. Kozono and H. Sohr
[11] V. Girault, P.-A. Raviart: Finite Element Methods for Navier-Stokes equations: The-
ory and Algorithms. Springer Series in Computational Mathematics 5 (1986)
[12] G. Grubb: Nonhomogeneous Dirichlet Navier-Stokes problems in low regularity L
p
Sobolev spaces. J. Math. Fluid Mech. 3 (2001), 57–81
[13] J.G. Heywood: The Navier-Stokes Equations: On the existence, regularity and decay
of solutions. Indiana Univ. Math. J. 29 (1980), 639–681
[14] E. Hopf: Ein allgemeiner Endlichkeitssatz der Hydrodynamik. Math. Ann. 117, 764–
775 (1940–1941)
[15] O.A. Ladyzhenskaya: The Mathematical Theory of Viscous Incompressible Flow.
Gordon and Breach, New York, 1969
[16] J. Leray: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math.
63 (1934), 193–248
[17] J.P. Raymond: Stokes and Navier-Stokes equations with nonhomogeneous boundary
conditions. Ann. Inst. Henri Poincar´e, Anal. Non Lin´eaire 24, 921–951 (2007)
[18] F. Riechwald, K. Schumacher: A large class of solutions for the instationary Navier-
Stokes system. J. Evol. Equ. 9 (2009), 593–611
[19] K. Schumacher: The Navier-Stokes equations with low regularity data in weighted
function spaces, FB Mathematik, TU Darmstadt, 2007, PhD Thesis Online:
http://elib.tu-darmstadt.de/diss/000815/
[20] H. Sohr: The Navier-Stokes Equations. Birkh¨auser Verlag, Basel, 2001
[21] V.A. Solonnikov: Estimates for solutions of nonstationary Navier-Stokes equations.
J. Soviet Math. 8 (1977), 467–529
[22] R. Temam: Navier-Stokes Equations. North-Holland, Amsterdam, 1977
R. Farwig
Department of Mathematics and
Center of Smart Interfaces (CSI)
Darmstadt University of Technology
D-64289 Darmstadt, Germany
e-mail: farwig@mathematik.tu-darmstadt.de
H. Kozono
Mathematical Institute
Tˆohoku University
Sendai, 980-8578 Japan
e-mail: kozono@math.tohoku.ac.jp
H. Sohr
Faculty of Electrical Engineering
Informatics and Mathematics
University of Paderborn
D-33098 Paderborn, Germany
e-mail: hsohr@math.uni-paderborn.de
Progress in Nonlinear Differential Equations
and Their Applications, Vol. 80, 233–249
c
2011 Springer Basel AG
Time and Norm Optimality
of Weakly Singular Controls
H.O. Fattorini
Abstract. Let ¯u(t) be a control that satisfies the infinite-dimensional version
of Pontryagin’s maximum principle for a linear control system, and let z(t)
be the costate associated with ¯u(t). It is known that integrability of z(t)
in the control interval [0,T] guarantees that ¯u(t) is time and norm optimal.
However, there are examples where optimality holds (or does not hold) when
z(t) is not integrable. This paper presents examples of both cases for a
particular semigroup (the right translation semigroup in L
2
(0, ∞)).
Mathematics Subject Classification (2000). 93E20, 93E25.
Keywords. Linear control systems in Banach spaces, norm optimal problem,
time optimal problem, weakly singular controls, costate.
1. Introduction
We consider two optimal control problems for the system
y
(t)=Ay(t)+u(t) ,y(0) = ζ (1.1)
with controls u(·) ∈ L
∞
(0,T; E), where A generates a strongly continuous semi-
group S(t)inaBanachspaceE. The first is the norm optimal problem, where we
drive the initial point ζ toapointtarget,
y(T )=¯y (1.2)
in a fixed time interval 0 ≤ t ≤ T minimizing u(·)
L
∞
(0,T ;E)
. The second is the
time optimal problem, where 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. The solution or trajectory
of (1.1) is the continuous function
y(t)=y(t, ζ, u)=S(t)ζ +
t
0
S(t −σ)u(σ)dσ . (1.3)
234 H.O. Fattorini
For the time optimal problem, controls in L
∞
(0,T; E)withu(·)
L
∞
(0,T ;E)
≤ 1
are named admissible.
Necessary and sufficient conditions for norm and time optimality can be given
in terms of the maximum principle (1.5) below which requires the construction of
spaces of multipliers (final values of costates). We summarize this construction
from [4] or [5], Section 2.3. When 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)
∗
canbeextendedtoanoperatorS(t)
∗
: E
∗
−1
→ E
∗
−1
, and Z
w
(T ) ⊆ E
∗
−1
consists of all z ∈ E
∗
−1
such that S(t)
∗
z ∈ E
∗
and
1
z
Z
w
(T )
=
T
0
S(t)
∗
z
E
∗
dt < ∞. (1.4)
The space Z
w
(T ) equipped with ·
Z
w
(T )
is a Banach space. All spaces Z
w
(T )
coincide (that is, Z
ω
(T )=Z
ω
(T
) for any T,T
> 0andthenorms·
Z
w
(T )
,
·
Z
w
(T
)
are equivalent). Z
w
(T )isanexampleofamultiplier space, an arbitrary
linear space Z⊇E
∗
to which S(t)
∗
can be extended in such a way that S(t)
∗
Z⊆
E
∗
for t>0. When A does not have a bounded inverse, the construction of the
spaces is modified as follows. Since A is a semigroup generator, (λI −A)
−1
exists
for λ>ωand 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
w
(T ) (and of multiplier spaces) is the same. See [5], Section 2.3
for proofs of these results and additional details.
A control ¯u(·) ∈ L
∞
(0,T; E)satisfiesPontryagin’s maximum principle if
S(T −t)
∗
z, ¯u(t) =max
u≤ρ
S(T −t)
∗
z,u a.e. in 0 ≤ t<T, (1.5)
where ·, · is 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 z(t)=S(T − t)
∗
z
the costate corresponding to (or associated with)thecontrol¯u(t). We assume that
(1.5) is nontrivial; this means S(T − t)
∗
z is not identically zero in the interval
0 ≤ t<T,although it may be zero in part of the interval (in which part (1.5) says
nothing about ¯u(t)). That (1.5) is nontrivial implies that z =0. The maximum
principle is especially simple when E is a Hilbert space; it reduces to
¯u(t)=ρ
S(T − t)
∗
z
S(T − t)
∗
z
(T − δ<σ≤ T ) , (1.6)
where 0 ≤ t<δis the maximal interval where S(t)
∗
z =0;ifδ ≥ T the interval is
0 <σ≤ T.
1
Without further assumptions, the semigroup S(t)
∗
may not be strongly continuous, or even
strongly measurable (consider, for instance, the translation semigroup S(t)y(x)=y(x − t)in
E = L
1
(∞, ∞)). However, S(t)
∗
is always E-weakly continuous, which guarantees that S(t)
∗
is lower semicontinuous, hence measurable. This justifies the integral (1.4).
Time and Norm Optimality of Weakly Singular Controls 235
A large part of the theory of optimal controls for the system (1.1) deals with
the relation between optimality and the maximum principle (1.5). 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 ¯u(t)
regular if it satisfies (1.5) with z ∈ Z
w
(T ).
Theorem 1.1. Assume ¯u(t) drives ζ ∈ E to ¯y = y(T,ζ, ¯u) time or norm optimally
in the interval 0 ≤ t ≤ T and that
¯y − S(T )ζ ∈ D(A) . (1.7)
Then ¯u(t) is regular. Conversely, let ¯u(t) be a regular control. Then ¯u(t) drives
ζ ∈ E to ¯y = y(T,ζ,¯u) norm optimally in the interval 0 ≤ t ≤ T ; if ρ =1the
drive is time optimal.
For the proof see [4], Theorem 5.1, [5], Theorem 2.5.1; we note that in the
sufficiency half of Theorem 1.1 no conditions of the type of (1.7) are put on the
initial value ζ or the target ¯y.
2
Following the terminology in [5] we call a control weakly singular if it satisfies
the maximum principle (1.5) but the costate does not satisfy the integrability con-
dition (1.4) (that is, z/∈ Z
w
(T )). The following question arises: is a weakly singular
control (norm, time) optimal? The answer to this question is “not necessarily” and
examples of weakly singular controls that are (or are not) optimal are known. It is
proved in [2] (see [5], Section 3.4) that for the (self-adjoint) multiplication operator
Au(λ)=−λu(λ)
in L
2
(0, ∞), which generates the analytic semigroup
S(t)u(λ)=e
−λt
u(t) (1.8)
there exist optimal controls for (1.1) satisfying the maximum principle (1.5) where
the growth of the costate z(t)ast approaches the final time T is ≈ C/(T − t)in
the sense that
(T − t)z(t)
E
∗
=(T − t)S(T − t)
∗
z
E
∗
→ C as t → T (1.9)
with 0 <C<∞. These controls cannot satisfy (1.4), thus they are weakly singular.
On the other hand there exist controls satisfying (1.5) and
(T − t)
α
z(t)
E
∗
=(T − t)
α
S(T − t)
∗
z
E
∗
→ C as t → T (1.10)
with α>1and0<C<∞ (thus weakly singular) that are not time or norm opti-
mal. We provide in this paper similar examples for the right translation semigroup
2
The statement on time optimality, however, needs additional assumptions on the initial condi-
tion ζ and the target ¯y. These conditions are satisfied if either ζ =0or¯y = 0 [5], Theorem 2.5.7.
We point out that the conditions are on the “size” of ζ ¯y, not on their smoothness like (1.7); for
instance, for ζ =0, ¯y may be an arbitrary element of E. We also need to assume that S(t)
∗
z =0
in the entire interval 0 ≤ t ≤ T.
236 H.O. Fattorini
S(t)inL
2
(0, ∞) defined by
S(t)y(x)=
y(x − t)(x ≥ t)
0(x<t) .
(1.11)
Although the technical means are totally different, the examples are of the same
sort as those in [2]; there are controls that satisfy (1.9) and are optimal, whereas
there are controls with faster increase of z(t) which are not optimal. What is
remarkable about the examples in this paper is that they resemble similar exam-
ples for semigroups as different as (1.8), analytic with (1.1) an abstract parabolic
equation. The semigroup (1.11) under study here is isometric, with associated
equation (1.1) having a finite velocity of propagation, thus qualifying as “abstract
hyperbolic”.
On the basis of this similar behavior of controls for very different semigroups
it is tempting to guess that there must exist some sort of classification of weakly
singular controls (as optimal or nonoptimal) which is based on the growth of the
norm of the costate z(t)=S(T −t)
∗
z as t → T and holds for arbitrary semigroups.
There seems to be no such general result except [3] Lemma 8.3, [5] Lemma 3.5.9
where the generator is self-adjoint in Hilbert space and S(t)
∗
z has “hyperpower
growth” as t → 0; this means
1
0
S(rσ)
∗
z
σS(σ)
∗
z
dσ < ∞ (1.12)
for some r in the range 1 <r<2 (the adjoint may be omitted since the semigroup
S(t) is self-adjoint). Under (1.12), the control corresponding to the multiplier z is
not optimal. Condition (1.12) cannot hold if (1.10) is satisfied for any α>0; in
fact, in this case
S(rσ)
∗
z
σS(σ)
∗
z
≈
Cσ
α
σC(rσ)
α
=
1
r
α
σ
making the integral infinite. However, there is a wide gap between hyperpower
growth and power growth like (1.9), and nothing is known for intermediate growths.
We mention in passing the results on multipliers in [6]. When the semigroup
satisfies
S(t)E = E (t>0) (1.13)
then every multiplier space satisfies Z = Z
w
(T )=E
∗
, that is, all multipliers in
(1.5) automatically belong to E
∗
; this makes moot the question of the growth of
z(t) as t → T. It is also shown in [6] that (under the assumption that E is
reflexive and separable) (1.13) is a necessary condition for all multipliers to belong
to E
∗
. Moreover, condition (1.7) can be dropped from Theorem 1.1 in case (1.12)
holds: all time or norm optimal controls satisfy (1.5) with a multiplier z ∈ E
∗
.
2. The right translation semigroup
The space is E = L
2
(0, ∞). Its elements y(x) (defined in x ≥ 0) are extended
as y(x)=0forx<0. The right translation semigroup S(t) defined by (1.11) is
Time and Norm Optimality of Weakly Singular Controls 237
strongly continuous and isometric in L
2
(0, ∞). The adjoint semigroup is the left
translation (and chop-off) semigroup
S(t)
∗
y(x)=
y(x + t)(x ≥ 0)
0(x<0) .
(2.1)
We have
S(t)
∗
S(t)=I, S(t)S(t)
∗
y(x)=χ
t
(x)y(x) (2.2)
where χ
t
(x) is the characteristic function of [t, ∞). The infinitesimal generator A
of S(t)is
Ay(x)=−y
(x) , (2.3)
with domain D(A)={all y(·) ∈ L
2
(0, ∞)withy
(·)inL
2
(0, ∞)andy(0) = 0},
the derivative understood in the sense of distributions. The semigroup S(t)is
associated with the control system
∂y(t, x)
∂t
= −
∂y(t, x)
∂x
+ u(t, x)
y(0,x)=ζ(x) ,y(t, 0) = 0 ,
(2.4)
in the sense that S(t) is the propagation semigroup of the homogeneous equation
(u(t, x)=0). Formula (1.3) for the control u(t)(x)=u(t, x)is
y(t, x, ζ, u)=y(t, ζ, u)(x)=
S(t)ζ +
t
0
S(t − σ)u(σ)dσ
(x)
= ζ(x −t)+
t
0
u(σ, x − (t − σ))dσ , (2.5)
thus the contribution of u(σ, x)toy(t, x, ζ, u) is the integral of u(σ, x)overthe
intersection with the positive quadrant of the characteristic line (σ, x − (t − σ))
joining (0,x− t)with(t, x), as shown in Figure 1.
kpvgitcvkqp nkpgu
y
yu
u
2
y
Figure 1