SIAM
J.
CONTROL
AND
OPTIMIZATION
Vol.
20,
No.
4,
July
1982
1982
Society
for
Industrial
and
Applied
Mathematics
0363-0129/82/2004-0009
$01.00/0
CONTROLLABILITY
FOR
DISTRIBUTED
BILINEAR
SYSTEMS.*
J.
M.
BALL,5"
J.
E.
MARSDEN:
AND
M.
SLEMROD
Abstract.
This
paper
studies
controllability
of
systems
of
the
form
dw/dt
w
+
p(t)Ydw
where
is
the
infinitesimal
generator
of
a
C
O
semigroup
of
bounded
linear
operators
e
t
on
a
Banach
space
X,
Y3’
X
X
is
a
C
map,
and
p
L
([0,
T];
[)
is
a
control.
The
paper
(i)
gives
conditions
for
elements
of
X
to
be
accessible
from
a
given
initial
state
Wo
and
(ii)
shows
that
controllability
to
a
full
neighborhood
in
X
of
Wo
is
impossible
for
dim
X
c.
Examples
of
hyperbolic
partial
differential
equations
are
provided.
1.
Introduction.
The
purpose
of
this
paper
is
to
discuss
controllability
for
abstract
evolution
equations
of
the
form
(1.1)
if(t)
sdw(t)+p(t)Yd(w(t)),
(1.2)
w(O)
Wo,
where
/generates
a
C
o
semigroup
of
bounded
linear
operators
on
a
(possibly
complex)
Banach
space
X,
Y3"
X
X
is
a
C
map,
and
p
L
1([0,
T];
R)
is
a
control
defined
on
a
specified
interval
[0,
T].
Usually
we
assume
that
Y3
is
linear,
so
that
(1.1)
is
bilinear
in
the
pair
(p,
w);
note
that
even
in
this
case
the
solution
w
of
(1.1),
(1.2)
is
a
nonlinear
function
of
p.
A
motivating
example
is
the
rod
equation
(1.3)
u,t+Uxxxx+p(t)Uxx=O,
O<x<l,
with
hinged
end
conditions
(1.4)
U=Uxx=O
atx=O,
1,
which
can
be
put
in
the
form
(1.1)
by
setting
w=(u",)
with
X=
(H2(0,
1)
f3
H(0,
1))
Lz(0,
1).
Here
the
control
p(t)
is
the
axial
load.
The
main
tool
used
in
our
analysis
is
the
generalized
inverse
function,
or
"local
onto"
theorem.
In
finite
dimensions,
the
well-known
controllability
results
for
bilinear
systems
have
been
obtained
in
this
way
(see,
for
example,
Brockett
[1972]
and
Hermes
[1974]).
In
infinite
dimensions,
however,
new
phenomena
arise.
Perhaps
the
most
interesting
of
these
is
our
result
(Theorem
3.6)
which
shows
that
for
linear
and
dim
X
az,
the
set
of
states
accessible
from
Wo
for
p
Llr.oc([0,
cx3);
[),
1
<
r-<_
o,
has
dense
complement
in
X.
Hence
we
can
never
expect
to
control
to
an
open
neighborhood
of
Wo
for
controls
in
Loc.
(Using
L
controls
doesn’t
help,,
at
least
for
examples
such
as
(1.3),
(1.4);
see
Theorem
5.5.)
This
stands
in
direct
contrast
to
the
available
positive
results
on
controllability
when
dim
X
<
o.
*
Received
by
the
editors
March
24,
1981,
and
in
revised
form
September
25,
1981.
"
Department
of
Mathematics,
Heriot-Watt
University,
Edinburgh,
United
Kingdom
EH14
4AS.
The
research
of
this
author
was
supported
in
part
by
the
U.S.
Army
Research
Office
under
contract
DAAG29-79-
C-0086,
the
National
Science
Foundation
under
grant
MCS-78-06718
and
a
United
Kingdom
Science
Research
Council
Fellowship.
:
Department
of
Mathematics,
University
of
California,
Berkeley,
California
94720.
The
research
of
this
author
was
supported
in
part
by
the
U.S.
Army
Research
Office
under
contract
DAAG29-79-C-0086
and
the
National
Science
Foundation
under
grant
MCS-78-06718.
Department
of
Mathematical
Sciences,
Rensselaer
Polytechnic
Institute,
Troy,
New
York
.12181,
The
research
of
this
author
was
supported
in
part
by
the
National
Science
Foundation
under
grant
MCS-79-02773
and
by
the
Air
Force
Office
of
Scientific
Research,
Air
Force
Systems
Command,
United
States
Air
Force
under
contract/grant
AFOSR-81-0172.
575
576
J.
M.
BALL,
J.
E.
MARSDEN
AND
M.
SLEMROD
Given
the
impossibility
of
controlling
the
system
(1.1)
to
a
full
neighborhood
of
w0
with
p’s
in
L
r,
we
investigate
two
alternative
procedures.
One
approach
generalizes
an
idea
of
Hermes
[1979];
we
show
that
it
is
often
possible
to
control
with
respect
to
finite-dimensional
observations
in
a
neighborhood
of
w0.
Our
second
idea
is
based
upon
the
concept
of
approximate
controllability,
i.e.,
we
identify
a
dense
subset
of
X,
depending
on
w0
and
t,
to
which
w(t)
belongs,
and
show
that
with
respect
to
a
strengthened
topology
one
can
control
to
a
neighborhood
of
etwo
(the
"free
solution"
of
(1.1),
(1.2)
corresponding
to
p
0)
in
this
set,
provided
is
suitably
chosen.
For
(1.3),
(1.4)
we
prove
that
>
0
can
be
taken
arbitrarily
small,
whereas
for
the
wave
equation
(1.5)
utt-Uxx
+p(t)u
=0,
0<x
<
1,
with
either
the
boundary
conditions
u=0
atx=0,1,
or
the
boundary
conditions
u=0
atx=0,
u+aux=O
atx=l,
a>0,
has
to
exceed
some
number
T
>
0.
This
study
of
local
approximate
controllability
involves
technicalities
concerning
nonharmonic
Fourier
series
in
the
spirit
of
Russell
[1967]
and
Ball
and
Slemrod
[1979].
The
delicacy
of
these
questions
has
the
unfortunate
consequence
that
we
have
only
been
able
to
obtain
positive
results
in
cases,
such
as
these
described
above,
in
which
(1.1)
is
an
abstract
hyperbolic
equation
that
is
"diagonal";
i.e.,
is
reducible
to
an
infinite
set
of
uncoupled
ordinary
differential
equations
(each,
of
course,
containing
the
control
p(t)).
Since
we
have
to
control
infinitely
many
ordinary
differential
equations
simultaneously,
however,
the
problem
is
still
not
trivial.
Nevertheless,
our
assumptions
exclude
some
important
nondiagonal
examples
such
as
(1.3)
with
clamped
end
conditions
u=u=0
atx=0,1.
In
special
cases,
such
as
(1.3),
(1.4),
our
local
approximate
controllability
theory
leads
to
a
global
approximate
controllability
result;
thus,
for
example,
for
suitable
initial
data,
we
prove
that
the
attainable
set
for
(1.3),
(1.4)
is
dense
in
X.
The
paper
is
divided
into
six
sections.
Section
2
assembles
the
machinery
for
studying
(1.1),
(1.2)
in
the
form
of
various
abstract
existence
and
smoothness
theorems.
Section
3
provides
an
abstract
controllability
theorem
and
the
result
on
noncon-
trollability
mentioned
above.
In
4
we
discuss
the
general
theory
of
control
with
respect
to
finite-dimensional
observers.
In
5
we
consider
abstract
hyperbolic
equations,
apply
the
theory
of
4
to
this
case,
and
develop
our
theory
of
approximate
controllability.
We
conclude
in
6
with
specific
applications
to
partial
differential
equations,
such
as
(1.3),
(1.4).
2.
Abstract
existence
and
smoothness
theorems.
In
this
section
we
give
some
basic
results
on
nonlinear
evolution
equations
which
will
be
useful
in
our
later
analysis.
Let
X
be
a
Banach
space
with
norm
I1’
II,
let
generate
a
C
O
semigroup
of
bounded
linear
operators
on
X,
and
let
:X
X
be
a
C
k
mapping,
k
->
1.
Let
Z
(T)
be
a
Banach
space
continuously
and
densely
included
in
L1([0,
T];
R),
where
T
>
0
is
given.
For
a
given
w0
X
and
p
Z
(T),
consider
the
initial
value
problem
associated
with
(1.1)
written
in
integrated
form,
i.e.,
(2.1)
w(t)
etwo
+
Io
ea(t-S)p(s)Yd(w(s))
ds.
CONTROLLABILITY
FOR
DISTRIBUTED
BILINEAR
SYSTEMS
577
Solutions
of
(2.1)
are
often
called
"mild
solutions"
of
(1.1),
(1.2).
The
question
as
to
when
solutions
of
(2.1)
are
actually
solutions
of
(1.1)
is
discussed
in
Remark
2.7
at
the
end
of
this
section.
PROPOSITION
2.1.
For
each
Wo
X,
and
p
Z
(T)
there
exists
to,
0
<
to
<=
T,
such
that
(2.1)
has
a
unique
solution
w
C([0,
to];
X).
Proof.
Let
={wf([O,
to];X)llw(t)-Wo[l<=R},
and
define
Tp:
C([0,
to];
X)
by
(Tpw)(t)
etwo
+
|
ea(’-S)p(s)(w(s))
ds.
ao
Since
Ile’ll
<-
M
e
t
for
positive
constants/3,
M,
an
easy
estimate
shows
that
T
maps
to
provided
-to
R,
0-<_
to,
Ile’w0
woll+Me’C
Ip(s)lds<-
<-
30
where
C
is
such
that
IIBwll
<-C
for
IIw-
Wol[
<--R.
This
condition
is
achieved
for
R,
to
sufficiently
small
via
the
continuity
of
,
eatwo
and
the
fact
that
p
LI([0,
T];
R).
Similarly,
T,
is
a
contraction
map
of
to
provided
that
KM
e
tt
fo
’
IP
(s)l
ds
<
1,
where
K
is
a
Lipschitz
constant
for
9
on
the
ball
IIw-
w0[I
R.
Again
this
holds
for
R
and
to
sufficiently
small.
The
result
now
follows
from
the
contraction
mapping
prin-
ciple.
Of
course
the
above
proposition
is
a
special
case
of
many
more
general
results
on
existence
and
uniqueness
of
solutions
to
semilinear
evolution
equations
(see,
for
example,
Segal
[1963],
Pazy
[1974],
Balakrishnan
[1976]
and
Tanabe
[1979b]).
The
point
for
us
here
is
that
use
of
the
contraction
mapping
principle
leads
to
other
important
features
of
the
solution
map
w,
as
we
now
see.
PROPOSITION
2.2.
Fix
Po
Z
T).
Then
there
exist
an
open
neighborhood
U
of
po
in
Z
(T)and
to
>
0
such
thatfor
any
p
U,
(2.1)
has
a
unique
solution
w(t;
p,
Wo),
0
<=
<-to.
Moreover
w(t;
p,
Wo)
is
a
C
map
from
U
to
C([0,
to];
X).
Proof.
The
proof
of
Proposition
2.1
shows
that
if
R
and
to
are
sufficiently
small
and
p
is
close
enough
to’p0
in
L
1-norm
then
T,
is
a
uniform
contraction.
Also,
Tp
is
a
C
function
of
w
and
p
on
the
interior
of
,
so
that
the
C
result
follows
from
Hale
[1969,
Thm.
3.2,
p.
7].
The
C
result
is
then
obtained
by
induction.
COROLLARY
2.3.
The
map
w(t0;
",
Wo):
U-+
X
is
C
.
Proof.
This
follows
from
the
chain
rule,
Proposition
2.2
and
the
fact
that
the
map
w(.
)-+
W(to)
is
smooth
(since
it
is
continuous
and
linear
from
C([0,
to];
X)
to
X).
In
the
same
way
we
see
that
the
solution
w(t;.,.)
is
a
C
function
of
w0
and
p.
However,
in
this
paper
we
are
primarily
concerned
with
differentiability
in
p.
The
proof
of
the
theorem
in
Hale
[1969]
cited
above
shows
that
the
derivative
can
be
obtained
by
formally
linearizing.
Thus
we
get
the
following
result.
COrtOt.LARY
2.4.
The
(Frdchet)
derivative
Dw(t;
Po,
Wo)’p
of
w(t;
p,
Wo)
with
respect
to
p
at
po
in
the
direction
p
is
the
unique
solution
of
the
equation
(2.2)
Dpw(t;
Po,
Wo)’p
Io
e*e(t-S)P(S)Ya(w(s;
Po,
Wo))
ds
(t
s)
+
e
po(s)D(w(s;po,
Wo))Dw(s;
Po,
Wo)’pds.
578
J.
M.
BALL,
J.
E.
MARSDEN
AND
M.
SLEMROD
Here
DY3(w(s;
po,
Wo))
denotes
the
Frchet
derivative
of
Y3
at
w(s;
po,
Wo).
In
particular,
at
Po
O,
Dpw(t;
O,
Wo)’p
is
given
explicitly
by
(2.3)
Dpw(t;
O,
Wo)’p
f
ea’-)p(s)Y3(eawo)
ds.
o
Next
we
show
that
solutions
are
globally
defined
under
a
sublinear
growth
condition.
TI-IEOREM
2.5.
If
there
are
constants
C
and
K
such
that
II (x)ll_-
<
c
/gllxll
for
all
x
X,
then
(2.1)
has
solutions
defined
]:or
0
<-
<=
T.
These
solutions
are
unique
within
the
class
C([0,
T];
X).
Moreover,
the
solution
w(t;
p,
Wo)
is
a
C
k
[unction
of
p
Z(T)
and
Wo
X
with
(Frchet)
derivative
in
p
given
by
(2.2)
(or
(2.3)
i]’
po
0).
The
proof
is
based
on
the
following
version
of
Gronwall’s
inequality
(see,
for
example,
Carroll
[1969,
p.
124]).
LEMMA
2.6.
Letp
Ll([a,
hi;
)
and
let
v
L([a,
b];
)
with
v
>-0.1[there
exists
a
constant
C
>-
0
such
that
for
all
[a,
b
v(t)
<=
C
+
|
Ip(s)lv(s)
as,
Jo
then
Proof
of
Theorem
2.5.
Suppose
w(t)
solves
(2.1)
and
is
defined
for
0
-<
<
a
-<
T.
Then
IIw(t)ll<-Meta(
I[wll+
Io
IP(S)I(C+KIIw(s)II)
ds),
and
so,
assuming
K
>
0
without
loss
of
generality,
we
get
IIw(t)[l__-
<
(MetallWoll+
CK
-)
exp
(MetK
Io
IP(S)l
ds)
CK
-
<-
C.
Therefore,
for
s,
[0,
a)
we
have
IIw(t)-
w(s)[llle’wo-ewol[+
e’-)p(’)(w(’))
dr
<-Ile’wo-ewoll+Me(C
+
gc)
I
[P(’)I
dr.
Thus
limt_,_
w(t)
exists,
so
that
by
Proposition
2.1
w(t)
can
be
continued
beyond
a.
Hence
solutions
are
defined
for
0
_-<
_<-
T.
For
global
uniqueness,
we
use
the
standard
argument:
suppose
w(t)
and
(t)
solve
(2.1)
for
0-<t
-<
T.
Let
S={a
[0,
T]lw(t)=
(t)
for
t[0,
a]}.
The
local
uniqueness
assertion
in
Proposition
2.1
shows
that
S
is
relatively
open
in
[0,
T].
If
an
S
and
an
a-<
T
then
a
S
since
limn_,
w(an)=
limn_
(an).
Thus
S
is
closed,
so
that
S
[0,
7"].
Thus
there
is
a
globally
defined
semiflow
Ff
(Wo),
Ft
(’)"
[+
x
X
X,
which
depends
parametrically
on
p.
Proposition
2.2
shows
that
F
(Wo)
is
C
k
in
p
and
Wo
for
sufficiently
small.
Let
{a
[0,
T]IFf
(Wo)
is
C
in
(Wo,
p)
for
e
[0,
a]}.
We
claim
that
is
open.
Indeed,
if
a
and
k
is
small,
P
Fa+h
(Wo)
Ff,
(FPa
(Wo))
CONTROLLABILITY
FOR
DISTRIBUTED
BILINEAR
SYSTEMS
579
is
C
k
in
p
and
Wo,
because
by
Proposition
2.2
F
(w)
is
C
k
in
p
and
w
for
w
near
FOa
(w0).
The
local
uniformit,y
of
the
time
interval
on
which
Proposition
2.2
holds
shows
that
,
is
closed,
and
hence
S
[0,
T].
Thus
we
have
shown
that
w(t;
p,
Wo)
is
C
in
p
and
Wo.
By
differentiating
(2.1)
we
obtain
(2.2).
1
Remark
2.7.
Suppose
woD(A)
and
pGCl([0,
T];
).
Then
w(t)D(A)
and
w(t)
is
differentiable
and
satisfies
(1.1).
This
assertion
follows
from
Segal
[1963,
Lemma
3.1]
or
from
Tanabe
[9,
p.
102].
If
merely
w0
X
and
p
L
([0,
T];
R)
then
w
is
a
"weak
solution"
of
(1.1)
(see
Balakrishnan
[1976]
and
Ball
[1977]).
3.
An
abstract
controllability
theorem
and
a
negative
result.
Define
the
linear
operator
LT"
Z
(T)
X
by
T
|
eT-sp(s)(eSwo)
ds.
LTp
o
Then
by
(2.3)
we
have
(3.1)
Dpw(T;
0,
Wo)"
p
LTp.
A
natural
consequence
of
Theorem
2.5
is
the
following.
THEOREM
3.1.
Let
be
the
infinitesimal
generator
of
a
C
o
semigroup
of
bounded
linear
operators
on
the
Banach
space
X,
and
let
3
X
-->
X
be
a
C
k
map,
k
>=
1,
which
satisfies
IlBx]l<_-C
+
KIIxll
]:or
all
x
X,
where
C
and
K
are
constants.
Suppose
that
Range(LT)
=X.
Then
there
is
an
e
>0
such
that
w(T;
p,
Wo)
h
for
some
pZ(T),
provided
Ilh
eTwo[]
<
e.
This
result
follows
easily
from
the
(generalized)
inverse
function
theorem;
a
convenient
reference
is
Luenberger
[1969,
p.
240].
The
p
that
controls
Wo
to
hit
h
will
be
in
a
neighborhood
of
zero
in
Z
(T).
We
note
that
if
1
generates
a
group,
surjectivity
of
LT
is
equivalent
to
surjectivity
of
T"
Z
(T)
X,
where
(3.2)
rp
e-’p(s)t(e’wo)
ds.
A
major
difficulty
with
Theorem
3.1
is
that
is
is
not
usually
an
easy
matter
to
check
the
surjectivity
of
LT
(or
T).
In
fact,
as
we
shall
prove
in
Theorem
3.6,
if
dim
X
eo,
LT
will
not
in
general
be
surjective,
though
it
may
have
dense
range.
This
prevents
us
from
applying
Theorem
3.1
to
partial
differential
equations.
We
now
present
a
basic
criterion
for
LT
to
have
dense
range.
PROPOSITION
3.2.
Suppose
that
(1,
e’-s)yd(eSwo))
0
for
all
s,
0
<=
s
<=
T,
where
X*
(the
dual
space
of
X),
implies
O.
Then
Range
(LT)
is
dense
in
X.
Proof.
Range
(LT)
is
dense
if
the
only
s
X*
annihilating
the
range
is
0.
But
T
(l,
LTp)=
fo
(l,
e’T-s)3(eSSWo))p(s)
ds.
If
this
vanishes
for
all
p
Z(T),
then
the
continuous
function
(l,
eSg(r-s)?(eSwo))
must
vanish.
This
follows
because
Z
(T)
is
dense
in
L
([0,
T];
).
Our
hypothesis
then
gives