Transcript ODE - National University of Singapore

INTERPOLATORY SOLUTIONS
OF LINEAR ODE’S AND EXTENSIONS
Wayne M. Lawton
Dept. of Mathematics, National University of Singapore
2 Science Drive 2, Singapore 117543
[email protected]
Yongwimon Lenbury
Dept. of Mathematics, Mahidol University
Rama 6 Road, Bangkok, Thailand 10400
SCOPE
I
G
G its Lie algebra, identified with TI (G)
is a connected Lie group with identity
We examine the following initial value problem

U  CU; U(0)  I
Continuous solution
Measure
U : R  G
C : R  G
ISSUES
Continuous dependence of solutions
U   (C )
Approximation & interpolation of continuous
U : R   G; U(0) I
~
~
~
by solutions U   (C ), C  C
functions
where
M
of
~
C M
is a dense subspace of the space
G-valued measures that vanish on finite sets
Applications and extensions
PRELIMINARIES
Choose a euclidean structure
, : G G R 
with norm
| | : G  R
and let
 : G G R 
be the geodesic distance function defined by the
induced right-invariant riemannian metric
PRELIMINARIES
M
space of
G- valued measures on R  without
point masses whose topology is given by seminorms
|| C ||k  0 | C(t) | dt, k  0.
k
P
topological group of continuous G - valued
functions W on R  that satisfy W(0)  I,
equipped with the topology of uniform convergence
over compact intervals, under pointwise multiplication
functions having bounded variation locally
BP
PRELIMINARIES
Lemma 1
A function
U : R  G
1
 U is in M,
CU
is in
B
then
(1.3) L(U)(t)  0 | C(s) | ds
if and only if
t
gives the distance along the trajectory in
G
and
(1.4) ρ(U(t),I)  L(U)(t), t  R 
PRELIMINARIES
exp : G  G
(1.5)
d
dt
exponential function
exp(tX) X exp(tX), t  R  , X  G
S M subspace of step functions
 0 : S  B map control measures to solutions
 0 (S) contains dense subset of interpolation set
(1.7) B(t 1 ,.., t n , g1 ,.., g n )  B  P(t1 ,.., t n , g1 ,.., g n )
0  t 1    t n  R  , g1 ,..., g n  G
RESULT
0
extends to a continuous  : M  B
that is one-to-one and onto. Furthermore, B
is a subgroup of P and it forms topological
Theorem 1
S M
is dense and
groups under both the topology of uniform
convergence over compact intervals and the
finer topology that makes the function
a homeomorphism.

DERIVATIONS
  ,   : G G G
for matrix groups  X , Y   XY  YX
Lie bracket
Ad : G  Hom ( G,G)
-1
Ad(g)(X)  gXg
Adjoint representation
for matrix groups
d
( 2.1)
dt
-1
Ad (U(t)) ( X )  [U U , Ad (U(t)) ( X ) ]
  0 such that
( 2.2) | [X, Y] |   | X | | Y |
We choose
DERIVATIONS
The proof of Theorem 1 is based on the following
Lemma 2 If
1
satisfy
and
U j  B, C j  U jU j , j  1,2,3,4
(2.3) U 3  U1U 2
1
(2.4) C4  C1  C2 ,
then
(2.5)
and
C 3  C1  Ad (U 3 ) C 2 ,
(2.6) L(U3 )(t)  L(U4 )(t)    | C 2 (s) | L(U4 )(s) K(s, t)ds
t
0
where
K(s, t)  e
 ( L(U2 )(t) - L(U2 )(s))
Proof Apply Gronwall’s inequality to the following
t
L (U )(t)  0 | C (s) | ds
3
t3 3
 0 | C (s) - Ad(U (s))(C (s)) | ds
1
3
2
t
 0 ( | C (s) - C (s) |  | C (s) - Ad(U (s))(C (s)) | ) d
1
2
2
3
2
t

 L(U )(t)  0 | C (s) - Ad(U (s))(C (s)) | ds
4
2
3
2
t
s d
 L(U )(t)  0 | 0 dv Ad(U ( v ))(C ( v ))dv | ds
4
3
2
t s

 L(U )(t)  0 | 0 [ C ( v ), Ad(U ( v ))(C ( v ))]dv | ds
4

 t s
)(t)   0 0 |
L(U
4
3
C
(v)
|  exp(  ( L ( U
3
 L(U
3
)( s )  L ( U
1
4
2
t

)(t)  0 L(U
)( s ))
1
3
)(s)f(s)ds
|
C
(v) |
2
dv ds
RESULT
Theorem 2 Let
C M
be a dense subspace.
,
Then for every positive integer n
and pair of
sequences 0  t 1    t n  R  , g1 ,..., g n  G
 (C) contains a dense subset of
P(t1 ,.., t n , g1 ,.., g n ).
DERIVATIONS
C  B(t 1 ,.., t n , g1 ,.., g n ).
It suffices to approximate C by elements in
( C)  B(t 1 ,.., t n , g1 ,.., g n ).
Lemma 3 Let f : D  M be a homeomorphism
m
of a compact neighborhood of 0  R
into an
N-dimensional manifold M. Then for any mapping
h : D  M that is sufficiently close to f,
f(0)  h(D).
Choose any
DERIVATIONS
Lemma 3 follows from classical results about the
degree of mappings on spheres. To prove Theorem 2
we will first construct then apply Lemma 3 to a map
m
n Define
n by
H:R G
: M G
 (C)  ((C)(t1 ),  , (C)(tn ))
We choose a basis B1 ,  , B d for G and define
X i  ( B1  i ,  , B d  i ), i  1,  , n;  i   ([t i-1 , t i ))
H ( v)   (C  
n
vi  Ad (U) (X i ) ),
1
d n
v  (v1 ,  , v d )  (R )

R
m
DERIVATIONS
We observe that H(0)  (g1 ,  , g n ). To show that H
satisfies the hypothesis of Lemma 3 it suffices, by
d
H ( v) | v0
the implicit function theorem, to prove
dv
m
n by
is nonsingular. We construct
F:R  G
F(v)  (C  (v1  X 1 )   (vn  X n ) ).
where we define the binary operation
C1  C2  C1  Ad ((C1 )) (C2 )
DERIVATIONS
A direct computation shows that
d
d
H ( v) | v0  F ( v) | v0
dv
dv
Furthermore, Lemma 2 and (2.5) imply that
thus
(C1  C2 )  (C1 )(C2 )
M and B are isomorphic topological groups.
Nonsingularity follows since
F(v)  (g1e1,..., g n e1...en ),
ei  exp((t i - t i-1 )X i  vi ), i  1,..., n.