Necessary and sufficient conditions are found under which operators of the form Sigma(m, j=1) X(2)sub j + X sub 0 can be made constant coefficient. In addition, necessary and sufficient conditions are derived which classify those linear partial differential operators that can be moved to the Kolmogorov type

Hunt, L. R.

Villarreal, Ramiro

English

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I. Introduction 
The emphasis o f  this paper is determining necessary and sufficient 
conditions under which the linear partial differential equation 
can he transformed tG become either constant ccreff i c i e n t  01- < z f  t h e  b o l i i l o a o i  0': 
r* I < L  
C 11 type. Here we assume that the A ( x )  and E3 ( Y )  are C and consider C 
j k  J 
r i  coordinate chanqes o n  IF: . 
more variables arE due t o  Cotton 1 4 3  artd Fredricts !51. TnFsF I t - C  1 1  1 
attention 1s paid to the f i r s t  order c o e f f i c 1 e ~ 3 t s  6 ( Y )  after thF i t : .  
coordinates are  introduced. 
J 
K, 
I f  there are C coordinates under which ( 1 )  becomes constant Coe'ff ic iEi- t t  
then Fourier transforms i n  the spatial variables yield 
* 
Research supported bv the NASA Ames Research Centei- under Grant NAG ?-3603 
Supplement No. 1. 
(YASdi*CR- 182486) CAYCPICAL CCOhDIDAZES PCE 188- 1 9 1 70 
E A 3 T I U  ~ I P P E G B I P I I L  ECUATICliS /%exas Univ. 
a t  Dallas) 15 p CSCL 12A 
Uaclas 
63/64 0124520 
https://ntrs.nasa.gov/search.jsp?R=19880009786 2020-03-20T08:01:29+00:00Z
ORIGINAL PAGE IS 
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1. 
.:::.J;. 
(3) 
n n 
Z ajk t jSkU- i  Z b . < . U  = F(t,t) du at - - -  J J  j 9 k = l  j = 1  
an ordinary differential equation in U. 
Similarly, i f  there are Cm coordinates in which ( 1 )  takes the form 
( 4 )  
where a and b are constants, then Hormander C61 shows that spatial 
Fourier transforms lead to 
j k  jk 
Under generic conditions, this first order linear partial differential 
equation can b e  sol\!ed bv the metpod o f  cha~azteristics ( 1 . ~ .  c r d ~ r ~ ; ~  
operator can be written as 
2 
R 
z x + x . .  
1 U 
J = 1  I 
For general partial differential o p e r a t o r 5  o f  t h e  f o r m  ( 6 )  Hormander D r o c e E d ?  
to prove neces5ar~ and sufficient conditions for hvpoelliptlcltv 11.e. C 
right hand sides imply Cu solutions). We require that our K‘olmoqorov 
equations be hypoelliptic. Weber 173 constructed fundamental SOlutionS f o r  a 
class of  equations related to those in ( 4 ) .  
In this light the problems consldered irk this paper are: 
i )  Given the partial differential operator ( 6 )  find necessary and 
... ... 
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00 
sufficient conditions so thst there exist nonsingular C coordinate 
changes (local-near the origin) on I f  under which (6) becomes a 
constant coefficient partial differential operator. Standard 
differential geometry results (e.g. Spivak CBI) are employed and 
the results are of no surprise. The conditions are derived here 
f o r  the sake of completeness. 
1 1 )  Given the partial differential operator ( 6 )  find necessary and 
sufficient conditions so that there erist nonsingular C coordinate 
changes (local-near the origin) on IF under which X 1 9 k 2 p - . . 9 X  i n  
( 6 )  are transformed to constant vector fields and Xo becomes a 
cc 
n 
Ill 
linear vector f i e l d .  This makes the partial differential operatcli 
( 6 )  o f  C:olrnoqcrov type, i f  the hvoaellipticity K 0 d i t i u i ) s  o f  
tiormandel ai-E sat i s f  1 ed ~ 
The spatial cp~ratnr i n  ( 4 )  is a Kolmoqo!-ov operato! it : - .  . , '  .  
hypoelliptic. 
, -. - WE remark that both problems 1 ! and i i ! c a n  be qenet -a i  1 z d  ; . , I  ,.,!<- 
IT1 - F . .  
- c  . . 'r' c 
+ X . ,  but we s h a l  1 concentr2i.c 
I :  
rjartial rjiffPrentis! operators ;i - - .  
,=! .l J " 1  
o n  the f o r m  : & I .  Moreover ,  w e  255~rne t h a t  XI , Y . ? ,  . . . , X  s r c  i 1 $ .+ -> '  '. . 
m 
i ndependciit . 
Our pi incipl~ tools are taken f i o m  the 
However, the purpose of this paper 15 
controllability of  cvstems o f  nonlinear o r d  
hypoelliplicity of partial differential 
field o f  systemc. ano co!itrol. 
not to draw a parallel between 
ria1-v differential equations and 
equations, as this has been well 
established in the literature and in conference presentations. 
As we mentioned previously, problem i )  is straightforward. Our w o r k  on 
problem i i )  is analogous to the study o f  coordinate changes to transform a 
nonlinear control system on R n 
t o  an n-dimensional controllable linear system 
.. 
I - 
(8) x = Fx + Gu. 
dx 
Here i = dt - , f,g ,g ,...,gm are C'.' vector fields on Rn, f ( 0 )  = 0, u =  1 2  
( u 1 , u 2 ,  ... 'u ) consists of real-valued functions, F is an nxn constant 
matrix, and G is ar! nxm constant matrix. A l s o  f,F, and u are obviously 
different objects in our control discussion than in our p.d.e. discussion. 
m 
We rely heavily on the results o f  Krener L9J and REspondek [ l o ] .  and, 11-1  
fact, our research essentially ; i ! [ ~ ~ e s  their results from the ordinar Y 
differential equation setting to the partial differential equatiop sett :r .c.  
Nonlinear control system i 7 i  15 replaced bv  eouation ( 6 )  with X tski1(1- '  +!-*c, 
place of  Q J = 1,; , . ... m .  and X t a i i n q  the 31act -  of f .  The li+=+r r\ ;c-?~.- 
( 8 )  is replaced by tne b . a i r ~ o a ~ r o ~ '  partial diffcrFrstial opf ; ra t .o r  
1 'J 
IT! c C - 
( 5 !  - 7 r  e X '  . 
{I ' 
where each is a constant vector field an3 X 1 %  linear. Here ;, , f  % .  . 
Correspond to the T columns of G and 'z corresponds Fic in (8 ! .  i.!? ;;c,i-,t t t l r  
span o f  the Lie brackets X ,CXo,X.l,...,(ad X C v X  . I ,  j=1,2 ,.... a tc .  tee F " .  
so we make the corresponding assumptions on the L i e  brackets of vector fields 
j +  .I 
J 0 3 L '  
0 
- 
- n-1- - - -  
I J - 1  
in (6). A s  noted before we also suppose that X , , X 2 ,  ... , X  are liriearl.. 
independent . 
N4 
Section 2 o f  this paper contains hasic definitions ahd consioel-aticj!-, o f  
those 1 inear partial differential operators wtlirh can be made constant 
coefficient. In section 3, necessary and sufficient conditions a r e  d w i v e d  
which classify those linear partial differential operators that can be m o v e d  
to the Kolmocjorov type. 
11. Constant Coefficient Operators 
We begin with a set of appropriate definitions. 
If X and Y a r e  Cm vector fields on R , then the L i e  bracket  o f  X n and Y 
i s  
FY ,* X 
ax if E [X,YI = - - x  - I - Y ,  
U Y  
B x  ?X where - and - " are Jacobian matrices, x heinq the '\,ariahl~ for iF". 
e t c .  can he taken. 4 standard notation i c -  
0 (ad X , Y )  = 1 
t h e  vec to r  field X ac- 
L h = ',dh,f ' .  x 
Successive L i e  derivatives a r e  
L X h  2 = LXLXh 
Moreover, the Lie derivative o f  t h e  one f o r m  d h  with respect to X is - ----- 
The three types o f  Lie derivatives satisfy the formula 
(10)  LX<dh,Y'> = <Lx idh),Y> + idhyCXyY1). 
We motivate our study b y  the followinq example. 
Example 2.1. Cons der the partial differ-evtial operator 
3 or F. . The l o c a l  coordinate change  ine.31- tnr- o ! - i g i n )  
( 1 3 )  
T h i s  is discovered in the followiliq w a y .  F i r s t  we write l i i !  < . -  
1' x o 3  Since X and X are linearly independent and the Lie bracket C X  1 0 
5 0 ,  standard differential geometry results (see C81) implvl that the 
transformation (12)  takes 
Hence, the partial differential operator ( 1 1 )  is moved t o  t h e  constant 
coefficient form ( 1 3 ) .  
We now prove our  result concerning transformation to constant 
coefficient operators. Again, this result is trivial from the differential 
geometry viewpoint. 
Theorem 2.1. G i v e n  the CrB partial differential cperator o n  IT’” 
( 1 7 )  
where X o , X 1  , X  ,... X , are linearlv independent f o r  mc,n and  X I  , F 2 ,  . . . 9 ar  E. 
2 m 
linearly independent for m=n, there erists a non-singular ( l o c a l  co~rdinatc 
change on F under w h i c h  X ( ) , X I > .  . . , X  become constant v e c t o r  field5 it 31-11:  
c j n i y  if 
n 
r 
. .. -:.- 
t a k  1 nq 
i .  
i .  
if and only i f  ( 1 8 )  is satisfied. 
I f  rn = n ,  we can move 
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OXGINAL PAGE IS 
OF POOR QUALITY 
i f  and o n l y  i f  E 0 for all l < r , s < m .  Setting 
we find the only possible nonzer~ colurrln of  
W E  now stud:. o u r  problem i i ) ,  the main considpration o f  this p a ~ e : .  
111. Kolraogorov Operators 
Fl 
We examine the partial differentlal operator ( b )  2- - X + where 
J 
1=1  
I !  xl,X2,...,X are Iinearlv 1ndependLni and X .  .vanishes dt the oriqirl 11-1 iT, . 
We derive conditions under w h i c h  C, coordinate c h a n g e 5  (local-near thE 
rn ( !  
t. 
o r i g i n ,  exist taking ( 6 )  to the KolmoQot-ov operatoi- ! 9 ) .  
6s stressed in the introduction, the main contrihulion of this paper 1 5  
realizing that the results of  Krener L73 and Eespondek L 1 0 1  1 1 - 1  the non l in t : a l  
systems ( 0 . d . e . )  and control area can b e  applied t o  partial differential 
.,... .   '- . .  
operators 
( 1 9 )  a 
n 
+ Z: B . ( x )  - a2 
n 
J dx 
j 9 k=l J j = 1  j 
A j k ( x )  dx.dxk 
The linear controllable system 
( 2 0 )  x =  
is replaced by the hypoelliptic KO 
(with a and b constant) 
- 
j k  j k  
n 
( 2 1 )  
FF + Gu 
moqorov partial differential operatoi- 
Of course we shall study operators ( 1 9 )  and ( 2 1 )  in o u r  v e c t o i -  field 
notation. F i r s t  WP wish to examine parallpis Letween s y q t n -  $2: )  ~ I I J  
operator ( ? I ! .  
For the control svstem t 2 u )  the t,ronecker indic~s and ~ i g r n . a i c i e . s  c ~ t  
F matrices a r e  invariants under coordinate changes. F o r  tbe  cpera toy  ( p i  b -  
._ . 
introduce kolmoqorov indices and note that these and t h e  ~ i o e . 7 . ~ : ~ ~ :  .f tQ. L ,  
t-oi-ms f o r  ( 2 0 )  will occur in o u r  w o r k . .  
- -  
If the matrix A = (a ) in ( 2 1 )  ha5 rank m, let i o , X l  , X 2 * . . . 9 i  ti- 
j k  lli 
m 2  - - 
linearly independerlt vector fiFlds 5.0 that ( 2 1 )  becomes 2 X .  + X - .  WF set 
..I j = t  J 
= Jacobian matrix o f  the vectoi- field X . .  I f  the partial differpntial 
'L) 
11-1 - 
operator i n  (21 )  is hypoelliptic (in this case i ,BY , . . . v ~  r v 1 = 
1729...9m9 span [P J we introduce t h e  following process: 
I !  
Y l  
’ORIGINAL; PAGE IS 
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1 )  Write out  the gr id  
- - - 
x1 x 2 . . .  x m  
Ex, BY, . . . BXm 
B2x ,  
g- 1 - 
B2x2 . . . B2Xm 
. Bn-l- 
X m 
Bn- 1 - x2 . . 
2 )  Start at x and move left to right across the first I - o w ,  thpn 5 , t a r t  
1 
at EX and move across the second r o w ,  etc. 
1 
3 )  Throw out any vector field that is a linear combination o t  the 
preceding vector fields in the qrid. Discard all vectcr f l E ! d ‘ :  i 3  
thF column below th!s vectoi. f ? E l G .  
a!: others hade tieen diccarded. 
t t-, 
E 
J )  Let i = number o f  entries remaining I,-[ the J colunli-t of t h e  0 1  1d  
J 
- - 
Lj  Renumber  r I 3 ’ X 2  ,..., X , i f  n e c e ~ ~ a r y ,  50 that t1.i.f I 2- . a . u . ( :  - TI . 
(I1 
Uef - The inteqez-s C 1  - t2) . . . , ._ ,’ a r e  called the k’cimogor-ov indices o f  +hi 
partial differential o p e r a t o r  ( 2 1 ) .  If the v e c t o r  f i e l d  notation f G ! -  (21) i.; 
(TI 
-
m 2  - - 
‘ “ . j  2 X + X w e  also have the integers t l , ~ u 2 ,  ..., C associated witkt ( 5 1 .  0 m 
J = 1  
By assumption 4’ + C +...+ ( = n. 
1 2 rn 
m -  
We ask if the partial differential operator ( 6 )  Z X‘ + Xo is a 
J j = 1 
coordinate change away f r o m  the Kolmoqorov  partial differentia! operator ( 9 )  
m 
2 x .  + x havinq given indices “ , t 2 ,  ..., Y . Before stating the qenerai 
j = 1  
-2 
3 0  m 
theorem we present an example. 
Example 3.1. Consider t h e  partial differential operator 
- x 2 + 2(x2-x3) 2 x ) a 
3d>! + - + (2+x ) - + 
I.2 3 1 
a2 a2 
3 ax3 4 3 6x2dx3 
( 2 2 )  4x - a2 + 4x 
ax2 
2 
3 on IF: . We write this as 
I + x + 2 ( x  - x  ) x - 3 dx 2 2 3 3 1 "  e x  (23) j2* 3 - dx 2 + q F x 3  ax 3- - (jrx+ 2 -j dx3. + jx 2 - x 3  r.7 3 d 
Letting 
x =  
1 
'- 0 1 
- "31 x o  - 
2 2 
K 2 3  - x  + 2 ( x  2 3 3 1  - x  ) x  
y,3 
w e  f i n d  ( 2 3 )  becomes 
2 
1 ( 2 4  1 ).: + XC) , 
The l o c a l  coordinate channer 
2 4  - 
- x 3  + 2 x  Y - 
1 = 3 2 2 '3 Y 3  
- 2 
= > : - x  = L  Y 
- 
1 
xo - x 2  2 3 
take ( 2 4 )  t o  
- 
x =  
1 
where and x = 
1 
ORIGINAL PAGE IS 
OF POOR QUALITY 
This yields the Kolmogorov partial differential operator 
( 2 6 )  
Since 
d2 - d + x - *  - 3  - + x 
dx - 2  2 - -  3 - -  ex 2 dX 1 3 
2- 3 - - 
XI, BX1, B XI span rT( and the single Kolmogorov index is e = 3 = n. 1 
In equation ( 2 4 )  we remark that 
and 
. .  . .  
We now state and prove our main result. The qeneral p r o o f  will follow 
from analogous results from C93 and C l O I ,  but we shall present a p r o o f  i l i  
the case r! = 1 for some sort o f  completeness. 
m 
2 with 0' Theorem 3.1. The linear p a r t i a l  differential operator (6) L X + X . .  + 
J = i  
T- x 1  9 X 2 9 . . . , F ,  linearly independent ar! $" and X .  (13) = O, can b~ transformed b, .  
nonsingular coordinate changes ! l o c a l - n e a r  the o r i g i n )  to the  C.01rnaqorcr~ 
partial differential operator ( 9 )  L '2 + x having indices ; , ( C  , . . .+ t if 
and only if 
IT1 C! 
IT1 
1 9 (r! j=l  
i i  
X o ,  ) i l  j , X 2 ,  I: x o ?  X 2 : l I , .  . I '  , - . I  the set I X  , C X O , i  I,".., (a6 
1 1  1 
1 5  linearly independent. 
and $) the Lie brackets o f  e v e r y  p a 1 1  o f  vector-s f i e 1 a . i  i r  
P r o o f ,  !For m = 1 ,  4 =n, and the operator Y , 2  + X ) :  
n- 1 
1 CJ -
) are 1 inearly independent and have 2ei-c. Since X ,, I X o , X  ,.. . (ad X O 9 X  
-12- 
L i e  brackets,  we have coordinates so t h a t  
x l  becomes 
-1 3- 
c 
( 2 9 )  ORIGINAL PAG3 Y3. 
OF POOR QUALITY- 
3 
- 
Hence X o  is a linear vector field as promised and x + x is a Kolmogorov 
1 0 
partial differential operator. 0 
In the  above p r o o f  
- 
( 3 0 )  x =  
1 
1. 
+ io 1 5  in a canonical f c r m .  I - -  ..E sei 1 and x 
that a second o r d e r  iiriear partiel differential operator he a coorrlinat~ 
change away from a k.olmogorov operator. 
Future research will be in two directions: 
1) Expand the transformations used to include "appropriate types" o f  
feedback. This research is presentlv underway, and first thour~hts 
were to include results in this paper. However, the p r o c e s s  o f  
feedback, a5 applied in transformation theory, is not well 
_' -14.- 
111 
c21 
C3! 
E4 1 
c 5 3  
C63 
E71 
C83 
c91 
c 101 
addressed in the partial differential equation literature, where 
coordinate changes are standard fare, Therefore, we decided that 
separate papers are appropriate. 
Extend all results t o  the discrete setting. A Ph.D. student o f  the 
first author is currently working on this project. 
REFERENCES 
A .  N. holmogorov, T r r f a l l i g e  Lv?wey~nqen, A n n .  o f  Math. 35 , -
116-1179 1934. 
P .  R.  Garabedian, Fartial DrEferent;al Eqrrat;ons, John Wilev and 
Sons, 1964. 
6. A. Fredricks, T h e  gc?cme?ry nf second c,r-de:- i 7 ~ ~ ~ 3 : .  G.I-? I& 
d i f f e r e n t  ia i operators , Comm I Part i a 1 Di f f  . Equas, . 3 .  .- N c j .  t 
1483, 643-655.  
M. Weber, ?%e fundamental S O ~ L ' ~  1011  , s i  a degenerate p a r t  la7 
different ra l  equation of paral6)l;c type, Trans. Arner. Math.  SOC 
- 71,  1951, 24-37. 
A. J. Krener,  On the equivalence of cor1trc.l cystoms and the 
linearization of nonlinear system,  SIAM J. Control - 1 1 ,  1973, 
670-676. 
W .  Respondek , Geonetric methc4s in 1 inearizat I I : ? ! ~  of control 
rysternr, in "Banach Center Publications," Semester on Control 
Theory, Sept.-Dec., 1980, Warsaw. 
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Canonical coordinates for partial differential equations

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