Canonical coordinates for partial differential equations

Systems & Control Letters i l (1988) 159-165 North-Holland

159

Canonical coordinates for partial differential equations L.R. H U N T

* and Ramiro V I L L A R R E A L

Programs in Mathematical Science, The University of Texas at Dallas, P.O. Box 830688, Richardson, TX 75083.0688, U.S.A. Received 12 January 1988 Revised 26 March 1988

Abstract: Suppose we consider the second order linear partial differential equation ~.~ Ou

"

- a-7+ ~

j.k=t

O2u ~ A~,(~)o~ja~, + ..

8u

S~l~)~-~j=/ j=t

where Aj, and Bj are real C *° coefficients on R". Assume that the matrix (Ajk) is symmetric, positive semidefinite, and of constant rea-,k'. When can we reduce an initial value problem for this equation to an initial value problem for an ordinary differential equation by using (i) coordinate changes on R n, (fi) Fourier transforms? We address this question by finding conditions under which coordinate changes exist that make the above equation either constant coefficient or of the Kolmogorov type. This latter equation can be attacked by Fourier transforms, resulting in a first order partial differential equation that can be reduced to ordinary differential equations.

Keywords: Partial differential equations, Partial differential operators, Canonical coordinates, Lie brackets, Coordinate changes.

1. Introduction

can be transformed to become either constant coefficient or ,~f the Kolmogorov [6] type. Here we assume that the Ajk(x ) and Bj(x) are Coo and consider C ® coordinate changes on R". Texts on partial difterenti~.~ equations develop the theory of canonical forms for second order linear partial differential equations in two variables (see Garabedian [4] and Courant and Hilbert [2]). Extensions to three or more variables are due to Cotton ~1] and Fredricks [3]. These results involve finding C °o coordinates in which the principal part coefficients Aj~, become constant. This is always possible in two dimensions with suitable necessary and sufficient conditions in three or more dimensions. Little attention is pai~ to the first order coefficiems Bj(x) alter the new coordinates are introduced. If there are Coo coordinates under which (1) becomes constant coefficient 8u - a-7 +

n

E o: j,k=l

0U

a2u

=/(~, t)

j=~

n

n

~. ajk~j~kU--i E bj~jU-F( ~, t), jffil

j,k=l

(3) an ordinary differential equation in U. Similarly, if there are C °O coordinates in which (1) takes the form

- 8--t + j,kffil ~ Ajk(x) 0xj 8x~

+ E Bj(~)

0u

then Fourier transforms in the spatial variables yield

The emphasis of this paper is determining necessary and sufficient conditions under which the linear partial differential equation

~

n

E b j"'V ~ =/(~, t),

(2)

Ot

au

02u a~j a~----~+

O)

jffil

0u

"

02u

- a t + j,kffil Y" ajk 0xj ax k "

* Research supported by the NASA Ames Research Center under Grant NAG 2-366, Supplement No. 1.

0u

E b:~jb-~~ -/(x, t),

j,k=l

0167-6911/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland)

(4)

L.R. Hunt° R. Villarreal / Canonical coordinates for PDE

160

where ajk and bjk are constants, then Htirmander [5] ~hows that spatial Fourier transforms lead to 0U

at

j,k=l

-- ~

ajk j kU

bjkl~k~j=F(l~, t).

(5,

j,k=l

m

P

E x?- E

Gaaer generic conditions, this first order linear partial differential equation can be solved by the method of characteristics (i.e. ordinary differential equations) and inverse Fourier transforms. The most important case involves a positive .~emidefinite matrix (aij) of constant rank m, a~ld we shall call the corresponding equation (4) a Koimogorov equation. H6rmander writes the spatial partial difI,'2 ferential operator in (4) as V'm ,.,j. 1-~j + X0, where Xj and X0 are Coo vector fields on R ~. In fact, if the (Ajk(x)) matrix in (1) is symmetric, ~-onstant rank m, and positive semidefinite, the spatial operator can be written as m

E x? + x0.

Kolmogorov type, if the hypoellipticity conditions of HiSrmander are satisfied. The spatial operator in (4) is a Kolmogorov operator if it is hypoelliptic. We remark that both problems (i) and (ii) can be generalized to the partial differential operators

(6)

j=l

j=l

but we shall concentrate on the form (6). Moreover, we assume that X 1, X2,..., X m are linearly independent. Our principle tools are taken from the field of systems and control. However, the purpose of this paper is not to draw a parallel between controllability of systems of nonlinear ordinary differential eqt~ tions and hypoelliplicity of partial differential equations, as this has been well established in the literature and in conference presentations. As we mentioned previously, problem (i) is straightforward. Our work on problem (ii) is analogous to the study of coordinate changes to transform a nonlinear control system on R n,

j=l

For general partial differential operators of the form (6) H~rmander proceeds to prove necessary and sufficient conditions for hypoellipticity (i.e. Coo right hand sides imply Coo solutions). We require that our Kolmogorov equations be hypoelliptic. Weber [10] constructed fl~:~damental solutions for a class of equations related to those in (4). In this light the problems considered in this paper are: (i) Given the partial differential operator (6) find necessary and sufficient conditions so that there exist nonsingular C ~ coordinate changes (local-near the origin) on R" under which (6) becomes a constant coefficient partial differenti.,d operator. Standard differential geometry results (e.g. Spivak [9]) are employed and the results are of no surprise. The conditions are derived here for the sake of completeness. (ii) Given the partial differential operator (6) find necessary and sufficient conditions so that there exist nonsingular Coo coordinate changes (local-near" the origin) on R ~ under which X~, X2,..., Xm in (6) are transformed to constant vector fields and Xo becomes a linear vector field. This makes the partial differential operator (6) of

m

+ E

(7)

j=l

to an ',.-dimensional controllable linear system .2.*

x =

+ Ca.

(8)

Here Yc = d x / d t , f, gl, g2, .... g,, are C °O vector fields on R n, f ( 0 ) - 0, u = (u 1, u2,..., u,,,) consists of real-valued functions, F is an n × n constant matrix, and G is an n × m constant matrix. Also f, F, and u are obviously different objects in our control discussion than in our p.d.e, discussion. We rely heavily on the results of Krener [7] and Respondek [8], and, in fact, our research essentially moves their results from the ordinary differential equation setting to the partial differential equation setting. Nonlinear control system (7) is replaced by equation (6) with Xj taking the place of gy, j = 1, 2 .... , m, and X 0 taking the place of f. The linear system (8) is replaced by the Kolmogorov partial differential operator n'i

y'. ~2 + ~2,

(9)

j----1

where each Xj is a constant vector field and X0 is

L.R. Hunt, R. Villarreal / Canonical coordinates for PDE

linear. Here X1, X2,_..., X,,, correspond to the m columns of G and X0 corresponds F~? in (8). We want the span of the Lie brackets Xj,

[~'o, ~ ] ,

161

The three types of Lie derivatives satisfy the formula Lx(,dh, Y ) = ( L x ( d h ) , Y)

..-, (adn-lXo, Xj),

+ (dh, [ X, Y]).

(10)

j - - 1 , 2,..., m, to be R", so we make the corresponding assumptions on the Lie brackets of vector fields in (6). As noted before we also suppose that X~, -¥2,--., Xm are linearly independent. Section 2 of this paper contains basic definitions and consideration of those linear partial differential operators which can be made constant coefficient. In Section 3, necessary and sufficient conditions are derived which classify those linear partial differential operators that can be moved to the Kolmogorov type.

We motivate our study by the following example.

2. Constant coefficient operators

yl = In(1 + xl),

We begin with a set of appropriate definitions. If X and Y are C OO vector fields on R n, then the Lie bracket of X and Y is

moves (11) to the constant coefficient form

ay X av [ X , Y ] = ax - ~ ' Y '

where aY/ax and a X / a x are Jacobian matrices, x being the variable for R". Successive Lie brackets such as

[x,[x, rll,

Example 2.1. Consider the partial differential operator (1 + X l ,

(ad°X, Y ) - Y,

+

ax--'~ (11)

on R 3. The local coordinate change (near the origin)

a2

+

y2 = x : - x~,

y~ = x3

i~

(12)

(13)

ay~ ay3"

This is discovered in the following way. First we write (11) as (1 + x,)~-~xa (1 + x,)~-~xt + 2x3~-~x2 + ~X 3 (14)

[r,[x, rll, [[x,[x, rll, Y],

etc. can be taken. A standard notation is

and set

(adlX, Y ) = IX, r ] ,

(ad2X, Y) = [ X, [X, Y]], ...,

x,=

(adJX, Y) = [X, (adJ-'X, Y)] We let ( - , -) denote the dual product of one forms and vector fields. Given a Coo function h on R n we define the Lie derivative of h with respect to the vector field X as Lxh = (dh, f ) . Successive Lie derivatives are

LOb=b,

+ (1 + Xl) ~-~X ! + 2x 3

o 0

,

Xo= 2 x 3 .

(15)

Thus (11) becomes

06)

x~ + Xo.

Since X1 and Xo are linearly independent and the Lie bracket [ X1, X0] --- 0, standard differential geometry results (see [9]) imply that the transformation (12) takes

Llh=Lxh,

L2h = L x L x h , ..., LJxh = LxLJx-lh. Moreover, the Lie derivative of the one form d h with respect to X is L x ( d h ) _ ( i}(dh)* )* iiX dx X + ( d h ) a x , where * denotes transpose.

X1 to

and

X0 to

.

Hence, the partial differential operator (11) is moved to the constant coefficient form (13). We remark that the partial differential operator in equation (11) is nol~.hypoelliptic. Only in Exam-

L.R. Hunt, R. Viilarreal / Canonical coordinates for PDE

162

pie 2.1 and the following result do we consider partial differential operators E~=1Xf + Xo which may not be hypoelliptic. We now prove our result concerning transformation to constant coefficient operators. Again, this result is trivial from the differential geometry viewpoint.

If m - n, we can move

X1 to

iOi11 Ill Iiil ,

. . . . , ~tmtO

X 2 to

if and only if [Xr, Xs]=-O for all l < r , s < m . Setting Theorem 2.1. Given the C °O partial differential operator on R n

3'2 X0

m

E X2 + X0,

(17)

j=l

we find [Xj, Xo] is where X o, X1, X z , . . . , X,,, are linearly independent for m < n and X 1, X 2 , . . . , X m are linearly independent for m = n, there exists a non-singular (local) coordinate change on R n under which X o, X1, . . . , X m become constant vector fields if and only i f

aV2/0x._j+ t a3'./Ox._j +

[ X r, Xs]---0

forallO
(18)

Proof. If m < n results from [9] indicate there are nonsingular coordinate changes taking

X 1 to

i0il i01 ,

Xm to

3. Koimogorov operators

( n - m )-th place,

We examine the partial differential operator (6) where X 1, X 2 , . . . , X m are linearly independent and X 0 vanishes at the origin in R n. We derive conditions under which Coo coordinate changes (localnear the origin) exist taking (6) to the Kolmogorov operator (9). As stressed in the introduction, the main contribution of this paper is realizing that the results of Krener [71 and Respondek [8] in the nonlinear systems (o.d.e.) and control area can be applied to partial differential operators

.

11 t

We now study our problem (ii), the main consideration of this paper.

X2 to

I0~0 X o to

for j---1, 2 , . . . , n. Then Xo is a constant vector field if and only if [ Xy, Pro] - 0, j - 1, 2, . . . . n.

(n - m - 1)-th place,

n

E ,4j (x) j,~-I

[}2

n

a

ax, + E ""

5

(19)

-

t-

,0, if and only if (18) is satisfied.

The linear controllable system x=F~

+Gu

(20)

163

L . R . Hunt, R. Villarreal / Canonical coordinates f o r P D E

is replaced by the hypoelliptic Kolmogorov partial differential operator (with %k and bj, constant) ~' 82 E aj, axj ax--~+

j,kffi l

" O E bj, xj ax,"

j,kffi l

(21)

Of course we shall study operators (19) and (21) in our vector field notation. First we wish to examine parallels between system (20) and operator (21). For the control system (20) the Kronecker indices and eigenvalues of the F matrices are invariants under coordinate changes• For the operator (21) we introduce Kolmogorov indices and note that these and the eigenvalues of the B matrix are invariants. Canonical forms which parallel controllable canonical forms for (20) will occur in our work• If the matrix A - ( a j , ) in (21) has rank m, let X1, X2,..., Xm be linearly independent vector fields so that (21) becomes Y'~'=1~2 + X0. We set B ffi Jacobian matrix of the vector field X0. If the partial differential _operator in (21) is hypoelliptic (in this case .~j, BXj,..., B"-1.Xp j = 1, 2,..., m, span R") we introduce the following process: (1) Write out the grid Q

m

Xl

x2

axl

BX2

Bn-Ix1

Bn-Ix2

-

--.

We ask if the partial differential operator (6) is a coordinate change away from the Kolmogorov partial differentifl operator (9) having given indices 11, 12,..., 1,,,. Before stating the general theorem we present an example. Example 3.1. Consider the partial differential operator

82

02

82

4x20x---~2 +4x3ax2 0x3 +

~.,'l'~ (22)

on R 3. We write this as

+

O

+

a

•

(23)

-

x+, aX,.

...

we also have the integers 11, 12 . . . . , I m associated with (9). By assumption i I +/2 + "'• +lm "--n.

Letting

Bn-lXm

(2) Start at Xt and move left to fight across the first row, then start at BX 1 and move across the second row, etc. (3) Throw out arty vector field that is a linear combination of the preceAing vector fields in the grid. Discard all vector fields in the column below tiffs vector field• (4) Continue until n linearly independent vector fields are found and all others have been discarded. (5) Let lj = number of entries remaining in the j-th column of the grid,__j = 1,2,..., m. (6) Renumber X 1, )(2,..., Xm, if n~essary, so that 11 >_ ! 2 >__ . . • >__! m. Definition. The integers !~, 12,•.., #m are called the Kolraogorov indices of the partial differential operator (21). If the vector field notation for (21) is (9)

we find (23) becomes

x ] + x0.

(24)

The local coordinate changes ~1

xl - x~ + 2x2x~ - x 4

~2 = x2 - x~ -- L x o x l , X3 "- X3 --

Lxo~2'

take (24) to ~:,2 + Xo,

(25)

where

I'nl if1--

P~21

i7i Lo and

X0 =

•

164

LR. Hunt, R. Viilarreal / Canonical coordinates for PDE

This yields the Kolmogorov partial differential operator 02

0

_

0

are linearly independent and have zero Lie brae. kets, we have coordinates so that: X 1 becomes

(26)

0.~='====~='[=.~2"=~i "at- X 3 0.~, 2 " XI =

[!1

Since B =

0 0

[Xo, XI] becomes

"~1, Bgl, B2XI span R 3, and the single Kolmogorov index is 11 = 3 = n. In equation (24) we remark that X 1, [X o, X1], and (ad2Xo, X1) span R 3 (near (0, 0, 3)? and [(adrXo, )(1), (ad'Xo, X1)] - 0

Ill i01 1

0 1 , 0

0

=

i

0

forO._-'r, s < 3 .

We now state and prove our main result. The general proof will follow from analogous results flora [7] and [8], but we shall present a proof in the case m = 1 for some sort of completeness. Theorem 3.1. The linear partial differential operator (6), with )(1, X 2.... , X m linearly independent on R n at:d Xo(O) = O, can be transformed by nonsingular coordinate changes (local-near the origin) to the Kolmogorov partial differential operato, (9) having indices i 1, l z , . . . , 1,, if and only if (or) tire set

...; (adn-lX o, X1) becomes

(adn-1~:o. X I ) =

If

81]

82 ] B

Xo=

•

I,

~_1 II

{ X,. [ X o, X,] . . . . . (ad t'- 'X o, X,),

X2, [X o, x , l .... ,(adt'-'Xo, ,Y2),

.... Am, [Xo, x,,l,...,(ad'--'Xo, Am)} is linearly independent; and

¢~~.a) .... tha Lie brackets of. every ~r'a-ir af vectors fields in

{ X1, [Xo, x l l , . . . , ( a d 6 X o , X1), X2, [ Xo, x , l , . . . , (adt'X o, X,) . . . . .

x,,,

[Xo,

Xm],...,(ad"Xo, Xm)}

is zero.

Proof. (For m = 1, 1~ = n, and the operator X~ + Xo): Since X,, [X o, All , ..., (ad--lXo, X1 )

then [ Xo, X1] = 0 implies D8,. 3~. = O,

i=l,2,...,n-2,

n,

08._i

~=1;

(adZXo, X~) = 0 implies 08; - ~

~,,'-1

= 0,

i=l, 2,...,n-3,

n-l,n,

08._2 ~ = 1 ; • "" ; (ad"-lXo, XI) = 0 implies 08~ O~--~=0'

i=2,3,...,n,

081 a~ 2 - 1 .

L R . Hunt, R. Viilarreal / Canonical coordinatesfbr PDE m

Now [(ad"Xo, .X'I), X l] = 0 yields (}2~i

=0,

i, j = l ,

2,...,n;

[(ad"Xo, X~), [ X o, X~]] = 0 yields a28~ =0,

i, j = l , 2 , . . . , n ,

...; [(ad"Xo, X]), (ad"-]Xo, X~)] = 0 yields a28~ ~ = 0 ,

i, j = l , 2 , . . . , n .

Hence Xo is a linear vector field as promised and ~2 + Xo is a Kolmogorov partial differential operator. E! In the above proof,

and btl b21

1 0

0 1

:

x0=

"."

0

0

b,1

0

0

0

Xl x2

".

b(n_l) 1

and sufficienll conditions that a second order linear partial differential operator be a coordinate change ,~way from a Kolmogorov operator. Future research will be in two directions: (1) Expand the transformations used to include 'appropriate types' of feedback. This research is presently ,mderway, and first thoughts were to include results in this paper• However, the process of feedback, as applied in transformation theory, is not weU addressed in the partial differential equation literature, where coordinate changes are standard fare. Therefore, we decided that separate papers are appropriate. (2) Extend all results to the discrete setting. A Ph.D. student of the first author is currently working on this project.

References

i0i! w

165

m

"'"

0

I

1

Xn-- l

0

Xn

.

I

and ~2 + Xo is in a canonical form• If we set /

Zl = ~'1, ~ Z2 = L~oXl, " " ,

z,, = L~o~,,_l,

we have a canonical form in which the matrix defined by t h e new Xo is in rational canonical form and X i is as before. If m > 1, we get the analogue of controllable canonical form. We have developed a theory giving necessary

[1] l~. Cotton, Sur ies invariams, diff~rentiels de quelques &luations Lm~aites aux d~riv~es partielles du secon{J ordre, Ann. Sci. Ecole Norm. Sup. 17 (1900) 211-244. [2] R Courant and D. Hilbert, Metho~ in Mathematical Physics, Vol. H: Partial Differential Equations (Interscience, New York, 1962). [3] G.A. Fredricks, The get, merry of second order linear partial differential operators, Comm. Partial Differential Equations $ (6), (1983) 643-655. [4] P.R. Garabedian, Partial Differential Equations (John Wiley and Sons, New York, 1964). [5] L. H~rmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967) I47-171. [6] A.N. Kolmogorov, Zfifallige bewegungen, Ann. of Math. 35 (1934) 116-117. [7] A.J. Krener, On the equivalence of control systems and the linearization of nonlinear systems, S l A M J. Control 11 (1973) 670-676. [8] W. Respondek, Geometric methods in linearization oJ" control systems, in: Banach Center Publications, Semester on Control Theo~, Warsaw (Sept.-Dec. 1980). PI [d. Spivak, A Comprehensive lntrodt~ction to Dz~rential Ceometry, Vol. I (Publish or Perish, i~,erkeley, CA, 1970). [101 M. Weber, The fundamental solution of a degenerate p~xtiai differemial equation of parabofic type, Trans. A mer. M~th. Soc. 71 (1951) 24-37.