MOTION PLANNING FOR A LINEARIZED KORTEWEG-DE VRIES EQUATION WITH BOUNDARY CONTROL

Copyright 0 2002 IFAC 15th Triennial World Congress, Barcelona, Spain

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MOTION PLANNING FOR A LINEARIZED KORTEWEG-DE VRIES EQUATION WITH BOUNDARYCONTROL BQatrice Laroche * Philippe Martin **

* Laboratoire des Signaux et SystZmes, CNRS/Sup&lec/Universite‘ Paris-Sud, 3 rue Joliot-Curie, 91190 Gif sur Yvette, France 1 a r o che @ 1ss , sup e 1e c ,fr ** Centre Automatique e t SystZmes, Ecole des Mines de Paris 35 rue Saint-Honor&, 77305 Fontainebleau, France m a r t i n @ c a s ,ensmp , fr

Abstract: Explicit motion planning of a linearized Korteweg-de Vries equation with boundary control is achieved. The control is obtained through a “parametrization” of the trajectories of the system which is a generalization of the Brunovsky decomposition for finite-dimensional linear systems. Copyright @ 2002 I F A C Keywords: boundary control, motion planning, Korteweg-de Vries equation, Brunovsky form, flatness, controllability

INTRODUCTION In this paper we illustrate a method for explicit approximate motion planning on the linearized Korteweg-de Vries equation with boundary control xt x,,, x, = 0 X(0,t) = X(1,t) = 0 (1) X,(l,t) = u ( t ) .

I

+

+

More precisely, given any final time T, any initial state EO E L 2 ( 0 ,1) and any final state ET E L2(0,1),we compute an open-loop control t E [O,T]H u ( t ) steering the system from EOto an arbitrarily small neighborhood of ET. In other words, we prove approximate controllability for every time T. Though a much stronger mathematical result can be found in (Rosier, 1997) (exact controllability of (1) and of the original (nonlinear) KdV equation), the interest of our approach is an explicit construction based on a “parametrization” of the trajectories of (1). The idea is reminiscent of flatness (Martin et al., 1997) and can be seen as a generalization of the Brunovsky decomposition for finite-dimensional 225

linear systems. The present point of view, introduced in (Laroche and Martin, 2000), is related to (Laroche et al., 2000), but the approach is rather different; in particular, it is not restricted to all the boundary conditions except the control being on the same side. It seems applicable to many linear boundary control problems (Laroche, 2000).

1. PARAMETRIZATION OF TRAJECTORIES AND MOTION PLANNING Definition 1. A linear boundary control problem on [0,1]x [0,TI is parametrizable if there exists a family ( a i ) i E N of functions in L 2 ( 0 ,I), a family (ai)iEN of real numbers and a subspace G of C”(0, T) such that: 0

Vy E

G, the series 00

i=O M

(3) i=O

0 0

It can then be shown that the open-loop control

are normally convergent and define a (classical) solution of the problem on [0,1] x [0,T] (j contains all the polynomials on [O, T ] span(ai, i E = L 2 ( 0 ,1).

+m

w}

i=O

exactly steers the system from no to IIT. Moreover, any C2 approximate control such that

If the subspace (j is big enough, namely contains Gevrey functions of order > 1, such a parametrization directly yields an explicit solution to approximate motion planning (provided the boundary control problem is well-posed), see (Laroche et al., 2000; Laroche, 2000) for more details.

in particular the truncated series N i=O

Recall that y E C m ( O , T ) is Gevrey of order o if

for N large enough, approximately steers the system from EOto 5:~.

If y 5 1, y is analytic (entire if y < l ) ,hence has a convergent Taylor expansion around any point. But if y > 1, y has around at least one point a divergent Taylor expansion; the larger o, the “more divergent” the expansion. A fundamental example for motion planning is the “smooth unit step” defined by aY(0) := 0, Q Y ( l ):= 1 and

f

ft Y\”J

-1

Z ( z , t ) := X ( z , t ) + G ( z ) ~ ( t ) , where

, -

, ), Considering controls t E [0, T ]c-) u ( t )in C 2 ( 0T (1) is a well-posed boundary control problem (see for instance (Curtain and Zwart, 1995, section 3.3)). Indeed, setting

1.

.-

loexp ( (( 1

2. THE PROBLEM IS WELL-POSED

G ( z ):= s i n z -

” -J-

) d7

t)ty

is the solution of

G“‘ + G‘

Whatever y > 0, QY is in C“O(0,l)and is Gevrey of order 1 l / y ; its Taylor expansion around 0 and 1 is divergent.

+

=0

G(0)= G(1) = 0 G’(1) = -1,

Assume now the system is t o be steered from the initial state

we get the abstract evolution equation

Z = FZ+Gu,

vx E [O, 11, X ( z , O )= Eo(z), Eo E L2(0, l ) ,

where the L2( 0 , l ) valued operator F is defined on its domain

to a filial state “arbitrarily close” to

D ( F ) = { f E H ~ ( oI), ,

vx E [O, 11, X ( m , T ) = E:T(z), E:T E , L ( O , 1).

f ( o ) = f ( i )= f’(i)= o}

by F ( f ) = -f”’ - f ’ . A simple but tedious computation (Laroche, 2000, section 6.3) shows that F is invertible, maximal and dissipative, hence that F is the infinitesimal generator of a strongly continuous semigroup of contractions (see for instance (Pazy, 1983)).

Thanks to the density of the q ’ s , there exists given any E > 0 polynomials n i=O n

&(z) = e q i a i ,

sin 1 (1 - cosz) 1 - cos 1

4i E

i=O

3. “DIRECT” APPROACH

such that IlEo - no11 5 E and l l E : ~ - I I T ~5 ~ E (11.11 denotes the usual norm on L2( 0 , l ) ) .On the other hand the function

Y ( t ):= x p i -ti (1 i=O

-

Q-/

(;))+%-a7(t

-

T)i

To determine the ai’s and the a i ’ s , we plug (2)-(3) into the system equations (1) and sort along the derivatives of y. Since by definition (j contain all the polynomials, we get the sequence of ordinary differential equations

();

Z!

is Gevrey of order 1 + l / y and satisfies

Y(i)(O) =pi, Y(i)(O) = 0,

Y ( i ) ( T= ) qi, Y ( i ) ( T= ) 0,

i

,

= 0,.. . n

vi E

i > n.

w,

a;’

+ a!;+ ai-1

a!i(O) = Ui(1) = 0

Ui(1)

226

=0

= ai,

where we have set and G this means

a-1

Using (5), the Fredholm determinant of F is thus

:= 0. By definition of F

A F ( A ) = f3(0, A). We now have compute the coefficients in the expansion of AF. In this example, we could do this directly by first computing f3(0, A) in closed form. However, this is very tedious and we use instead a more general alternative approach. Indeed, it can be seen that

hence

M

As pointed out in (Laroche and Martin, 2000; Laroche, 2000), this construction is very similar to the Brunovsky decomposition in finite dimension. Indeed, consider the finite dimension system

X

=FX

+ FGu,

i=O

where f3(z): = f 3 ( z , 0 ) = 1 - c o s ( 1 - z )

X E R”,

and W is the operator defined on L 2 ( 0 ,1) by

with F an invertible n x n matrix. This system is to be thought of as (1) since it is transformed by

Z ( t ) := X ( t ) + G u ( t ) in other words

into

1

Z=FZ+GU If we want a parametrization of the form,

(Wg)(z.)=

n-1

n i=O

a1, . . .

a0 =

a2. -

-

, an in R”we readily find

-CLOG F - l a .2-1

-

-

l)g(M.

Moreover, taking ai := (Wif3)(0) ensures the series (3) is normally convergent when y is Gevrey of order at most 3. The proof is exactly similar to that of proposition 2 in the following section. We are now in position to compute the a i ’ s . Unfortunately, we have not yet been able to directly prove the convergence of (2) and the density of the a i ’ s , because our present proof requires F to be a Rieszspectral operator, a fact we have not been able to establish. In the following section we prove by an “indirect” approach that (1) is parametrizable. We conjecture (and have experimentally checked the first terms using Maple) that the a i ’ s obtained by the “direct” and the “indirect” approach are identical.

i=O

with

(cos(z - 8

u ~ G , i = 1,.. . , n - 1

Moreover, the a i ’ s must be (up to a constant factor) the coefficients of the characteristic polynomial of F . If a1, . . . , an form a basis (i.e., if the system is controllable), the system written on this basis is in controller canonical form. The intuitive idea in infinite dimension is thus to take for the a i ’ s the coefficients of the Fredholm determinant of the operator F , i.e., the entire function AF whose zeros are the eigenvalues of F (such an entire function, which obviously generalizes the characteristic polynomial, exists because F is the inverse of an integral operator). Let f i ( z ,A), i = 1 , 2 , 3 be the fundamental solutions at z = 1 of f”’ f ’ Af = 0, (4)

+ +

05t

05

0

I 0002

0004

0006

0008

001

0012

0014

0016

0018

IPrnPF

Fig. 1. Open-loop control U ( t ) .

fl(0, A) f 2 (0, A) A F ( A ) := fl(1,A) f 2 ( 1 , A) fi(L A) f81, A)

f3(0, A) f3(1, A)

Before going further, we illustrate the relevance of this motion planning method with a numerical simulation. The goal is t o steer the system from the initial steady state X(z,O) = 0 to (a

=0

f 8 , A) 227

is the solution of

neighborhood of) the final steady state X ( x , T ) = sin 1 (1 - cosx) - sinx, with T := 0.02. Notice that every steady state is a multiple of ao, hence of G; in other words, we do not need t o compute the a i , i > 0 to steer the system from rest to rest. We have thus taken (with the notations of

B”’+ B’= 0 B ( 0 ) = B(1) = 0 B’(1) - B’(0) = -1, we get the abstract evolution equation

Z=AZ+BU where the operator A is L 2 ( 0 ,1) valued, and defined on its domain

0.8 0.6

D(A) = { f

commande (pente au point 1) 0.4 0.2 u

0 0.2 0.4

E

~ ~ (11,0f ( ,o ) = f ( i )= f/(i)-fyo)

( K g )(x):=

0.8 1 0.8 0.015 0.4

0.01 0.2

0.005 0

0

temps

k ( x , 6 ) :=

Y ( t ):= @-/

();

and as the approximate control the truncated series i=O

The results (with y := 1) are displayed on figures 1 and 2.

We now tackle the same problem but first perform the boundary feedback

vi E w,

+ v(t),

which symmetrizes the boundary conditions and transforms (1) into

I

where we have set

+

xt x,,, x, = 0 X ( 0 , t ) = X( 1,t ) = 0 X,(l,t) - X,(O,t) = v ( t ) .

. 1-x sin 2 . 1-6 sin 2 ~

. if 0 5

a!;’

+ a!;+ ai-1 = 0

q ( 0 ) = q ( l )= 0 a!;(l)- a!;(o)= u i ,

a!-1

a0 =

-uoB

a!i =

KCei-1

:= 0. In other words

-UiB,

i 2 1,

(6)

The motivation for this feedback is t o give rise to an operator with antisymmetric hence normal algebraic inverse, easily shown to be Riesz-spectral. This feedback is just a technical intermediary and need not be actually performed. As mentioned earlier, we conjecture it does not change the a i ’ s . The problem (6) is also well-posed: setting

Z ( x , t ):= X ( x , t ) + B ( x ) u ( t ) , where sin 1 1 - cos 1

228

el (0, A)

e2 (0, A)

AA(A):= el(1, A) 4(1,A)

e 2 ( L A)

ei(L A)

-

e3 (0, A)

e3(1, A) ei(0, A) .;(I, A)

65 2,

. ifx.<<<.

Exactly as in the previous section, we get the sequence of ordinary differential equations

4. “INDIRECT” APPROACH

+

(1 sin 2

~

K is compact (Curtain and Zwart, 1995, Theorem A.3.24) and normal. Therefore (Curtain and Zwart, 1995, Theorem A.4.25), A is closed, its eigenvalues are isolated and its eigenvectors form an orthogonal basis for L 2 ( 0 ,1).Since moreover (Laroche, 2000, section 6.3) its eigenvalues are simple, A is a Riesz-spectral operator. Since K is antisymmetric, all the eigenvalues of A lie on the imaginary axis and are pairwise conjugate; therefore (Curtain and Zwart, 1995, Theorem 2.3.5), A is the infinitesimal generator of a strongly continuous semigroup of contractions.

, 1 < 1 + l / y 5 3.

u ( t ):= X,(O,t)

6 . x-6 sin - sin 2 2 x x-6 sin - sin2 2

sin

Fig. 2. Motion of the system. section 1)

k ( x ,6)g(6)d6,

with the antisymmetric kernel

0.02 0.6

{

s

1

état initial

0.6

x

= 0:

by A ( f ) = -f”’ - f ’ . Its algebraic inverse is the integral operator K on L 2 [ 0 1 ,1 defined by

état final

(7)

On the other hand f3 = clel+c2e2+c3e3 for some real numbers el, c2, c3; elementary computations show that C l = e2(L

A)eb(l, A)

-

and third derivatives. It thus defines a function in C3, which is easily seen to satisfy (4) with the required initial conditions. The proposition is now obvious, the Weierstrass order resulting from (Saks and Zygmund, 1965, chapter 7). 0

e3(L A)ei(l, A) = f3(0, A).

This implies

AA(A)= es(1, A)

+ ~ ( 1-A).,

As in the previous section, we compute the coefficients in the expansion of AA in term of iterates of E3(z) := e3(z,O) = 1 - cosz by the operator V defined on L2(0,1) by -f”’ - f’ = g f=Vg H f ( 0 ) = f ’ ( 0 ) = f ” ( 0 ) = 0,

Proposition 3. Vy E C-(O, T ) with Gevrey order (at most) 3, the series

in other words

are normally convergent and define a classical solution of (1) on [O, 11 x [O, TI.

00

i=O M ..

{

u ( t )= E a ; ( l ) y ( i ) ( t ) i=O

PROOF. The key point is t o establish the L2

Proposition 2. AA is an entire function,

estimate

M

i=O

Since A is antisymmetric, its eigenvalues are purely imaginary and pairwise conjugate. Moreover 0 is not an eigenvalue. Denoting the eigenvalues by X i , we can therefore index them on Z* so that A-i = X i and

where the coefficients ai are defined by

Moreover the ai’s satisfy the estimates

3 M , R > 0 , V i E W,

ai

Ri 5 M-(3i)!’

hence AA is of Weierstrass order (at most)

xi -<

Xi+l

a a

i.

(remember all the eigenvalues are simple). Let ( u i ) i t ~ be * the associated Hilbert basis of eigenfunctions of (this basis exists since A is Rieszspectral). By proposition 2, AA is entire of Weierstrass order 5 1, hence (Saks and Zygmund, 1965, chapter 7) can be factored into the normally convergent infinite product

Identifying this product with the expansion of AA, we readily find for 1 E W

=M

[3i+3

(3i + 3)! Using the same technique, we get similar estimates for (Vo E3 ) , (Vo E3) ’, (VoE3) ” and (Vo E3) ”’. ’

On the other hand for i E

W i

We then claim

Z ( V ~ E ~ ) ( ~ )where A ~the.ck’s are the coordinates of ao, i.e., 00

e3(z, A) =

i=O

Indeed, the previous estimates ensure this series is normally convergent, as well as its first, second

a0 =

c

kt%*

229

CkUk.

readily obtain (up to renaming M and R ) similar estimates for a: and a:. A similar estimates for a:' is then derived from (7).

Indeed, assuming (8) holds true, ai+1 =

Kai

ai+1 + -a0

a0

i+1

i=O M

2 IICeill

=

1

2

c

i ICIcl2

IctZ"

la01

Ixii12

tion for any y Gevrey of order at most 3, which ends the proof. 0

c a j x ; j=O

We recognize in the last factor the expansion at order i of AA(&). Since a21+1 = 0 for 1 E W, 21

c4

Proposition

21+1

ca3xi

=

j=O

= la01

1+

j=O

c

%l,,,,%l+o P i l < ...< i l

i 1

I

span{ai, i E

W} = L 2 ( 0 ,1).

xi ' ' '

I& I

2

%l,,,,%l+o P

i l < ...< i l

2

2000, theorem 1) (see (Laroche, 2000, section 7.4) for more details).

lh12

=1a011-c i 1

I

' ' '

IXil

I

2

Curtain, R.F. and H.J. Zwart (1995). An Introd u c t i o n t o infinite-Dimensional L i n e a r Syst e m s Theory. Text in Applied Mathemtics, 21. Springer-Verlag. Laroche, B. (2000). Extension de la notion de platitude & des systkmes dkcrits par des kquations aux dkrivkes partielles linkaires. PhD thesis. Ecole Nationale Supkrieure des Mines de Paris, France. Laroche, B. and Ph. Martin (2000). Motion planning for a 1-D diffusion equation using a Brunovsky-like decomposition. In: Proc. of t h e International S y m p o s i u m MTNS 2000. Perpignan. Laroche, B., Ph. Martin and P. Rouchon (2000). Motion planning for the heat equation. Int. J. Robust Nonlinear Control 10, 629-643. Martin, Ph., R. Murray and P. Rouchon (1997). Flat systems. In: Proc. of t h e 4 t h European Control Conf.. Brussels. pp. 211-264. Plenary lectures and Mini-courses. Pazy, A. (1983). Semigroups of L i n e a r Operators a n d Applications t o Partial Differential Equations. Vol. 44 of Applied Mathematical Sciences. Springer-Verlag. Rosier, L. (1997). Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. COCV 2, 33-55. Saks, S. and A. Zygmund (1965). A n a l y t i c Functions. Paristwowe Wydawnictwo Nankowe.

2

This implies

IlwlI , I I ~ 2 1 + 1 l l 5

la211

Thanks to property 2, we finally get (up to renaming M and R ) the estimate

We next find a similar estimate for the uniform norm. By the Cauchy-Schwartz inequality

1i. 1

II(K+II

4.

j=O

21

cajxi

Finally, we have:

=

5 llk(x, ,111 II%-lll

k ( x , b ( W 6 1

5 SUP

[0,1l2

MY .)I II%-lll ,

which implies (up to renaming M and R ) Di

Since the kernel k is twice differentiable with respect to x with a bounded second derivative, we

230