Variational and Hamiltonian Control Systems

Lecture Notes in Control and Information Sciences Edited by M.Thoma and A.Wyner 101 P. E. Crouch A. J. van der Schaft ...

Author: P. E. Crouch | A. J. Van Der Schaft | A.J. van der Schaft

78 downloads 913 Views 7MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

Lecture Notes in Control and Information Sciences Edited by M.Thoma and A.Wyner

101 P. E. Crouch A. J. van der Schaft

Variational and Hamiltonian Control Systems

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

Series Editors M. Thoma • A. Wyner Advisory Board L D, Davisson • A. G, J. MacFarlane • H. Kwakernaak J. L. Massey • Ya Z, Tsypkin • A. J. Viterbi

Authors Dr. P. E. Crouch Dept. of Electrical and Computer Engineering Arizona State University Tempe, AZ 8 5 2 8 7 USA Dr, A. J. van der Schaft Dept, of Applied Mathematics University of Twente P. O. Box 217 ?500 AE Enschede The Netherlands

ISBN 3-540-18372-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-38?-18372-8 Springer-Verlag New York Berlin Heidelberg Library of Congress Cataloging in Publication Data Crouch, R E. Variational and Hamiltonian control systems. (Lecture notes in control and information sciences; 101) Bibliography: p. 1. Control theory. 2. Calculus of variations. 3. Hamiltonian systems. I. Schaft, A. J. van der. II. Title III. Series. QA402.3.C74 1987 629.8'312 8?-26421 ISBN 0-387-183"72-8 (U.S.) This work is subject to copyright, All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin, Heidelberg 198'7 Printed in Germany Offsetprinting: Mercedes-Druck, Berlin Binding: B. Helm, Berlin 2161/3020-543210

PREFACE

This monograph

grew out of a combined

the second author,

concerning

in terms

variational

rated

of their

by the Hamiltonlan

Classical

Mechanics.

conjecture

and

the

adjolnt

control

of independent

The main

In

extension, control

interest

and results

concepts

whose

the

considering

on

of this monograph

(0) we give

a brief

emphasis

the relations

where

(theoretical)

study physics

chapter

and

Hamlltonlan system

wlth

presented,

as well as some open problems.

Enschede,

June 1987

are

introduction

monograph

Tempe

(7)

of

was moilProblem

version

of

the

the

prolonged

variational

system,

in

of this

system and

are, as we

to

the

believe,

theory.

In chapter

the

concepts

to any nonlinear

particular Indeed,

conjecture

by

systems

interest seems not to be confined

based

(6).

notions.

Thls

formulated

control

Problem as well as by the Inverse

wlth

on

behavlour.

particular

systems

to control

a conjecture)

of Hamlltonlan

of proving a slightly modified

concepts,

systems.

Hamiltonlan varlablonal

input-output

Realization

some

effort to p r o w

characterization

In the course

we developed

Hamiltonlan

the

discussing

control

some

control

physical systems theory

possible

control

and control Is

(I)

in chapters

to Hamiltonian

between

and

contained

one

meet.

extensions

of

We to

to

systems, theoretic

the

places

conclude the

the

theory

CONTENTS

Chapter 0

INTRODUCTION

Chapter I

THE HAMILTONIAN REALIZATION PROBLEM

12

Chapter 2

VARIATIONAL AND ADJOINT VARIATIONAL SYSTEMS

33

Chapter 3

MINIMALITY OF THE PROLONGATION AND HAMILTONIAN EXTENSION

39

Chapter

THE SELF-ADJOINTNESS CRITERION

47

Chapter 5

THE VARIATIONAL CRITERION

60

Chapter 6

GENERAL NONLINEAR SYSTEMS

85

Chapter 7

FINAL REMARKS AND SOME OPEN PROBLEMS

96

References

116

0.

Of

central

importance

INTRODUCTION

in the modelling

Euler-Lageange or Hamiltonian equations. a

very

large

class

of

conservative

of

physical

systems

are

the

classical

These equations describe the dynamics of

physical

systems,

Including

mechanical

and

electromagnetic systems, and l le at the heart of the theoretical framework of most physics. eases

Although the conservation of energy is usually an idealiZation,

in many

the neglectlon of dissipation of energy (friction, damping) forms a natural

startlng point. Let us consider, for example, a conservative mechanical

system with n degrees of

freedom, locally represented by n (generalized) configuratlon variables ql,...,qn. The Euler-Lagrange equations are the following well-known set oC second-order dlfferentlal equations

(0.1)

d

dt

(eL)

_ B__~L = F I

a~l

i

=

1 .....

n

aqi

where L(q,q) - L(q|, .... qn,ql ..... qn) is

the

Lagrangimn

of

the

system.

In most

mechanical systems the L a g r a n g l a n is the dlfference of a kinetic energy T(q,q) and a potential energy V(q)

(0.2)

L(q,q) - T(q,q) - V(q)

where T(q,q) iS quadratic In the generalized v~locitles

T(q,q) = ~I qTM(q)q

(0.3)

for some posltlve-definite matrix M(q). In this case the Euler-Lagrange equations specialize to

(0.4)

and

~ (3T) d~ ~qi

3T aqi

_ av

represent

the terms

~

av + Fi, ~ql

the i n t e r n a l consorvatlve

p o t e n t i a l ) forces i n the system. (generalized)

external

forces

i = I ..... n

( i . e . derivable from

a

F l n a l l y the vector F = (F I . . . . ,Fn) denotes the

acting

on

the

system

(ql . . . . . qn )" In the (mathematiCal) physics literature the external

while

in

forces Usually

configuration

are seen as

2 Riven functlons two parts:

of

time. Consequently,

one "maximal"

the external

component which

forces are often split into

is derivable

from a potential

function,

and so can be added to the internal forces, and remaining non-conservative forces. Alternatlvsly,

in stochastic mathematical physics the external forces are modelled

as stoohastlo variables [Bi]. In systems and control theory the approach, however, is quite different.

Usually

control Or input variables;

(some 0£) the external

derlng the Influmnce o f the environment interested

in

the

way

the

forces will be interpreted as

i.e., "arbltrary" functions of time. Instead of consl-

system

will

on the system as given, one is primarily react

to

difiCerent

external

forces.

Of

course this is intimately related to the fact that in control theory one wishes to prescribe the beha vlor of the system, instead of only describing It [BI]. In general

not all degrees of freedom are dlrectly accessible to control action,

resul~ing in Euler-Lagrange equations of the form

(0.5)

d (B_L__) BL

i = 1,...,m

(' Ui

i = m+1,...,n

3q i

where now u - (uI,...,U m) are the controls or Inputs

(i.e.,

"arbitrary"

functions

of time). We call (0.5) a Lagranglan control system. As is well-known,

the Hamlltonlan equations of motion are obtalned from the Euler-

Lagrangs equations

(0.6)

In

most

Pl oases

(0. I) by defining the generalized momenta

a6i the

transformation

from (41 . . . . . 4 n) t o (Pl . . . . . pn ) i s

a

(local)

diffeomorphlsm,

allowing us to transform the Lagrangian L ( q , q ) I n t o the Hamiln tonlan H(q,p) = ~ plql - L ( q , q ) (the Legendre transformation), and the (secondi=I order) Euler-La~range equations into the set o f flrst-order differential equations

~i = ~Pl &M(0.7)

i =

~ql which are

+

I ..... n

Fi

called the Hamiltonlan

equations

of motion.

gi%~en as in (0.2)-(0.3) the Hamlltonlan becomes

In case the Lagranglan

is

H ( q , p ) = ~1 pTM-1 ( q ) p + V(q)

(0.8)

and so denotes the

(Internal)

energy.

Finally

t h e Lagranglan c o n t r o l

system ( 0 . 5 )

results in the Hamlltonian control system

=

i = I ...,n

aH

(0.9)

+ful

~H

Pl =-~ql For

I = 1 ..... m

i=m*l

Lo

(0.5) as well as (0.9) we have assumed

1,...,m,

are

directly

coupled

to the

particular form Is not invarlant

.....

for simplicity that the inputs u i, i =

first m degrees

under a nonslngular

of freedom.

Of course

hhls

change o£ configuration coor-

dinates

(o.1o) with

% = ~i(~, ..... ~n ) the

checked,

i : I ..... n

Jacoblan De(q) everywhere under

such

a

coordinate

non-singular.

In

transformation

the

fact,

as

Lagrangian

can

b~

easily

control

system

transforms into

- ~

(o. tl)

EL-~-iJ

=

a~I

uj

j~1

~

i

=

1,...,n

a~ i

with L(q,q) = L(q,q), while the Hamiltonlan control system

(0.12)

aH

where ~(~,~)

=

m

a~j

(0.9) becomes

I = 1 . . . . ,n

H(q,p).

Let us from now on concentrate on Hamlltonlan

control systems.

Notice that

suggests

control

to systems

form

we

enlarge

the

class

of Hamiltonlan

systems

(0.12) of

the

~Ho (0.13)

~H 0

m

÷

Uj

I :

~Hj

1,...,n

with HO(q, p) the internal Hamiltenian, and H~(q), j ~ 1,...,m, arbitrary (smooth) functions. In particular, this form iS clearly invarlant under a change of configuration coordinates. We shall even go a little bi~ further. Part of the power of Hamiltonlan formalism is to regard

the

generalized momenta Pl on the same footing as the generallzed

configuration coordinates qi" Consequently, one does not only allow for transfor8L motions of the configuration coordinates ql (with P l ~ --T resulting in a bransforma~lon

of

t h e pl), but

(q,p), which leave canonical

one considers

the Hamiltonlan

transformations.

Under

~ql trans~v ~^rma tlon s all coordinate

form of

such

a

tha equations

general

Invarlant,

canonical

{q,p)

i.e. the

transformation,

the

functions HI,.o.,H m become functlons of q .~nd p, and therefore we define a general (afflne) Hamiltonlan control system as

3H O :

m -

~Hj

~

U j 3p i

i :

(0.14) ~H 0

n +

where

the

1,...,n

~Hj Uj

"~

functions H0,H I,...,H m are

all

arbitrary

functions

of

q and

p.

(See

Example 3 for a physical interpretatlon.) Notlc9 that a general Hamiltonlan system tlme-varylng

Hamlltonlan

dlf ferantlal

(0.14) c a n equations

also be regarded as a set of governed

by

the

tlme-varylng

Hamiltonlan m

40.15)

Ho{q,p)-

~. uj(t)Hj(q,p) j-1

This can be interpreted in the foilowlng abstract way (see also [BuI,Bu2, VI])° The possibility

of oontrolllng the system

with HamIltonlan (internal energy) HO(q, p)

rests on the ability to exchange energy wlth the environment along some external channels.

This can be regarded as the physical basis of conbol. The exchangeable

energy along the j-th channel i s

of the form Hj(q,p), and ui denotes the strength

of thlS energy exchange. Indeed, It is ~ s i l y

deduced that

dHo -

(0.16) In

dt

physics

the

~ uj(t)

j~1 Hamiltonians Hj(q,p), j

are

called

interaction

or

coupling Hamlltonlans.

Up to now we have not yet defined outputs y of a Hamlltonlan course

control system. Of

nothing forbids us to consider as outputs arbitrary functions of the state

x : (q,p). However, there Is a natural set of outputs associated to every Hamlltonian control system (0.14), namely the interaction Hamlltonians themselves:

(0.17)

yj : Hj(q,p)

There

are many

J = I ..... m

good reasons

for

doing

this.

First

of

all

with

this

choice

of

outputs we obtain From (0.16) the energy balance dHo d---~=

(0.18)

~

" Ujyj.

j=1

Hence the decrease or increase of the internal energy of the system Is a function of the work

inputs and

of Wlllems

(the tlme-derlvatlvos on dlsslpatlveness

of) the outputs only_.

[W3].)

(Compare with the

For example in the simple case (0.9)

where yj = qj, J = 1,...,m, we have dH0 d--'~"

(0.19)

~

" ujqj

j=1 m and ~ ujqj equals the instantaneous external work performed on the system. j=1 Secondly, with thls particular choice of outputs we obtain the following symmetry or reclprcclty fluence

the

between

system

H I, . ..,Hm, which

are

same llne of action.

inputs

and outputs.

via

the

external

the

"dlsplac~ments"

For example in case

The

external

channels caused

"forces" ul,...,u m

corresponding by

these

to

in-

the outputs

excitations

Hj(q) = qj, j = 1,...,m, the

along

the

input

uj

~quals the external force corresponding to the J-th degree of freedom qj. Hence If qj Is a Cartesian coordlnate,

then uj will be a translational force,

while if qj

is, say, an angular coordinate,

then u. will be the corresponding external torque 3 (see also Example 2). Notice also that in the original Euler-Lagrange or Hamilto-

nlan equations

(0. I) and (0.7) the vector F - (F I .... , F )

represents

the external

forces as measured in the configuration (ql,...,qn). Hence in order to define the external are

forces

we need

to

know

the

configuration

coordinates

{ql,...,qn ) which

(if we interpret F i as lnputs) just the natural outputs of the system.

also remark that in order to define

a general coupling Hamiltonian

Let us

we need this

6 to

boa

function

natural

of the observations

OUtpUts.

A third argument

made on the system,

theoretlo flavouP. Consider for the Hamiltonlan ui = ai(q, p) + v~, with v~ the new inputs. Hamlltonlan?

This is the case

i.e.

function

a

of the

for choosing yi as in (O.]6) has a more system-

([VI])

system

40.13) a state feedback law

When is the system after

if and only if there exists

feedback again

a

function S such

feedback with respect

to the natural

that

aS ~j(q,p) : ~-~j(H1(q,p) ..... Hm(q,p))

40.20) l.e.,

if and only

if the feedback

outputs and furthermore

ooutput

is

has the special

j = I ..... m

form as in (0.20).

From a mathematical

point of view this dlscusslon

the space

and natural

of Inpubs

outputs

can

bs

can be summarized

by noting that

given the structure

of a cotangent

bundle T'Y, where Y is the output manifold wlth local coordinates where

the

coordinates

of the

forces) U i. Concludlng, system

as

we

a Hamiltonlan

fibers

define

control

of a

this

b~idle

general

system

(0.14)

are

the

(yl,...,ym),

end

inputs

(or external

(afflne)

Hamiltonlan

input-output

together

with

the natural

outputs

(0.17). Finally

let

us

notice

networks wlth external Variables,

current and

(in-)equallty.

In our dual

the

close

ports.

similarity

voltage,

which are

formallzatlon

variables

the

description

%

also needed

of Hamlltonian

involve

the

details,

including a partial treatment

ionian input-output

wlth

In thls case each external

("force")

of

port carries for stating

systems

electrical two an

the external

channels

and yj = Hj ("displacement").

of the theory of interconnectlons

"dual" energy

For

more

of Hamil-

systems we refer to IV1].

Some examples

I. Consider the following linear mass-sprlng system

kI

(without frlotlon)

k2

m2' 1 ql The

Hamlltonlan

q2 = y H 0 here

is

the

sum

of

the

both masses m I and m 2. If u is the external output

y

is

the

first mass m I.

displacement

q2 of

this

kinetic force

second

and

potentlal

energies

of

on mass m 2 then the natural

mass.

The

same

holds

for the

2. Consider the following simple robot manipulator

Both joints are equipped with an actuator 1,2.

Again H 0 is the total

outputs

~o be the

dlng inputs would In general, arbltrary as

vered

number

by

Cartesian

the

friction,

of freedom systems

u i,

i -

that if we choose

the

of the end-polnt, external

robot manipulators

of degrees

an e x t e r n a l

Notice

and horizontal

Input-output actuators

delivering

of the system.

coordinates

be the vertical

neglecting

Hamlltonlan

energy

with two revolute joints

torque

then the

(rigld or flexible) with an

(revolute or prismatic)

csn be described

wlth u = (u I .... ,um) the

and y = (yl,...,ym)

correspon-

forces on this point.

the

forces

corresponding

as

dell-

configuration

(joint) coordinates.

3. Consider

the same system as in Example

kI

wall

controlled,

k2

ql

tha~

Now assume

to now we h a w

u] ..... um enter

q2

and t h a t

The natural

to the inertial

so uy is again the external

Up

"

>

the wall is movable,

i.e. u = v.

(with respect

I

Notice

momentum of mass

that y is the reaction

mI

force, and

power.

in

in arbitrary This

Hamiltonlan

(not necessarily means

we have

a linear

systems where the control

afflne)

to consider

(agflne)

variables

Equlvale,ltly, we assumed m all t l m e - ~ r y l n g Hamlltonlans to be of the form Ho(q,p) - Z ujHj(q,p). H ~ e ~ r , j-1 in its most general form the controlling action for a Hamlltonlan system consists

trois.

equations

the velocity v of the wall can be

output y is now the

frame).

only considered Hamlltonlan

the

>

dependence general

control systems as originally

way.

of

the Hamlltonlan

Hamiltonlans

proposed

~

Brookett

H(q,p,u) [BI]

on the

con-

resulting

in

~H ql " ~ (q,p,u)

(o.21) u : ( u l . . . . . um)

P! : - B ~ ( q ' P ' ~ )

BqI

The natural

outputs

~H

(0.22) System

thls

in

yj - cj ~ (0.21)

sy:~tem, see

[V1,

(q,p,u),

outputs

wlth

V2].

ease are glw~n as

j

(0.22)

Nohe t h a t

Is

called

=

a

as a speelal

1,...,m,

~neral case

O .J

~

-+

Hamlltonlan

we recover

I.

Input-output

statle

reciprocal

systems

~H yj = ~

(0.23)

(u)

j

:

1~.,o,m.

Examples ( o o n t l n u e d ) 4. Consider

k pointmasses

m I, with

q i e R 3, i -

positions

1,...,k,

field corresponding to the potential m i mv(ql . . . . qk) = [ 3 Suppose that fihe controls are the

in

£hel¢

own

gravltatlonal

• the f i r s t

1<j l q i - qJ I" Z p o i n t masses (£ < k ) . The s t a t e

then c o n s i s t s

momenta o f the r e m a l n l n g (k - £) p o i n t masses. Th~ f u l l

H(q %tl , . . . , q k , p £ + 1 , . . . , p v(ul,

. , u £ q£+I

i

..,qk)

+

- q i ~ R3, i ~ I , . . . . £. The r e s u l t l n g

q• j

i

• I pj

the p o s i t i o n s

k 7.

I

i-~+I ~

]2

]pl

Hamlltonlan control

system i s

= - -~-H~ 3pj -

_

I = i+1,...,k

J = 1,2,3

i : I .....

j - 1,2,3

@H ---{

~qj

with outputs

yj

i

@H

o . ---f

of

and

H a m i l t o n i a n i s g l v e n by

k ,u 1 , . . . , u £) : :

. . . . wlth u

of

posltions

~

9

which are the forces experienced

Apart from being strongly class

of

systems

Hamiltonian should

be

general nian the

with

control

to

vectorflelds.

vectorflelds; other

hand

physically very

with

the

M,

theory

which

V9].

directly

The main

problem

we

theory,

Inverse

Problem,

he phrased of

as

external

equations? rewritten

as

the

Notice

the

The

interest

per so,

lies

a

to be Euler-Lagrange

tical

in particular

In the same

be applied

basis

vein,

set

techniques. [VI,

law

V3, be

In a generalization

of

synthesis structure

concerns

procould

Problem

in

can we single

Problem

has

from general second-order

two

aspects.

differential

First,

what

differential equatlons

apart

then all the powerful machinery

quantizatlon

of theoretical

zero

theory

physics,

([Sa1,Sa2]):

and,

be

to a conservative

aspect,

motivation

can

(with

if we know a set 0£ secondlorder

to it. A more general

in The

out among all sets

second-order

second

physics,

equations

it "corresponds"

of this

Problem

mechanics.

in (mathematical)

Euler-Lagrange

so that

the characteriza-

by the Realization

Inverse

How

the

of

in the fact that

may

tems are at the

On

vectorflelds,

can

the Hamiltonlan

Interest

importance

tial equations

to quantum mechanical

to

structure.

of

and structural

motivated

Inverse

equations,

practical

Such equations physics,

systems

on Hamilto-

conservation

resulting

that

classical

equations

can

as Euler-Lagrange system?

class

of

regard

the same tendency

and

in this monograph

equations

that

when

physical

with

such as linearization

symmetry

form as follows:

the Euler-Lagrange

Secondly,

control

particular

critical

indicates

shows

systems,

by

differential

forces)?

position

way also in this context.

in Its simplest

distinguishes

a

systems,

which has a longstanding

second-order

on their

form

control

systems

(nonlinear) vectorfields

work on stabilization

Input-output well

heavily

of

systems form a

The

is much theory available

systems

wish to address

tion of Hamlltonian

there

notions

control

be used in an advantageous

system

control

Preliminary

for Hamiltonian

general

to common approaches

fundamental

theorem.

to

control

properties.

Hamiltonian

relies

extended to Hamiltonian

Noether's blems

The

of

vectorflelds

work on Hamiltonian

V8,

regard

As is well-known

Hamiltonlan

Hamiltonlan

mathematical

position

which is often not amenable Previous

motivated,

peculiar

systems

compared

by the first £ point masses.

comes

from

developed

for

from theore-

Euler-Lagrange

in principle,

its

differen-

sys-

can be quantized

systems.

the question

can be posed of determining

when a nonlinear state

space system with inputs and outputs m

(0.23)

:x" go(x) + Z ujgj(x) j=1 yj = Hj(x)

can

be written as

a Hamlltonlan

j = I ..... m input-output

system.

A second closely

related

I0 question, which ties in wlth the Realization lowing.

Given

behavior order

[WI,

that

output

can

system?

Notice

The

by

Hamiltonlan

maps,

thesis

purposes.

control

systems

consbructlve

as

an

behavior

monograph control

conditions system

are

found

{0.24), two

control

speaking,

the

flrst

dual

to

system

isomorphic

each

other

in

is completely

systems.

From

conditions

element

certainly

can be interpreted

classical

Inverse

end of the last century then proceed to

deduce

o~ some

is

having

of

in

itself.

and

system

in itself.

a

Let us

{minimal)

and outputs

We

reallz~tlon

theory

we

together

first

with

all

system

to-

extensions

Hamlltonlan

input-

dual extensions

are

know this to be equlbehavlor.

of the condltions

in their

to each

and outputs.

two canonical

(minimal)

be

and sufflhow

Z~ inputs

system

These

Now a

show

non-

can

Can be associated,

is the origlnal

the same Input-output

directly

the

an analogous question

by the fact that these

which,

as

or rotatlo-

can be realized as

inputs

original

systems.

sense.

a

equations

sense)

interest

behavior

2n states

the

(see for a discussion

find conditions

physical

(translational

The way these necesssary

as generalizations

Problem,

also contains

differential

under which

number

interest

wlth

to a

and syn-

elements).

conditions

the second

prsclse

second

to quantum mechanical

and m inputs and outputs,

determined

this

control

aspect

theory there is

an equal

systems

system,

system llke a robot manipulator.

with electrical

or

respectively

Again

for

is of some

input-output

network

(nonllnear)

to the two extensions

a

(in

variational a

of

llke

while

the ~ o l n t

set

elements

system

systems,

a

reallzed

with n states

way,

all

We

iS

be quantlzed

by

Input-output system.

as a Hamlltonlan

First, to

input-output

system.

that the second

be

with

Hamiltonian

systems,

systems,

information

physical

first

(0.24)

may

given

of a mechanical

we shall

system

gefiher with

the

This

a

input-output

which

Hamiltonlan

extra

Notice

map,

linear electrical

control

of

etc.?

this

system

input-output

variational

valent

a

of transfer matrices

in a natural

output

Can

by

Problem.

have two aspects.

determine

systems

Ta].

Problem,

map

input-

if we know a system to be equivalent

one may ask if a particular

{realization

Roughly

V7,

external

of a Hamiltonlan

input-output

of a Hamiltonlan

since use

of some specific

that in

control

map

its

behavior have to satisfy in

Realization

by a

is the fol-

generally

behavior

by Hamlltonian to

realized

Also Hamiltonlan

the input-output

written

be

then we may

masses

For instance,

In this

an

possessed

can

(or more

external

Secondly,

importance,

element:

springs,

linear

the

as in the Inverse

[BuI,Bu2,

interconnection

remark

Problem in system theory,

a system

realizing

input-output

the

system,

or

of

maps.

is of praotical

{0.23),

of

does this external

as

properties

input-output

respectively

clcnt

realized

problem

Input-output

aspect

map

system is called the Hamiltonlan

the special

their

which

are

be

that both questions,

specify

hal)

input-output

VII); what conditions

it

Input-output

by

the

The resulting

for solvability

form, were stated at the

[Sal ]). in terms of

the variational

input-

11 output

bahaviour

of the control system (0.23).

In [Vl, V2] already a conjecture

was stated, which formed the original inspiration for thls monograph. As a matter of fact, we shall be able to prove a sllghtly modified version of thls conjecture. From

a

mathematical

point

of

view

the

conditions

of

this

conjecture

can

be

expressed, roughly speaking, as follows, We can define a "symplectic form" on the pairs of variational

input and output functions with compact support.

A minimal

control system is Hamlltonlan if and only If for the system thls symplectlc form equals zero. The

explicit

approach

consideration

rather

distinct

or

from

problem as hays been taken

varlatlonal

other

input-output

approaches

to

b~.havlors

the Hamlltonlan

makes

our

realization

by Crouch & Irving [CI, C2] and Jakubczyk [J1, J5].

Here one gives conditions in terms of the Input-output map or the control system. For a dlseusslon of these results, especially of the expllclt condltlons found by Jakubozyk, we refer to chapter (I).

Finally let us put the theory discussed above in a historiCal perspective. During the last

century

the main

emphasis

in the study

of Hamlltonlan and Lagranglan

systems has completely shifted from the original equations of" Newton, Hamilton and Lagrange,

including external

forces,

bO the

equations

with

zero external forces

[Sa1,Sa2]. Consequently the scope of the theory, and espeolally of new developments such as the geometrlzatlon of Hamiltonlan systems, has been narrowed to analytical mechanics. Since in most mechanical systems external forces do play a prominent role this entails quite a loss of generality. We are of the opinion that the system theoretlo approach to Hamlltonian and Lagranglan systems, as taken In this mono~aph, external

forces.

can again

enlarge the scope of current

Furthermore

the

Hamlltonlan

Realization

theory by reintroducing Problem

in particular

seems to be capable of adding new insight into classical and quantum machanlcal physics;

we mention

moment map

the

theory

of

[VII and

the

Introductlon of

a generalized

in [JT] (for system theoretic applications of some familiar tools in

ge.ometric quantlzatlon theory see, for example, Crouch and Irving [C2], Goncalvss [GI],

and

Bloch

[BI]).

Indeed we

hope

that

this monograph

cross-fertlllzation between physics and system theory.

will stimulate

the

I. THE HAMILTONIAN

The present chapter posltlon

of

the

Hamiltonlan This

is

[J5],

would

seem

main

to have

necessary

here

and

the

since

completely

in which

[V5],

deflnltlons

SECTION

problem

of the inverse

results

[VI],

present

at

[JT] and to a far less

any mention

~hen

we

solved

relation

inverse first

sight

that

the

remaining

the

the significance

known

in

results

classical

work

by Crouch

the Hamiltonlan

valldate

to

problem

by

and

on

the

mechanics.

Jakubezyk

and Irving

reallz~tlon

After describing

essentially the

in

extent

problem.

review

PROBLEM

the results needed to understand

we

raallzation

espeelally

[J2],

outlines

results

REALIZATION

[J1],

[CI],

problem,

[C2],

without

these results we outllne conjecture

chapters

of

the

of

Van

der

monograph.

our

Schaft

Complete

and results are given as they are needed in the main body of the work.

1.1

The nonlinear

systems

we deal with Call roughly

into two classes,

the afflne

(in

control ) systems m

= go (x) +

X ulgi(x) i=I

(i.~) yj

~ ~(x)

,

and the general nonlinear

I ~j

,

~p,

x(O) = x o

u~

,

x e M

(uI,-..,%)~

a=

Rm

systems

0

x

= f(x,u)

,

x(O) - x O,

x e M

(i .2) yj = hj(x,u)

Here

M

is a

dlfferentlable

each u s 9, f(.,u) on M, and We

will

for be

analytic

(C ~)

, 1 ~ j < p,

manifold,

concerned

but

for the

present

smooth

(C=). These

systems

will

System

(~.2) will

real

situations

be

x and u.

Let U be the space on [0, =). An element

element

u, we may solve

suppose

that

vector

of all

equations fields

where

called

plecewlse

u ~ U is called (1.1) or

gi'

vector

fields

I < j ~ p are real valued

we also admit

then

be called Jointly analy$1e,

functions

the

are

with

~ c Rm

gl' 0 g i ~ m are

is a vector field on M, Hj,

each u e 9, hj(.,u)

mainly

u = (u1,...,Um)e

on

M,

for

valued functions

functions on M, I g j < p. all

the

above

the situation analytic

data

where

or smooth

is

real

all data

if f and hj are jointly analytlc constant

right

contln~ous

an input or control.

(1.2) on some interval

0 < i ~ m and f(-,u),

in

~ valued

Given such an

[O,e). We shall

u ~ O, are

that we may take E = ®. In this case we call the system complete.

is

respectively.

complete,

so

Thus if Y is the

13 space

of

R p valued

unique y e Y. function.

continuous

functions

on [0,®), to each

where y(t) = [Y1(t),..°,yp(t)) Is

Each

initialized

system Z either

u e U we obtaln

a

called the output or observation

(1.1)

or

(1.2)

therefore

a

yields

causal input-output map

@z(x 0) : U--> Y defined by @~(Xo)(U)(t)_ - yj(t), u(t) = v(t)

map

for

t ~ [O,T] then Cz(Xo)(U)(t) = @E(Xo)(V)(t)

arising

a

from

system

reference to a p a r t i c u l a r The

is clear that we a r e referlng

slmply write @ when i t

shall

system Z, either

realization

the

of

(causal refers to the fact that If u, v e U

at

Z initialized

Xo;

for t ~ [O,T]). We to t h a t

when ¢ is

or

Input-output given

exist

an

(1.2), may be vlewed as a n

(external)

input-output

Inltlallzed

system ~ such

(internal)

map Cz(Xo). The

state space

basle

~hat @z(x0) = ¢, wl~h

realization Y

the

when does restrlotlon

class of systems. Clearly we may choose

Z m~st lie within some predetermlned

that

without

system.

(1.1) or

problem is now simply stated as: - given a causal mapping @ : U ~ > there

and

many classes of system and Bach would have a correspondlng realization theory. We concern ourselves equations The

(1.1) or

global

here only with those (1.2), or suitable

realization

problem

classes of system which are described generalizations

as deSCrlbed

here

descrlbad

in chapter

has a pleasing resolutlon,

by

(6). as

given In Jakubczyk [J1], [J2] for the class of analytic systems glven by equations (1.2). We therefore use hls termlnology in the sequel. Let

(1.3)

a = (ml,tl)(m2,t2)

represent

the

pleoewlse

.....

( m k , t k)

constant

function

on

[O,T k)

defined by setting,

Tr

~ tl, t I > O, and a(t) = ~r ~ ~ for t ~ [Tr_I,T), T O - O. Let a~ reprei=l sent the function on [O,T k] which coincides wlth e on [O, Tk), and has value ,., e at

T = t k. The

totality

input-output

map

@ = Cz(Xo)Is

now

completely

determined

by

the

of posslble values ass~ned by

(1.a) WO note t ~ t I ~ j < p,

<¢,a~> - ~(a,.,)(Tk) dealing w l t h systems ( 1 . I ) , or (1.2) where ~ h i ( ' ' u ) - h j ( . ) , it

s u f f i c e s t o consider the inputs a i n ( 1 . 3 ) alone, r a t h e r than the

inputs au. An Input-output map ¢ is said to be (jointly) analytlc

if all the followlng maps Rk ,

e r e analytic and have analytlo eontlnuatlons to all Of (Rkx~k+1),

14

[(t l,''',bk,m 1,-'.,mk,mk.l) --> <#,(m1,t 1)'''(m k,tk)mk+ I>) (tl...tk) ~ >

< ¢, (ml,tl)---(~k,tk)mk+l > respectively.

is said to have finite rank if rank where

ak,bl...b k are

plecewlse

a k = (ml,t~)'''(mk,tk), k > I, ak, bj~j, and

# = sup rank {~---~tÀ<#qJ,a k" bjwj>}~,j:1 < ®

constant

~j e n, and

integers

qj,

the

inputs

of

supremum

the

is

form

taken

glven in (1.3),

o~r

all possible

1 g qj < p, I < j g k. Note a k • bj~j is

just

the concatenation of the two inputs.

THEOREM 1.1 JAKUBCZYK [J1]

A

causal

jointly

analytic

input-output map ~, with ~ compact and c o n ~ X ,

has a

jointly analytic and complete reallzatlon by a system (1.2) if and only if rank @ is finite.

The

original

paper

hj(-,u) = hj(-),

by

Jakuhczyk [J2] deals wlth systems of the form 41.2) wlth

I K j K p. Sllghtly

modified

definitions

are

tlcular analytlclty with respect to m ~ 0 and the condltlons obtain analytic systems rather

than jointly analytic.

flcultles

results

in

restrictionS and Gauthler

extending

these

to

the

used,

and

inpar-

on ~ are removed, to

There are significant dlf-

smooth

case,

unless

significant

are placed on the system, see Sussmann [$I], Hermann and Kroner [HI, and Bernard

[Ga];

or new definitions

are introduced,

see Goncalves

[G2]. Since these difficulties are basically of a global nature this does not rule out the existence the local

of, even C k, local realizations.

theory for systems

Flless

[FI],

[F2], works

(1.1), with analytic data, and Jakubczyk

out

[J3] works

out the general C k local theory. Since our theory is largely of a global nature we do not glve any details here. Although theorem

(l.l) and similar results

tions, an important question, ness of the realizations.

especially

demonstrate

the exlstence of realiza-

in our monograph,

is that of the unique-

In general there is no reason to b e l i e r

any realization

of a particular input-output map should be unique. However if we restrict to those uealizatlons,

in which the dimension of the state manifold M is minimiZed,

is a type of unlqu0neos.

These reallzatlons

the existence

and uniqueness

goals

of the

papers

[J2],

and

all

realizations

the

[$I], local

described

of these minimal realizations

[H],

[Ga],

theory.

above

In

[G2], although fact

correspond

In more

realizations defined In the above references.

there

are called minimal realizations,

the

is one of the prlmary

it is implicit

global

closely

and

situation with

the

in the work the

minimal

quasl-mlnlmal

15 SECTION 1.2

The realizablllty conditions descrl~d in

theorem (1.1) may also be interpreted in

terms of series expressions for the input-output map, and these conditions are nicely reviewed in [J6]. The two main series representations of the input-output map are the power series in nonoommu~Ing variables, and the VoltePca sgrles. The power series approach initially relied on tho fact that, at least for analytic systems (l.l), the Input-output map may he defined by a oonvorgent series

(1.5)

Yio(t) = alo(X O) +

~ 11 ...,ik

aioll " ik(xO) (t) "" 511...i k

k~1 where I ~ I 0 K p , 0 ~ lj g m , for I K j ~ k , and setting Uo(t) ~ I

ft

~i(t)

=

0

Ui(S)dS' ~11 ....ik (t)

ft

=

0

Ulk(S) ~11 .... Ik-1(s)ds'

ale = Hie, ~ioi I ..... ik = gik[al0il ..... ik_1 )"

The latter expression is just the Lie derivative of aloll ... field

gi " If X0,XI,-..,X m are k series is simply

+ 11,...,I [ k

alo(XO)

See Fliess tation.

[FI],

[F2],'for

The Volterra series

the

noncommutlng

variables

the

by the

veotor

corresponding power

h01 , .....ik(xo ) XloXl I .....Xlk

(local)

reallz~tlon

theory

based on this represen-

representation of the Input-output map of system

(1.1) is

basically a recombination of certain terms in (1.5)o For m = p = I, write u,(t) = u(t), g1(t) = g(t), HI(t) = H(t), to obtain

y ( t ) = wo(t,x O) + I t Wl(t,el,x O) u(al.)da 1 (I,6)

o t

J f 01 W2(t,~1,~2,Xo) U(C I) U(a2)d~ido 2 + ....

+ 0

0

For analytic systems the series converges to the input-output map of (|.I), (in a suitable uniform sense) to the output function y. The following formula for the Volterra

kernels Wk(t,al,...,Ok,X O) developed

in

Lesiak and Kroner it], clearly

16 shows the analytic dependence of the kernels on their arguments.

(1.7)

Wk(t,~1,''',ak,xO)

= X[-Ok) , g ( X ( O k ) X O ) ( X ( - O k _ l ) , g(X(Ok_1)')){''" • ..[7(-.i) , g(X(Ol).))(H°X(t).)... ) where (t,x) --> Y(t)x is the flow of the vector field go" For

systems

Jakubozyk

(1.2)

the

following

more

general

expansion

is

appropriate,

see

[J~]. We let fi denote the free monoid generated by fl. Thus elements of

fl are thought of as noncommutlng variables, and fi is the set of words ~I~2 ..... mk ' ~J e 9.

A formal power

series In 9 is just a map P : fl --> R, but

we are usually interested in p series together, so we view F as a map

P : fl* ~ >

Rp" We w r i t e

P =

Z ,

Given an analytic input-output map ¢ : U --> Y we define a

(1.8)

a

~ ~1 . . . . . Wk at1

ark-1

It I

0

<¢,(~1,tl)

.....

(mk_l,tk_l)~k >

NOW ¢ defines a formal power series

(1.9)

.Z

P -

=I ' " " "'mk Eft

¢~1 . . . . . ~k ml . . . . . ~k

ka I Conversely a formal power series

P "

~

c¢I,"" .mRS fl

< P ' m l ..... m k >

ml .... "•k

such tha~ each series i]

ik_ ]

iI Ik_ I t I ..... tk_ ] i I [ ..... ik! 1 1 , "'',! k

= ¢

.... tk_ I ) ~I ..... '"k(t] "

converg9s in a neighbourhood of 0 ~ R k, and has an analytlo continuation to all of R k, defines an analytic input-output map @, (here we use the notation ii

ik_1

a}1 ..... mk-1

i I times mk=

ik_ I times

w] .... '~I ..... ~k-1 ..... ~k-I ~k )" Indeed ¢ is

(1.4), by the values

speolfled,

as

in

17

<@'(ml'tl ) .....

(mk-l'tk-1)mk

>

=

#

I'~I" .... (~k

(tl ..... tk-1)'

I t @= @r(Xo), f~(x) = f(x,~), h~(x) ,, hj(x,ta) then

(1.t0)

mj {~1 . . . . ' ~ k

. f

[f~2( . . . . . ( f ~ k - t ( h

Note the similarity bo-twean the eoefflolents the

ooefflolents

Of

the

expansion

(1.5)

works out a local realization theory, terms Of the

SECTION

As

~eries

k) . . . . .

)(xo )

(1.I0) of the power serle8

in the

affine

ease.

(1.9),

and

In [j4] Jakubczyk

in the analytic ease) For systems

(1.2) in

(I.9) and operations on it.

1.3

already explained in the Introduction, our speeifle interest in this paver lies

In systems

Lagrang~.

and

(I.I)

and

(1.2)

with added

motivated

by

the equations

0£

Hamilton, namely a ddt (~Taqi L(q,(~)) - ~ i

(1.11)

strUCtUre

b(q,<~) - Fi

,

I < I ~ n

and

(1.12) Here

qi " a~pi(q'P) ' Pl " - @--qi@(q,p) H + F1

,

I ~ i ~ n.

we Include exbernal forces F I, as in the oPlginal coneeptlon by L a g r a n ~

and

HamllSon, see Santilll [Sa2], and q - (q1' ' ' " q n )' ~ " (ql' " ' " q n ) or q = (ql,-..,qn) , p = (p1,...,pn) lle in some open subset U of R ~ , We notice that i ' a2L l " .n If the matrix wlth oomponen~s l.-q---q--.l is nonslngular on U, we may use the

I,j LeEendee

transform

to rewrlts

equations

- I

(1.11) in the form of

equations

(I.12).

Sines we prefer to work wlth flrst oPdeP dlffePentlal aquatlons we shall therefore concentrate on squations In the systems inputs,

(I. 12).

theoretic setting we

and write

ui

Vlew the external

components F 1,,..,F m are non zero, equatlons

(1.12) become

aH 0

4i = ~ 1 (1.13)

foroes F i as oontPols, or

= F I , I ~ i ~ n. If we also assume that only the first m

(q, p)

. aH 0 Pi " " aq~; (q'P)

* UI

,

I ~ i ~ n

,

I ~ i g m

18 8H o ,

This

situatlon may be generalized

m+

1 g lgn

(see the Introduction)

by introducing m obser-

vation, or output maps H. : U --> R , I < j g m and defining the system J

i0i i

=

(I. 14)

~Ho

j[ I

yj = 5 ( q , p )

Notice

is

that

by the observations iV5],

discussion

and

concerning

The equations

we can

we make.

Willems

1
= qj we recover

the system

exert

forces

external

See the Introduction

Van

der

Schaft

the introduction

([VII,

[W2]

(1.13).

The idea in this

in the dlrectlons

and Van der Schaft and

Brockett

determlned [VI],

[BI])

for

iV2],

further

of systems such as (1.14).

(1.14) lead to the following

(locally) Hamiltonlan system

,

I < j g m

that by setting Hj(q,p)

generallZ~tion

iV3],

,

aHj

uj

iV3],

coordinate

free definition of an afflne

[V~])

m

= g0(x) - i=I[ U I XHi(X)

,

x(O) = x O,

(1.15) -i y" = HI(x)

(M,m) i s

where

a

' I ~ i ~ m , x e (M,~)

sympleetlc

globally Hamilbonlan

manifold

, u • n = Rm

([A]),

vector fields determined

wlth

symplectlc

form m, XHI are

from the relationships

m(XHl'') = -dHI , I ~ i g m and

go

is

states

an infinitesimally

that L

~ - 0 where L

gO go is a locally Hamlltonlan

symplectlc Is

the

go vector

vector field. Lle

fleld;

derlvatlve,

The latter

condltlon just

so it follows

that is locally

there

([A])

that

is a function

H0

such that

~(go' ")

If

go

afflne

Is

globally

globally

that about Cot

a

M in

=

any

-dHo"

Hamiltonlan

Hamiltonlan

vector

system.

In

fleld any

we

case

say

that

the DarbOux

system theorem

(1.15) ([A])

is an shows

point In (M,m) there are local coordinates (ql,...,qn,Pl,--.,pn) n which ~ - [ dPl A dql. It follows that in these coordlnates, system 1=I

19 (1.15) is indeed Although

most

given by the equations

of our work

concerns

(1.14).

Hamiltonlan

systems

in the

(1.15) we also wish to consider more general Hamiltonian general

nonlinear

~ystems

(1.2).

Thus a general

form

systems,

(globally)

of equations

corresponding

Hamlltonlan

system

to is

defined by the equations

= XH(x,u)

,

x(O) - x 0

,

x • (M,~)

(1.16) aH Yl = el ~ i (x'u)

where

(M,m) is a symplectlc

vector

general

point,

and a

Instead

enough.

global all

in

external

W. This

local

coordinates

physical

input

thls and

symplectlc

and

system

for

W,

(locally)

manifolds, with

takes

such

that

the

local

in

evolve

of

to YI"

in some

of a nonlinear

form

equations

system (1.2).

in which (M,m) and

(Darboux)

coordinate

not this

variables

to

system

local

Is

demonstrate

output

assumed

the

Hamiltonian

definition

it is not possible

u I and and

is a Hamlltonian

[BI]

systems

variables combined,

are

(y,u)

(W,m e) are

coincides

[Vt]

rise to the general definition

we define a general

the

However

[V2],

in many

between

Similarly

M

[V]],

variables

gives

and for each u • £, XH(X,U)

[W],

that

e i = ~ 1,

u • £ c R m,

functlon H(.,u).

in

show

distinction

the

other manifold whlch,

Work

Inpartlcular

I < i ~ m,

manifold,

field with Hamlltonlan

quite

make

'

~rslon

coordinates of

for W

system

(1.16),

general

problem

namely aH

ql (I

17)

= aT i

BH ~qi

Pl

"

Yl =

The

(q,p,u)

(q,p,u)

aH

cl ~

Hamiltonian

(q,p,u)

realization

defined

in section

systems

descri~>.:d above.

maps

is now

evident.

In the

The

Without

class

has

(globally)

Hamlltonian

r e a l i z a t i o n differing

(globally) coordinates. guaranteed.

Hamiltonlan, Thus

I ( i < n

,

I ~ i < m ,

is

importance

a

special

to a

In fact this

and large

of

of

there will

nonlinear

of

the

within a class of Hamlltonian Of input-output

realizations

constraint,

even

if an

Input-output

exist many non Hamlltonlan

systems

described

[GI], and Van der Schagt quasl-minimal

c I = +I.

case

minimal

the minlmallty

realization,

general

known, see Goncalves

a

,

problem

already

minimal

~ ~

(1.1), where we seek realizations

map has a Hamiltonlan tions,

i ~ .

,

above.

realiza-

However

as

[V4], if an Input-output

realization,

then

any

other

is map

quasi-

by a (global) change of cobrdlnates, is in fact also the

two

extent

situation

system

differ

uniqueness Is completely

of

by the

a

sympleetlc

Hamiltonlan

resolved

change

of

structure

is

in our work here, but

20 ses also the work by Jakubezyk

Remark

(see

[J1])

No~ice

[J1], [JS], [J7].

that

without

(I.~5) or (].17) the Hamiltonlan fact)

any nonlinear

the restriction

realization

input-output

map

an

=

-

as

in

is much less well-deflned.

to natural

In

with a realization

x e M, also admits the pseudo-Hamiltonian

=

problem

outputs

x - f(x,u)) y = h(x,u),

fealizatlon

(x,p,u) y - h(x)u) aH

6 with H(x,p,u)

SECTION

(x,p,u)

= pTf(x,u).

Of course such a realization

is not minimal.

1.4

Although

the

uniqueness

previously

of

realizabllity

THEOREM

-~

(quasi)

cited work minimal

oondltions.

[01] and

Hamiltonlan

The oclglnal

[VI],

iV4], deals

realizations,

with existence

It does

not

contain

and any

result in this area is as follows.

1.2 BROCKETT AND RAHIMI [B2]

The Input-output map

y(t)

;t

=

W(t,q)u(~)do

0

Of

the

linear

system % - Ax + BU, y - Cx, x(O) - O, m - p) has

nlan realization

(1.18)

This

a linear

W(~,e) = -W(a,t) T

result

was in

E~nerali~d Crouch

[C4],

U

to systems

Define,

as

kernels,

for the case m = p - le

the

with

following

Wk(t,~l,..-,[ei,al+1],...,ak) -

Hamilto-

if and only if

-

Wk(t,al,...,ai+1,ai,...,qk).

and hence inductively

via

finite ~acket

Volterra operation

series on

as

follows.

the

Volterra

Wk(t,el,...,ai,Ol+1....,~ k)

21

W (t,a k

,-'-,[o

,-..,o

1

i

],-..,o j

) - W (t,c, , . . - , [ o k

k

I

,--.,~, i

Ill (J ! "''~(I

j

J-1

)

k

(t.19)

- Wk(t, 01, . . . , 01_I, (~j, [ e l , • • . , oj_1], Oj +i, . . . , Ok). THEOREM 1.3 CROUCH AND IRVING [CI].

An i n p u t - o u t p u t

map ¢ which has a r e a l i z a t i o n

by an a n a l y t i c ,

affine

and complete

system(1.1) in which gO(Xo) = O, and has a representation as a finite Volterra series of length N, has a realization by an afflne, analytic and complete globally Hamiltonian system (1.15) if and only if

(1.20)

Wk(Et,a1,...,Or],...,o

for I ~ r ~ k

A

similar

I ~ k ~ N.

and

result

Hamlltonlan

obtained

is

system are required. varying

k) : (r + I ) W k ( t , o 1 , . . . , o k)

O if g0(x0) # O, but

then

time

varying

Hamlltonlan

See Gonoslw.s [01] for the b~st available exposition of time realization

theory.

We

defer

until we review some recent work of Jakubczyk,

any

discussion

of this result

[J1], [Jt], [J7]. Indeed he consi-

ders both the global and local theory, but we only Consider the global case here, Jakubczyk

first

introduces

another

class

of systems related

t o these defined in

equations (1.16), namely those defined by equations of the form

- XH(x,u)

,

x(O)

- x0

, x •

,

u e fl c R m .

(M,~)

(1.21) y = a(x,u)

where

again (M,~) Is a symplec~Io manifold and XH(-,u) Is the Hamiltonlan

field

with

Hamlltonlan H(-,u) for each u • ~. Note

the

output

space

for

vector these

system is always R. To state the first result we must also introduce more notation related to the quantities ~

defined in equation (1.8) for an Input-output

ml ..... ~k map ~, and similar t o that in (1.19). Let

~m1 . . . . . [ m l , m l + l ] . . . . . mk = ¢ml ..... ~i~I+I ..... ~k -

¢

~I ..... mi+Iml ..... ~k and define inductively

22 @

-

~1 ..... [wl ..... wj ] . . . . .

(1.22)

¢

(~k

~I . . . . . ~l[ml+1 . . . . m j ] . . . . . mk -

¢

THEOREM 1.4 JAKUBCZYK [J1], [J5]

A causal, jointly

jointly

analytic

analytic

input-output

complete realization

map @, wlth

by a system

£ Compact

(1.21)

and convex,

has a

if and only if the rank

Of @ is finite and

(1.23)

¢[~I . ."'mk . ]. -. k .#ml

"'mk' k ~ 2, mle £, I ~ i ~ k

[] To

overview

the

proof of this result note that wlthout the conditions

obtain a minimal realization In thls

case

p - I set h

In the form of a system = h, and h

(1.23) we

(1.2) by theorem (1.1). Since

= f~1[f ~2 ..... (f~k-1(h~k) ..... ) as 9 1 ..... ~k

J

in equation

(1.10). Now (1.23) implies that

(1.24)

h[w I ..... ~k](xo) - k h i ..... ~k(XO)

for k ~ 2, wl e £, I g i ~ k, where the definition of h[w

..~ ] iS analogous to

that of ¢[~I ..... Wk ]. The £ollowlng interm~dlat, result Is of independent interest and should be compared with our theorem 4.2 of chapter 4.

THEOREM 1.5 JAKUBCZYK [Jl], [J5].

A minimal analytic system identltles manifold,

are satlsfied

(1.2) satisfies identically

f(.,u) is a Hamiltonlan

the identities

on M, and

(1.24) if and only if the

If and only

!f M Is a symplectle

vector field XH(.,u) with Hamiltonlan H(.,u) for

each u e £.

We see

immediately

that

(1.21)

as desired.

To motivate

our system

(I,2) can be rewritten

the conditions

Polsson bracket on a symplectlc manifold

in the form of system

(1.23) we recall

(M,w), ~ t w e e n

the definition

two f u n e b l o ~

of

~ and ~,

{B,a} = ~(x6,x a) = X~(a) where X

and X B are

tion of theorem

the associated

(1.5) we have

Hamlltonlan

vector

flelds.

Thus

in the sltua-

23 f

{h

] = XH(',= I) H(',=2) = { H ( ' , = I ) , H ( ' , = 2 ) / = ~

}

,h

and more generally eel

.

- {h

,{h

m2

,{ .....

,lh=k-I

,h

~k

} .....

}.

~1"'.''=k The fact that h must s a t i s f y the eondltlons of theorem (1.5) Is now Just ~1"''''~k a dlreot result of another result by Dynkln-Speoht-Wever, which eharaeterlzes Lie monomlals in nonoommutlng variables. See Ree [R] for a nle@ exposition of tbls result, and Lamnabhl-Lagarrlgue re presentatlorm conditions

of

and Crouoh [La] for f ~ t h e r

Input-output

maps.

(1.20) on the Volterra

Volterra kernels may b@ rewritten

Thls

kernels.

result

also

implloatlons for series explalns

Indeed the expresslons

for a Hamlltonlan system (1.15),

the

analogous

(1.7) for the In t~e form

wk(t,a I, '",~k,XO) - {g(ak),{g(ak_1),"'''{g(~1),g(t)I.....}(x0) where g(o) = H o T(o). (We use the ~ermlnology Introduced f o r the expresslon ( I . 7 ) . ) To obtaln a r e a l l z a b l l l t y r e s u l t f o r systems of the form (1.16) r a t h e r than (1.21) Jakubozyk proceed~ as f o l l o w s . Glven an Input-output map @deflne a causal mapping : U - - > Y, where Y is now j u s t the eontlnuous functions on [ 0 , - ) , by s e t t i n g k s=l ~

i=I

s

where a s = (ml,tl) ..... (~,ts).o That thls Is well deC1ned, i.e. that the integrals are Independent of path, requires that the components @I satlsfy

(1.26) Another

au--~ <¢J ' aku> = ~ Intermedlate

result

<~i, akU> ' I ~ i , j ~ m. states

that

realization by a Hamll~onlan system (1.16)

an

analytio

Input-output

map @ has

a

If and only if @ has ~ reallzation by a

system of the Corm glven in (1.21). Necessity is olear since

~--~u <@, akU> = <#, akU>. Conversely If @ has a reallzBtlon by a Hamlltonlan system (I. 16) then < e l

J = ~u~H---~ {x(~j),u) where Tj = iZ1= t i, aj = (~i,ti).....(~j,tj) as before. Inpartloular (1.26) Is satisfied 8fld we obtain from (1.25)

eju> -

24 k

<~, a k = k + l > - s=O [ But

(,(.%).%+,)

- H(.%).%)).

H(x(Ts),mS) : H(x(Ts_I),%) slnoe H(-,m s) is

x(t), t • [Ts_I,Ts), which is governed by t h e

constant

along

the

trajectory

Hamiltonlan vector field XH(-,m s) by

con3truotlon. Thus

<~, ak~k+1> = H(x(T),Uk+I) - H(Xo,Uo) Since H(x0,u 0) Is just a constant we see that ¢ is

the input-output map of system

41.21). Combining this result with theorem (1.4) the Following result is obtained.

THEOREM

A

1.6

[J1], [J5]

JAKUBCZYK

causal Jointly analytic

Input-output map ¢, wlth fl compact

and convex,

has a

jointly analytic complete realization by a system (1.16), If and only if the rank of @ is finite, @ satisfies

(1.26),

and ¢

given by 41.25)

~I

satisfies

(1.23)

where

for

m1'''''mk

. . . . . mk

atl . . . . . )_ ark-1 t tI = a__

= 0 <$' %%>"

D

The only work remaining in this result, after theorem 41.4) has been established, is to show that $ is of finite rank if and only if ¢ IS also. In Jakubczyk [J7],

[J8],

a realization theory is worked out, for systems

formal power series

representation of the Input-output

41.2).

which

Thls theory

dimensional" ¢

introduces

version of an

in t h e

idea

power series

a

[JS],

based on the

map introduced in section

generalized moment map, is an "infinite

in Goncalves [GI]. (1.9)

(I.21),

exactly

are

Note that those

that

the coefficients appear

in

the

~1 ..... ~k conditions

(1.23) Of theorem (1.4).

SECTION 1.5

The

Inverse

problem

in

olassloal

mechanics

in

its

original

form

concerns

th=~

system of equatlons without external forces

(1.27) where

Rt(q,4,~)-o q - (ql'

"'%,)

, '

l~i~n

~i - 'dr ' " "

,

q~

d~ ,

Rn (~-~ ' " "

d - ~ J"

We

a ddl

-

25

tlonaly

assume

that

the

matrlx

i]

is

nonslngular

on an

open

domaln

aqj t , j - 1 U c R 3n, and R i Is C I on D. One

asks

under

what

frothed

condition8

on

exists a funotlon L(q,q) such that after a possible reordering of the

Ri

there

indices we

ha ve

(1.28)

d {~ (q,4)) dt ~qi

The s o l u t i o n related

of thls

- aL ( q , q•) = R t ( q , q , q )

problem as d e t a i l e d

in Santllli

[Sal],

[ S a 2 ] , a l o n g w i t h many

toplcs, Involves the following deflnitlons and constructions. Rn be a C 2 solution

q : I ~> I c R, and

define r : I - - >

of

the

such

that

= (q(t,e),

equations

(1.27),

- > Rn where V is

a2°" ( t , ¢ ) ) at 2

then a2

ar t ---) ~ - ( t , O )

Let

some open interval A

variation

of

q

an open nelghbo~hoo(t of 0 in

q(t,O) = q(t) for t ¢ I, and if r(t,e)

aq ( t , a ) , ~ - = aq

on

U c R3n by r(t) = {q(t),q(t),q(t)).

IS a map (t,g) ---> q(t,e), I x V R,

1 ~ i ~ n.

"

~i

is exists

defined and

by is

r(t,e) = continuous

a

Write q(t,e)

~-~ (t,e), q(t,¢) - ~ (t,e) and 6q(t) = ~-g q(t,O), at 2 6q(t) = ~ a q(t,o), 6q(t) = ~-a ;(t,0). 6q is called the variational field

along q.

For n o t a t i o n a l convenlenee we set 6 r ( t ) = [ 6 q ( t ) , 6q(t), 6 q ( t ) ) . I f q(t,¢) is

a

variation of a solution q(t) of (1.27), such that for each g e V

t ~ q(t,¢) IS a C 2 solu~lon of (1.27) then

Ri(q(t.,). It

follows

that

4(t.E).

by

differentiating

variational

equations

(1.29)

M(r(t)}

6r(t)

components R i. As i n

also

~ependlng

q2(t,c) of

= o. with

1 ¢ t ~ n. respect

to ¢ we

obtain

the

so

called

= 0

where M i s a n x 3n m a t r i x the

q(t.,)}

d e p e n d i n g on r and cor~91stlng o f p a r t i a l Santilll

on r, with

the

derivatives

property

that

given two variations q1(t,e) and

a solution q(t) of (1.27), with corresponding v a r i a t i o n a l

fields 61q,

62q, there exists a unique function Q(r,61r,62 r) satisfying

(1.30)

62Tq(t)M(r(t))61r(t)

Note t h a t q i s b l l l n e a r (1.31)

M*[r(t))

*

- 61Tq(t)M ( r ( t ) ) 6 2 r ( t )

in (81r,62r).

6r(t)

= 0

of

[Sal] there exists a unique n x 3n matrix M*,

The e q u a t i o n s

d

= ~-~ Q ( r ( t ) , 6 1 r ( t ) , 6 2 c ( t )

).

26

are

called

books

described

THEOREM

The

the ad~Lolnt

on

equations,

equations

(see

and are well treated

e.g.

[Me]).

The

desired

in many textresult

is

now

if

the

Ri(q,q,q) , i - 1,...,n,

are

in the following

1.7 (see [Sal])

inverse

variational

The

varlatlonal

differential

problem

in

classlcal

mechaniCs

has

a solution

if

and

only

equation is self adjoint i.e. M(r) = M*(r) for any r.

conditions

resulting

in

terms

known as the

Helmholtz conditions

She essential

constructions

of a symplectlo

form,

of

the

[Sal].

In Santllll's

from which

functions

We would llke to point out proof of sufficiency

a Hamiltonlan

and hence

given to the equetlons.

In our t h e o r y , especially

is essentially

showing

repeated

that Santilli's

theorem

that one of

is the construction

Lagranglan

structure

is

(4.2), this oor~truotlon

construction

is even more

general

than perhaps was appreciated. As noted result

in

Takens

IT]

also solves

and

in more

a restricted

system of Newtonlan

detail

in Van

Hamlltonlan

der

realization

Schaft

[Vl],

problem.

[VS],

Consider

this

first a

equations

Rl(q,4,,~)

-u

i

,

I ~ i ~ m

,

m+1 ~ I ~ n

( I . 32)

RI(q,G,,~) = o under

the

same

restrictions

possible

to

write

external

forces?

this

Clearly

As noted

9L aq I

d BL dta¢:!

aL aq i

in section

Hamiltonlan

system

I ~ i ~ m. Thus (1.6) provides

imposed

a

the

conditions

since once we have equations d aL dt a~ i

as

as

set

of

on

equations

L~gPanglan are exactly

(1.27).

We

ask when

Hamlltonlan

it

equations

the same as In theorem

is

wlth (1.6),

(1.28) we also know that

ul

'

I ~ i ~ m

0

m+1 ~ i ~ n

(1.2),

if we are to write

(1.13)

the

adding

or

these

natural

equations

these

outputs, to

us with an exact solution

those

or of

equations

in the

observations, (1.31),

to the corresponding

we

see

form

of a

are Yl = ql that

Hamiltonlan

theorem realiza-

tion problem. Note the

however

that

corresponding

we have

not solved

input-output

map

the

problem

directly,

as

by examining Is the

case

the

properties

in theorems

of

(1.2),

27 41.3) and 41.5). Rather the problem is solved in terms of properties of variations in the state trajectories.

In the case of system 41.31) with OUtpUts Yl = qi' each

variation of the input function u, yields a corresponding variation in the state and output trajectories x and y, via equations the form

Ri[q(t,e), ~(t,e), q(t,~))

-

ui(t,~),

Ri(q(5,a), q(t,e), q(t,¢)} - 0

yl(t,e) - qi(t,¢)

u(t,O) It is assumed

=

that

m+1 g i < n

,

,

1 g i ~ m

u ( t ) , y(t,O) = y ( t ) , q(t,O) the

I ~i ~m

variations

q(t).

=

u(t,¢) are such that the corresponding

varia-

tional fields

6r(t), ~u(t)

au - ~

a

(t,o), 6y(t) - ~ y ( t , o )

exist and are continuous. By differentlatlng wlth respect to ¢ we obtain

[M(r(t))6r(t)] i - 6ui(t)

,

I g i ~ m

[M{r(t))6r(t)] i - O

,

m+1 g I g n

,

I

6yi(t) - 6qi(t) In case

the

i g m

~

variational system iS self adjolnt equation (I.30) can now ~

written

as

41.33)

6)(t)61u(t ) _ ~Ty(tl%u(t) " EE d Q[r(t)'61r(t)'6~(t))

A generalization

of this result for general Hamiltonian systems has already been

given in Van der Schaft

[VII,

iV2],

iV5].

In this paper we

generalize equations

(1.30) the~u~elves, for general nonlinear systems, see lemma (2.1). Now

if ~i u and 61y, i = 1,2, have

true, compact

given

sufficient

support.

conditions,

and

There

compact

dlfferentlabillty, Is sufficlent

support that

reason

to

in (--,-), --then it

(61qj,~i4j,6iql),__ believe

inparticular m = n, 6jr, i = 1,2 also

this case equatlon (].33) yields

that

have

is

1 ~; j

under

compact

clearly

< m, have

appropriate support.

In

28

Before further investigating equation (1.34) in the next section, we would llke to mention an open

problem

problem originates

which

from

the

will

following

not be dealt with in thls mOnograph. Thls generalization of

RI(q,~,G)

-

o

Inverse problem In

of solutions of

classical mechanics. Notice that the set

(1.35)

the

i - 1,-..,n

is not changed by pre-multlplicatlon of thls set of equations by a non-slngular

matrix (81j (q,(~));,j=1 . Hence given the equations

(1.35) the question can be asked: When does there exist

a non-slngular multiplier matrix 81j(q, 4) and a Lagranglan L(q,q) such that n

(1.36)

-~" g l j ( q , q ) j=1

Rj(q,(~,q)

d ° E

(a~L) a"l q . ag

I = 1,...,n?

An excellent discussion of thls problem is given i n [Sar], from which It is clear that finding explicit conditions for this problem seems very hard In general. In the

framework

following.

GIwn

of Hamlltonlan reallzatlon a

control

system

theory the

(1.1), when

does

problem amounts

to the

there exist a non-slngular

transformation of the inputs m

(1.37)

uj : k!l Bjk(X)Vk

j : 1,...,m

with Vl, • ..,vm the transformed inputs, such that the transformed system m

(I .38)

m

= go (X) + Z v k [ Z Bjk(X)gj(x)) k:1 j:1 ' yj = l l j ( x )

j = 1,...,m

x(O) - x 0

is Hamlltonlan? From a system theoretic point of vlew thls suggests the even more general question: when does there exist a feedback m

(1.39)

uj ,. a j ( x )

+

m

[ Bjk(X)Vk , [ 8 j k ( x ) } j , k = 1 k=l

non-singular

such that the feedback transformed system m

= gO(x) + Z c~j(x)gj(x)+ j=1

(I .4o)

yj : Hj(x)

j = 1,---,m

m

Z

m

vk

k=1

{j~

Bjk(x)gj(x)}

29

is Hamlltonlan? Thls open problem fits in very well Into current research of finding normal

forms for nonlinear systems by applying feedback and coordinate

transformations; see In this context also [St]. SECTTON 1.6 Before we state the conjecture of Van der SchaCt, motivated by the observations of section 1.5, i~ Is clear that we must consider the more general situation of non initialized systems (i.e. x(0) is arbitrary). Given a complete system Z, described by equations (1.2) we say that the behaviour Of the system Zl is the set of time responses (u(t),y(t),x(t))

t--~

,

R--~

~x

@~

M

satisfying the equations dx" ~-t-
(1.41) and

such

assumption

that

u Is right

y(t) = h(x(t),u(t)) , t e R

continuous and plecewlse

constant.

Our

just projection of Z I into the set of input and output responses t - - > +

Let

completeness

ensures that this definition makes sense. The external behavlour ~e is [u(t),y(t)).

+

El(T), [Ze(T) ) be

the

time responses obtalned from Ei(Ze) by restriction to +

IT,'). Because

of

Invarlance of the definlng equatlons (l.~O), Zi(T |)

the tlme

÷

+

+

÷

÷

[Ze(T|) ) differs from ZI(T 2) {Ze(T2) ) only by time translation. ZI(T) (Ze(T)) is a +

+

union of subsets ZI(T)(x T) [Ze(T)(XT) ) corresponding to those responses satlsfylng +

+

x(T) = x T. [Se(T)(x T) is Z~(0)(x 0)

just

the

projectlon

of

ZI(T)(XT). )

Note that

may be identified with the input-output map eZ(x0).

We define a variation of an element (J,{,x) ~ Z i, In the same fashlon as before, as ~or

a mapping each

(t,~) "---> [u(t,c),y(t,e),x(t,¢)), R × V ----> R x Rp x M

~ ~ V

t--~

satisfying,

[u(t,E),y(t,c),x(t,~)) ~ Z i, and [u(t,O),y(t,0),x(t,O))

=

= [u(t)){(t),X(t)). Moreover we asstmle the corresponding variational field t ~

(6u(t),6y(t),~x(h)) exists,

oontlnuous

S by

and 6x is

absolutely

and

that 6u is

plecew!se

satisfy

6y Is

continuous. We define variations of elements in

projection, and those In Z~(T), Z~(T) s i m i l a r l y .

ZI(T)(x T) [Z~(T)(XT) ) must

constant,

an extra

Variations of elements In

constraint x(T,e) - x T for e • V, so

In thls case (6u(T),6y(T),6x(T) I - O. A weakened version of the conjecture by Van der Schaft is as follows.

30 CONJECTURE 1.8 VAN DER SCHAFT [vii, [VS].

If

Z

of a general

represents the external behavlour

e

system

is Hamiltonlan

nonllnear system, then the

if and only If given any element (u,y) e re , any two varia-

tions (61u,61y), i : 1,2, of (u,y), such that (61u,~ly) have

compact

support,

satisfy

- aiTy(t)62u(t))dt

(a~(t)61u(t)

Although thls conjecture

has

: 0

been inspirational,

we have to change its statement

for technical reasons and because we are not yet able to charaeterlze non-mlnlmal Hamiltonian

systems.

system

The

result

by (1.I),

Z, deSCribed

¢£(x 0) has

main

of

this

monograph,

theorem

(5.11)

may

be

: - If Cz(x0) is the input-output map of an analytlc, complete,

stated as follows.

a minimal,

which satisfies

analytic,

complete

an

additional

Hamiltonlan

assumption, then

realization Z',descrlbed

by

÷

(1.15),

if and

only

if for any (u,{) ~ Zo(O)(Xo) , any two admlsslble variations

(61u,61y), t = 1,2, Of (U,y), Such that (6lu,61y) have

compact support in (0,-),

satisfy

f

[6~(t)61u(t)

- 61Ty(t)62u(tl)dt

= O.

o The addltlonal as assumed admissible, yielding

assumption is satisfied

In theorem

(1.3).

The

is given in section

theorem

(5.9).

if for example

preclse statement, (5). If Z is minimal,

This result

go(Xo) ~ 0 in system (1.1), as well

as a definition

of

then we may take Z' = Z,

is also true foe general systems

(1.2), see

theorems ( 6 . 3 ) and ( 6 . 4 ) . We now describe, a result more in the spirit of the original conjecture by Van dee Schaft. It is necessary to consider certain infinite dimensional manifolds of maps R ---> n x and

Rm, [~ c Rm, t ---> [u(t),y(t)), where t ---9 u(t)

right

continuous,

C ® or C w it put

is not

on these

shall

and

"manifolds",

therefore

t ----> y(t) is

clear what

eonslder

topological

since

the

these

only

Is

plecewlse

constant

continuous. Even if both u and y were of

domain

dlfferentlable

of

formally

the

as

structure should

be.

Is not compact.

We

functions

manlfolds,

and

derive

formal

results about then, which hopefully may b~ rlgorlzed at a later date. Consider first the manifold of maps IM,Q,m, defined as the union of all behavlour sets Ze as Z ranges over all minimal,

afflne, analytic and complete systems

(1.1)

with state space M, control constralnZ set ~ c Rm, and outputs in Rm, i.e. m = p. On

this

consists admissible

manlfold of all

we

suppose

variational

vaflations

of

the tangent space to it at (u,y), T(u,y ) NM,~, m,

fields (6u,6y), of compact support,

(u,y). We

define

a

(weak) sympleetle

corresponding to form on MM,~,m as

31 suggested by (1.33), by setting

(1.42)

~(u,y){(61u,61y),(62u,~2y)]-

;

[~2Y(t)T61u(t)- 61y(t)T62u(t))dt

-w

We now

the

make

usual

definitions.

A submanlfold M = NM,Q, m is Isotroplc

if p

restricted to M is identically zero. We say M is a Lagranglan submanifold if it is Isoteoplc and co-lsotropie. To be. precise, M is co-isotroplc if given (u,y) e M, and (Du,Dyl e T(u,y I HM, n,m then p(u,y)[(6u,6y),(Du,Dy)) - 0 for all (Su,6y) T(u,y)M

implies

consisting tions

that (Du,Dy) e T(u,y)M. Let ~Z be. the

submanifold

of

of the behavlour set Ee of a Hamlltonlan system ~ dcscrlb~d

(1.151.

Our

result,

theorem

(5.17),

by equa-

which is closest in spirit to that

expressed In the conjecture of Van der Schaft may be stated as :

Every submanlfold ~

is a Lagrangian submanlfold of NM, R,m

D

These ideas can easily be. extended to the systems (1.2). However we would llke to present these results for the g~nerallzed systems introduced in chapter (61, as In the original conjecture, but we have not yet resolved all problems In dealing with such systems. Inpartioulsr it is not clear how to deal with existence and t~%lquoness

of

minimal

realizations

when

the

external

variables

belong to a

general

manifold, as discussed earlier.

SECTION 1.7

It is clear from the statement of our main results, that our work does not solve the Hamiltonlan (1.4);

realization

but rather

problem aS do

they characterize

those

the results

of Jakubczyk

Input-output maps,

In section

or external

beha-

v!ours which have Hamlltonlan realizations, as do theorems (1.2) and (1.3). However Jakubczyk's results are comprised of two parts, one part guarantees a ~eallzatlon,

and the other part provides extra alg~bralc conditions which ensure that

the realization may

be_ taken to be Hamiltonlan.

One therefore naturally expects

the extra alg~bralc conditions to ba 9quivalent to the conditions we give. Indeed Van

der

Schaft

[V2] shows

the equlvalenee

in the

case

of

linear systems.

The

general situation will be. dlscussed in chapter 7.

In chapter

(2) of

this monograph we

introduce

our

~rslon

of

th~

~riational

systems and adjolnt variational systems, in the context of a control system (I.1). At a global level thls involves the introduction of two new systems derived from (1.1), which we c811 the Hamiltonian extenslon, and the Prolongation. (3) we

consider

Prolongation,

as

the mlnlmelity a result

of

In chapter

properties of both the Hamiltonlan extenslon and minimallty

properties

of

the

original

system.

In

32 o~pter

(4)

we

Introduce

th~

concept

of self

adjolntness

systems, and establish an important Intermediate result,

for

the

theorem

varlatlonal

(4.2).

This IS

simply shared as : - A mlnlmal system is Hamiltonlan if and only If It's varlatlonal theorem

systems

are

self

adjolnt.

This

(I,5) in Jekubczyk's work

[J1].

adjointness and the criterion

(1.33)

result

plays

In chapter

roughly

the same

role

(5) the equivalence

as

of self

is established, along with a compilation of

our results. In chapter (6) we outline the work required to generalize oUr results to

systems

equivalence dltio~

desorlbed of

(theorem

our

by

eq~atlons

self-adjointness

].6),

and we

presented in thls mono~aph.

(].2).

Finally

condition

discuss

some

in ohspter

wlth

possible

(7) we

Jakubozyk's extensior-9

show

algebralo be the

the eon-

theory

2.

We are concerned

VARIATI(~NAL AND ADJOINT VARIATIONAL SXST£MS

with nonlinear

control systems Z wlth an equal number of Inputs

uj and outputs yj m " gO(x) + j~IY uj£j(x)

(2. I)

, x e M

, x(0) - x 0

Z: Yj

-

5(X)

, j = I,...,m

, u = ( u 1 , . - - , u m) g (~

c

Rm

As In section 1.1 of chapter (I) M denotes the state space, which is assumed to be a

k-dlmenslonal

slmpllclty the

dlfferentlable

manlfold,and

~ is

the

control

space,

whlch

for

is taken to be an open subset of Rm. eontalnlng 0, The uj appearing in

right-hand

slde

of

the

differential

equatlon

belong

functions of time t, called the admissible controls. is admissible

If the

corresponding

solution

to a

certain

class

of

Baslcally a control function

of the differential

equation

is de-

fined. For our purposes we may restrict the admissible controls to the ~lecewlse constant right continuous functions. Moreover we assume the vectorflelds gi' I = 0,1,---,m, to

be

complete.

This

implles that for every control function the

solution of the differential equatlon Is well-deflned for every t ~ R. Finally Hi, j = 1,..-,m, are functions from M to R. Our major assumption wlll be that all data involved, l.e. M, g0,gl,,..,g m, HI,..-,H m, are real-analytlc. A

subclass

systems

of

as

the

nonlinea~ systems (2.1} is formed by the Hamiltonlan control

discussed

in

the Introduction

[VI] for a detailed treatment and references. symplectlc

form

Necessarily

M Is even-dlmenslonal,

called

Hamlltonlan

~(C,-) = - dH. We dlnates dlnates

~ (i.e.

If

then

u Is

there

and section

1.3 of chapter

a non-degenerate

two-form

such

that

say dim M = k - 2n.

A vectorfleld

exists

H : M --> R

a

write f = X H. Darboux's

(I) see

Let M be a symplectlc manifold wlth

function

d~ = 0). f on M is

such

that

([A]) there exist coorn (ql,-.,qn,Pl,-*,pn) for M such that locally m = Z dPl ^ dql. Such cocoI-I are called canonical. In canonical coordinates a Hamil~onlBn veOtorfleld

X H has the familiar form ql = --ap ~H I, Pl = - ,~~H __

theorem

i - 1,..-,n.

Now assume that the state space M in (2.1) is a symplectio manifold (M,~) and that the

vectorflelds

loca ll 7

there

requiring

gj

exists

are a

glven

as - XHj, j - 1,.-',m.

function H 0 such

that gO " XH O"

Furthermore

that

(This Is equivalent

that Lg ~ ~ 0, el. [A]). go Is called a ~ocally Hamiltonlan

and wlll be denoted by X 0. Then the resulting system

suppose

to

vectorfield

34 m

2n

=

Xo(X) - j-~IT uj XH.(X)j

,

=

5(X)

,

x e(M

,m), x(O)

x0

=

(2.2) yj is

j

=

1,---,m

u

=

(u1,-.-,U

m)

eal

c

{~

a Hamiltonlan system. If there exists globally a function H 0 such that

called

go = XH 0 then the system is called globally Hamiltonlan.

We wish

to

explained (2.2),

gi~

later

necessary on)

and

nonlinear

sufficient system

conditions

(2.1) to be actually

(as will

a Hamiltonlan

be

system

i.e. for the exlstence of a symplectlc form on the state space M of (2.1)

such that (2.1) equals the Input-output

42.2).

These conditions will be glven entlrely in terms of

behavior of any variational

system of (2.1) and the Input-output

behavior of a related linear system, called the ~ We

for a minimal

shall

now

nonlinear

deflne

system

the

(2.1).

variational For

any

system.

and adjolnt

initial

state

system

x(O)

along

= x 0 we

a solution take

a

of a

coordinate

nelghbouchood of M contalnlng x 0 and let x(t), t ~ [0, T], be the solution of (2.1) correspOnding

tO an input function u(t)

x(O)

that x(t) remains within this coordinate neighborhood.

=

x 0 such

resulting variational

output

by

system

y(t)= along

=

[u1(t),---,Um(t) ) and the initial state

[Y1(t),-.-,Ym(t))

the

with

stat~-input-output~

yj(t)=

Denote the

H~[x(t)).4 Then

the

trajectory [x(t),u(t),y(t)) is

glven by the time-varying linear system

~(t)

ag O

° ~ {x(t))v(t)

m

agj

+ o~ uj(t) ~ {x(t)}v(t)

(2.3)

m

+ ~ ujVgj{x(t)}

j=1

j=1

~k

aH. y (t) = ~ (x(t))v(t)

,

ag i where ~ denotes the k x k Jacoblan

j - 1,..-,m , v(0) - v 0

matrix of gl : Rk ~ >

aHj Rk and~-~ is the

I x k

Jacobian matrix of Hj : Rk --> R. Furthermore u v = (u~,...,u~) e #~m and Yv , (ylm .,.,y y v) ~ Rm denote the inputs and outputs of the variational system. The system (2.5) is called variational because of the following. Let {x(t,g),u(t,g),y(t,~)), torles

of

(2.1),

t e [a,b], be

parametrlzed

a

family

of state-lnput-output

trajec-

by ¢, such that x(t,O) = x(t), u(t,O) = u(t) and

y(t,O) = y(t), t e [a,b]. Then the quantities (2.4)

v(t)

satisfy

ax(t,o) , uV(t) a~

au(t,o) , yV(t) De

ay(t,o) De

42.5). We note that in ease of a fixed initial state x(0) = x 0 the varla-

tlonal state v(0) at time 0 is necessarlly 0. The

ad~olnt (variational) system along

the same trajectory

(x(t},u(t),y(t)l

is

35

obtained by "duallzlng" the variational sysbem ~,o the linear time-varylng system @gOT m agj T m BH.T -p(t) = (~-~ ] (x(t))p(t) + Y u (t) (~-~) Cx(t))p(t) + Z u a ( 3 ) (x(t)} j-~l- j j-l J "

(2.5)

yj(t)

-

pT(t) gj(x(t

) , j : l,...,m

,

p(0) : PO ~ Rk

with i n p u t s u a - {u~,

transpose). The fundamental lemma connecting variational and adjolnt systems is LEMMA 2,1

of the variational and adjolnt system corresponding to the same state-lnput-output trajectory the following Identlty holds Along solutions

(2.6)

d---pT(t)v(t) dt

.

[uV(t))Tya(t)

-

[ua(t))TyV(t)

Furthermore the adjolnt system is uniquely determined by (2.67.

Proof By d i r e c t

differentiation

we o b t a i n

dt pT(t)v(t) ~ pT(t)v(t) + pT(t)v(t) " m @gj @go Ix(t)) - j~l Y uj(t) pT(t) ~-~ (x(t)) = {-pT(t) ~-x T @gO + p (t)(~-~ (x(t))v(t) + m

= NOW

m ~

u (t) j-1 j

@gj

m

a

BHj

Z u(t) ~

(x(t)))v(t)

j~1 j m

(x(t))v(t) +

u.V(t)g. (x(t))) j:1 a J

m

j:l

ua.(t)y .vCt) + a a

v a ~ u.(t)y.(t). a

j~l J

let p(t) • F(t)p(t) + G(t) ua(t), ya(t) = H(t)p(t) be

varying

an

arbitrary

time-

linear

system. Suppose It satisfies (2.67 for any uV(t), u a(t). Then @go T. m agj necessarily F(t) ~ - ( ~ } Lx(t)] - ~ uj(t)(~-~ }T(x(t)), the j-th column of G(t) aH. T j-I equals - (~-~J) (x(t)) and the j-th row Of H(t) is gT(x(t)). SO the system equals the adjolnt system.

Q

We may also add the variational or adjolnt system to the orlglnal system (2.11 and regard them as one system. We call the original system together wlth the variatlonal system, i.e.,

36 m

x(t)

- gO(x(t)) +

uj(t)gj[x(t)]

~(t)

- ~-£ [x(t))v(t) +

[

j=1

8g0.

m [

m

Bgj

uj(t) T~ (x(t))v(t) ÷

J:1

(2.7)

Ujv (t)gj (x(t))

j:1

y j ( t ) ~ %Ix(t)) j - 1,...,m

@H.

y]'(t). with

inputs

uj and u~, o u t p u t s yj and y~ and

42.1)

or prolonged system. The original system together with the adjolnt system m [ uj(t)gj(x(t)) j=] m

~(t)

- gO(x(t)) +

!~(t)

- - [~-~ ) ( x ( t ) ) p ( t )

2go T

~

j : .l yj(t)

(x,v),

the

prolongation

of

~gj T

[ U.(t)(~-~ ] ( x ( t ) ) p ( t )

j:1 j

8H. T

m

(2.8)

-

state

a uj(t)(T~

,~

) (=(t)]

= Hj(x(t)) j - 1,-..,m

yj(t) = pT(t)gj[x(t))

with

inputs

extension

(u,ua),

of

output

(2.1).

This

and

state

(x,p),

terminology

(y,ya)

wlll

become clear as we will now g i ~ a

is

called

the

Hamlltonlan

coordlnate-free definition of both systems (2.7) and (2.8), which also shows that the prolongation and Hamlltonlan extension are globally (not just in a coordinate nelghbourhood) defined systems. First we give the definition of a prolongation (or complete llft, cf. [Y]) of a function and a veetorfleld. Let H : M - - >

R, then the prolongation H : TM---> R is

defined by

(2.9)

~(x,v)

- dH(x)v

,

v a T M x

Given local coordinates (x],..-,xk) for M we obtain natural coordinates (xl,...,xk,vl u xl'''''vk = Xk ) for TM. In thes~ coordinates, H is

(2.10) Let

f be a

H(x,v)-

k X I=1

vectorfleld

__(ft), : TM---> TM is the

just

g i v e n by

33-~Hx(x)v.. j J on M, w i t h integral

the above natural coordinates

integral

flow of the

flow f t

: M---> M~ t e [ 0 , ~ ) .

prolonged

wctortleld

Then

f on TM. In

37

k

(2.11)

f(x,v) -

~ fl(x )

+

a

I

k

~fl

Z

~.

(x) vj

a

i,jo,

Denote the natural projection from TM to M by ~, Then for any function H : M-->

R

(2.12)

In

we dgflne the vertical llft (cf. [Y]) H ~ : TM---> R of H slmply by

H£ = H o

local

coordinates H£(x,v) * H(x). For any veetorfleld f

on M we l e t

the

vertical llft f£ be the vectorfield on TM such that

(2.13)

rZ(A) : {r(~)}~

rot any

M : M--~ R

In [Y] it Is shown that thls determines f£ uniquely as a vectorfleld, and moreover that f£ in natural coordinates is simply given as k

<2.~4>

f~<x,v> =

[

a

f~<x>

After these preparatlons, we define the prolongation of (2.1) as the system

m

.~p (2.15)

m

" (Xp) + ~ v - ~0(Xp) ÷ j=~l ujgj j~l ujgj(Xp)

yj - H~(Xp) v yj ~ Hj (Xp)

Xp E TM, Xp(O) - (Xo,VO) j = ~-,,~m u

- (ul,...,u m) ~ o ~ Rm

V U

It is easily seen that in natural coordinates For

the

bundle

definition has

a

=

(x,v) for TM (2.15) reduces to (2.7).

of the Hamiltonlan extension we note that T M as a cotang~.nt

canonically

(xl,-..,xk,Pl,..-,pk) for

defined

sympleotlc k T'M, fl Is given by [

form

vectorfleld f on M we associate

a

0. In

natural coordinates

dp i ^ dx I. Furthermore wlth

l=l

(2.16)

V V (UI,''',U m) ~ Rm

any

,

function H f from T M to R by setting

Hf(x,p) - - pTf(x) ,

p a T: M

where <,> Is the natural pairing between TxM and TxM. For notational ease we wlll often

write

pTf(x) instead of H f. Finally denote the projection from T*M to M by

w. Then the vertloal llft of a function H on M is again defined by

38 (2.17)

H E = H o 7.

The Hamlltonlan

extension of (2.1) is now given as m

(2.18)

YJ

m

+ ~" u.x ( x ) + ~' u a X ~(xe) j'1 J Hgj e J,~1 2. Hj

xe " XHgo(Xe) (xe)

xe ~ T M, Xe(0)

a gj yj = H (x e)

u

= (u l , - - - , u m) ~ n c Rm a

a

of

course

XH

denotes

the

(~(XH,-) = - dH , H : T M---) R. It (x,p) the

(2.18)

Hamiltonlan

keep a

-y

for T*M

extentlon

is

the same sign convention instead

"adjoint"

of ya. Notice

a and outputs yj,

see also chapter We

reduces

conclude on

system

(2.5)

the

TM, resp. have

TM,

resp.

T*M.

trivial bundles) The

above

easily

seen

that

in

defined by

natural

coordinates

given

before.

We note that

itself

Hamiltonlan

system.

(In order

a

globally

as in (2.2)

furthermore

one should take -u a instead

that

the "adjolnt"

prolongation T*M.

been

the

inputs

uj

inputs u? J correspond

and Hamlltonlan

In contrast,

only

of u a and

correspond to the

to

to

the

outputs y~,

defined

However

the on

actually

in general

extension

variational

coordinate

are globally

system

nelghbourhoods

of

M.

can be extended to trlvlailzlng

(excep~

for

the

case

defined

(2.3) and adJolnt

that

TM and

It

is

charts T*M are

the variational and adjoint system cannot be globally defined.

coordlnate-free

definitions

us to give a coordinate-free

(2.19)

d dt

x

of

prolongation

and Hamiltonian

version of Lemma

ya (t)]

je

(t) = [uV(t)

symplectic

along the prolongation

pairing between T X M and T X M.

form

Im

extension

2.1. Namely

ua(t)]

0 je is the linear

ferentiation

T M

(2.8)

also enable

where

W

on

to the expresslon

easily seen that these definitions of

is

veetorfleld

(6)).

that

systems

Hamiltonlan

a

= (u1,.--,u m) E d a

u where

(Xo,P0)

-

1,---,m

j =

o

and Hamiltonian

on

Rm×

~ , ~o

extension.and

denotes

dif-

<'>x denotes the

3. MINIMALITY OF T H E PROLONGATION

Consider

a nonlinear

system

AND HAMILTONIAN

EXTENSION

(2.1). Denote by L the Lie algebra

generated

by all

the vcctorflelds g0,gl,..,g m under Lie bracketing. Denote by L 0 the ideal in L generated

by

the

vectorflelds

g1,...,gm. L0 is called

and L 0 the strong accesslb111ty algebra.

the accessibility

algebra

It is well-known [$23 that the system Is

strongly accessible if and only if dim L0(x) - dim M for any x ~ M, where Lo(X) c TxM. (S~rong accessibility means t h a t

= spanRlf(x)If~L0}

the set o f points which

can be reached at any time T > 0 from any point x e M by choosing different input functions contains a non-empty interior with respect to M ([$2]).) Furthermore denote by H the linear space of functions with

fr' r = 1)...,s) equal

([HI), we may take f

of the form LfiLf2...Lfs 5 ,

to gi' i = 0,1 ..... m, and j = I ..... m. Equivalently

of L.H is called the observation r It follows from the analyticlty assumption that the system is observable if

space.

and only

to be arbitrary elements

if H distinguishes

there exists

an H e H

points in M,

such that

i.e. for every xl,x 2 ~ M

with x I ~ x 2

H(x]) ~ H(x2), el. [H]. (Observabillty

means

that for every pair xl,x 2 ~ M with x I # x 2 there exists an admissible control such that ~he output functions resulting from the x(0) - x2, are different).

initial

conditions x(0) - Xl, rasp.

The system is weakly observable

if H only distinguishes

nearby points in M.

We

call

a

observable. requires [HI.

nonlinear

minimal

if

observabillty and accessibility,

(The reason

implies

system

is that for Hamlltonlan

strong acoesslbiliby;

system Is called quasi minimal We will

it

is

stronglz, accessible

as

well

as

Thls definition Is slightly stronger than the usual one where One only

see Remark

1.8. dim L(x) = dim M for any x ~ M, or. systems observabillty 2 after

Proposition

and acoesslblllty 3.4.)

A nonlinear

if it Is strongly accessible and weakly observable.

now relate the accessibility

and the obsecvablllty

properties

of a non-

linear system (2.1) and its prolongatlon. The basic connection is contained in

THEOREM 3- l

Consider a nonlinear system Z g i w n

by (2.1) with strong accessibility algebra L 0

and observation space H.

a.

The

strong

accessibility

algebra

of

the prolongation

is

given by

LoP = ~o + Lo ~' where LO~ . {f~{f ~ % 1 and L° . {~lf ~ Lol. b.

The observation space of the prolongation is given by IIp - H + H ~,

.~

=

IH~l. ~ .I

and H -

{~IH ~

hi.

where

40

In order to faollJEate ths proof of thls theorem and for later use we flrst list some ldentltles for Lie dgrlvatlves vectorflelds and functlons.

of

prolongations

and

vertical

lifts of

LF.~.A 3.2

Let M be a manifold. Then for any vectorflelds f, fl'f2 on M and any function H : M ~ R the following Identltles hold:

a.

[61,f 2] =

b.

[fl,f~] = [t" I , f 2 ]~

o.

[fl 'f2 ] = 0

d.

~(~)

e.

i:(H~) - (f(H)) ~ = fZ(l~)

f.

f£(H £) = 0

Proof

~1';f2]

-

a. In natural coordlnates (x,v) for TM af. = fj -...+

v ~

af2 [fl 'f2 ] " I ax af2

, j - 1,2. Hence

0 ] af2

~{~-~ v) ~ - j

[

af2

~ ~

rl

_

all

i all aTa

all

af I af 2 v) Y~-3 [ b-x--

all

rI - ~

Ill 'f2 ]

r2 =

af 2 af I v] r 1 +

a-~"

D-T"

a v - -~

afl (bf~- v) r 2

af I af 2 ax ax

,f2]vJ

41 b. Since f2 = f2 ~-v we have

"0

fl

0

i Ir L~-- o jL-~ af 2

0110}

If: Ia

all

all

at I

(~'~- f l

- ~'-'~ f 2 ) ~'v " [ f l ' f 2

]~

Om can be proved slmllary, while d,e,f can be proved using the local coordinate expressions for H and H ~.

[]

Proof of Theorem 3.1 a,

The

accessibility

torflelds g0,gj,gj,

[gl,gj]" "~ -

~

we c o n c l u d e

algebra L p of

j=1 ....m,

that L p = L + L l ,

observation

LkiLk2.. .LksHj£ i - 0, I, ....m,

prolongation

[~i,g~] - [gl,gj]~ and [gl,%]~ ~

,

It follows that the ideal

b. The

the

and

is

generated

by the

vec-

on TM. Since by lemma 3.2 a,b,c,

where Of Course i -

L~ c Lp

space H p of

-

o, I-o,I . . . . .

{fir

prolongation

Lk]Lk2...Lksg,

wlth

j-l, ..... ,m

Is spanned by

kr, r - I ..... s,

.....

{f~lf

G L} and L £ -

generated by gj,gj," £

the

re,j-1

is

all

~,

~ L}. given as

functions

equal

to

gi'

or gj£, j = l,...,m. Using Lemma 3.2 d., e, f it follows that H p Is

spanned by the FunctionS (LfILC2"''LCpsHJ)£. andH £. Lf ILc2...L£sHj , wlth fr' r - 1,..,s equal to gl' i " 0,1 ..... m. Hence H

- H +

[]

The main use of Theorem 3.1 is contained in

COROLLARY 3.3

Consider a nonlinear system Z given by (2. I). Then

a.

E is

strongly

accessible

if

and

only

if

its

prolongation

Is

strongly

accessible. b.

Let Z be strongly accessible.

Then Z is (weakly) observable if and only if

its prolongation is (weakly) observable. c.

E is (quasi-)minlmal if and only if its prolongation Is (quasl-) mlnlmal.

42

Proof

a.

By Theorem

3.1 LoP = LO+ LO£. H e n c e

TM, if end only if dim Lo(X) - dim M for e v e r y

b.

By

dH(x):constant

Theorem

3 . 1 1 ~ = H + H ~. Since E is

spanR{dH(x)IH g H I has dimension

for

constant

dim LoP(x p) = dlm TM foe

any Xp

x • M.

(strongly)

dimension

accessible and

([H]).

analytic

Similarly dHP(xp) has

every Xp ~ TM. Furthermore E Is weakly

observable

If and

only dim dH(x) = dim M([H]) and the prolongatlon is weakly observable if and only If dim dH p= dim TM. Hence E Is weakly observable if and only If its prolongation is weakly observable. Now suppose the prolongation is observable. Take two xl,x 2 e M, with Xl~X 2. Then (Xl,O) and (x2,0) ~ TM. Hence there exists a H p e H p such that HP(xI,0) W HP(x2,0).

Since

points function

H(x,O) = 0 for every x e M

H • H it follows that there exists a functlon H ~ H

and

such that H£(xl,O) ~ H£(x2,0)

or equivalently H(x I) M H(x2). Hence X is observable. Conversely assume that E observable.

Take

two

polnts

(xl,v ~)

and

(x2,v 2)

HP(x2,v 2) fop every H p ~ H p. Since H £ c H p this yields

is

in TM. Suppose HP(xl,v I) = x I = x 2 = x.

Then,

slnc~

~H (x)v 2 for all H E S. Because dlm dH(x) = dim M = H p we have that ~~H (x)v| = ~-~ for any x e M, we obtain v I = v 2.

c. Follows immediately from a. and c.

For the connections

between strong accessibility and observablllty

system and of its Hamlltonlan properties. the

extension,

It appears

This is due to the fact that for a Hamlltonlan system

Hamlltonlan

observablllty.

extenslon)

strong

accessibility

o£ a nonlinear

that we have to combine both

is

almost

(and hence for

equivalent

Recall that on a sympleotle manifold (M,~) the Polsson

wlth

bracket of

two functions l s defined as

(3.1)

{F,G) ~ ~(XF,X G) = XF(G),

F,G:M ÷ ~,

and satlsCies the basic identity (3.2) This makes

ixF, X G] - XIF,G } C~(M) Into

a

Lie

algebra

under Polsson brackets and by (3.2) C'(M)

modulo the constant functlons is Isomorphic to the Lie algebra (under Lie track-

43 ets) of

Hamlltonlan

v ~ o t o r f i e l d s on (M,~). In canonical coordinates

(q1,..,qn,

pl,..,pn ) the Polsson bracke~ equals the famlllar expression n

(3.3)

{F,G} = ~

( aF

i~

aPi

aO

aF

~O

aqi

aql

aPl

PROPOSITZON 3.~ ([Vl,V4]) Consider a Hamiltonian system (2.2). The observation space H is equal to the ideal generated

by

the

functions HI,.., H m within

the

Lie

algebra

(under

Polsson

brackets) generated by the functions HO, HI,.., % . Furthermore

the strong accessibility

algebra

L 0 is equal

to the set of vector-

fields X H with H • H. Consequently dim Lo(X) - dim M <::> dim d~(x) = dlm M

sp~nR{dH(x)IH ~ aJ).

(de(x)

-

Proof

([V1,V4]) H is generated by functions Lfl Lf2..Lfs Hj with j=1, .., m, and

fr' r=1,..,s, equal to XHi, I=0,1 .... m. By (3.1) thls is equal to

{5,{~2,{...{F,5~...tJ}, .... m. with

within1

Furthermore L0 is generated j=1,..

, m,

and

r,1

. . . . m, a n d F r,

.... s, equaltoH

i ,

I-0, I,

by veotorflelds [fl,[f2,[ .... [fs' XHj]'']]]

fr' r=1 ..... s, equal to XHi, i-0, I ..... m. By (3.2) this is

equal t o a H a m l l t o n l a n v e c t o r f l e l d

c o r r e s p o n d i n g t o a f u n c b l o n I n H.

Remark I Note that even if H 0 is locally defined, the obser~dtlon space H consists of globally defined functions.

Remark

2 For an accessible

analytic

system

the

dimension

of

dH(x)

is constant

([H]). Hence it follows from Proposition 3.4 that an accessible, weakly observable Hamlltonian system is automatically stf0ngly accessible.

THEOREM 3.5

Consider a nonlinear system Z given by (2.1) with strong accessibility algebra L O and observetlon space H. Th~n the observation space of th~ Hamiltonian extension

is gt~n

by He = pTL 0 + H~, where p T L 0 = t p T f ( x ) l r

For the proof we again need some identities.

a L0t and . ~ = [H~IH e U].

44 LEMMA 3.6.

Let M he a k-dlmenslonal

T*M with its natural symplectic k form ~, in natural cooedlnates (Xl,..,xk,P1,..,p k) given by ~ = I dpiAdxi, and i=I the corresponding Polsson bracket k

manlfold.

aF

aG

I=I

Consider

- aF aG axl

Then for any vectorflelds f,fl,f2 on M and functions H,H~,H 2 on M we have a.

XpTf1(x)(pTf2(x) ) - {pTf1(x), pTf2(x)} = pT[fl,f2](x)

b.

XpTf(x)(H£)

c.

x

-

{pTf(x),H£) o (f(H)) £ = -XH£(PTf(x))

~(H2 ~) - {HI~,H2~I

- o

HI

(a) d.

[XpTf 1(x)' XpTf2(x)] " X{pTf 1(x),pTf2(x)}

e.

[XpTf(x), XH£] - X{pTf(x),H } =

f.

[x~, xH2~] - xlH1~,~2~!

(b)

X(f(H )

"

XpT[f 1,f2](x)

)~

0

Proof: Use the local coordinate expression (3.4) for the Polsson bracket.

Proof of Theorem 3.5

Gr, r:1,...,s, equal to a function pTgl(x),l~0,1 .....m, or Hj ,...,m, and G equal to a function pTgj(x) or 5 £, j=1,...,m. Therefore uslng Lemma 3.6, He is spanned by all functions pT[f1,[f2,[f

with fr' r=1,...,s,

equal

3 .....

[fs,f]..]]]

to a

vectorfleld gi' i-0,1,..,m,

vectoefleld gj, j-1 .....m, and all functions

and

f equal

to a

45 {pTr 1(x),(pTr2(x),{pTf3(x)...(pTfs(x),Hjg"(x e)~...I}} "

(LfIL f LC ..Lf Hj) & 2 3 s

j~1 ..... m,

with fr' r=1,...,s equal to gi' i=O,1,...,m. Hence M e - pTL 0 + H £.

COROLLARY 3- 7

Consider

a nonlinear

system Z given by

(2.1).

Then E is (quasi-) minimal

If and

only If its Hamiltonlan extension is (quasi-) minimal.

Proof Let Z be minimal. Take (xl,p~) and (x2,P2) in He(x2,P2 ) for Furthermore

since pTL 0 c H e

we

Because Z is strongly accessible sion is observable,

then

have

and M e = pTL 0 + H £ every

He(xl,pl) =

plTf(x) - p2Tf(x) for every f e L 0.

this yields pl-P2.

Hence the Hamlltonlan

exten-

and by Proposition 3.4 strongly accessible, Con~rsely

assume

the Hamlltonlan extension to be minimal.

for

T*M such that

all He~ H e. Since H- c He and Z is observable this implies x1=x2=x.

this implies

Since the prolongation

Is real-analytlc

that dim dBe(x e) - dim TWM and dim Lo(X)= dim M

Xee T'M, x E M. Furthermore

because .~c He the

system Z is

obser-

vable.

Remark

Note

that

observabllity

in

properties

8eneral of

the

relation

a nonlinear

between

the

and

variational

system

Its

acoesslblllty

and

or ad~olnt

systems (both viewed as tlme-varying linear systems) Is not so clear, On the other hand

it can

be easily

observable, I f

proven

and only i f

that a variational

system

Is controllable,

resp.

the corresponding a d j o i n t system Is observable, resp.

controllable.

For l a t e r

use we remark that

regarding the

observablllt~_ properties of

the

prolongatlon, resp. Hamlltonlan extension, we actually do not need the additional Inputs uv , resp. Ua , to distinguish between two states. Consequently we may put v ua and equal to zero:

u

46 PROPOSITION 3.8

Consider

the

system

(2.1)

wltb

observation

space

H

and

strong

accessibility

algebra L 0. a. The observation space of the truncated prolongation m

~p - ~o(Xp) + Z uj ~(Xp)

XpE TM

j=l

(3.4)

yj

- --Hi£(XP) j=l, ...,m

yjV = I~j(Xp) Is equal to the observation space of the prolongation, i.e. H + H ~. Hence If (2.1) is ( q u a s i - )

mlnlmal, then (3.4) iS (weakly) observable.

b. The observation space o f the t r u n c a t e d Hamlltonlan extension m

Xe = X (3.5)

[

j=l

uj X g

H3

(x e)

x E T*M

e

y j = Hi£(x e)

yj equals

HgO

(x e) +

a

j=1, ...,m

gj

= H (x e)

the observation

space of the Hamiltonlan

extension,

i.e. pTI~ + H £. Hence

if (2.1) is (quasi-) minimal, then (3.5) is (weakly) observable. P r o o f Conslder the p r o o f of Theorem 3.1 a

and

££(H), with C E L0and h E H, are superfluous

note

that

the

terms

in the construction

of ~ .

fE(~E) and Hence the

observation space of (3.4) equals H p = H ÷ H £, Slmllarly, observe in the proof of theorem 3.5

that for

the eonstructlon

XH£(pTL0), with H E H, are redundant~ rasp. Corollary 3.7.

of M e = pTL 0 * H ~ the 0bservab111ty

follows

terms

XH£(~D and

as in Corollary

3.3, []

4. THE SELF-ADJOINTNESS

In

this

chapter

we

HamIltonlan system

shall

show

that

a

CRITERION

minimal

nonlinear

system

(2.1)

is

a

(2.2) iC and only if every variational system is self-a~joln~±

which will be Interpreted as meaning that the Input-output maps of the variational system and the ad~olnt system coincide. So let us consider the analytlc nonllnear system (2.1)

=go(X)

*

(4.1)

m [

yj = % ( X ) ,

in which all local

uj gj(x)

, x ~M,

x(O)

= xo

j=1 J=1 ..... m,

the associated

coordinate

chart

(x,v)

u

e

~

¢ ~m

vectorflelds go + ~ ujgj, u ~ ~, are for TM the

variational

system

complete.

In any

along an adm[sslble

control function u(t) (e.g. plecewlse constant right continuous) is given as

~(t) - A(t)v(t) ÷ B(t)uv(t),

v(o)

- v0

(4.2)

yV(t) - C(t)v(t)

and in a local coordinate chart (x,p) for T*M the adjolnt system is g 1 ~ n

(t) = -AT(t)p(t) -cT(t)u a(t),

as

p(O) ,. Po

(4.3)

ya(t) = BT(t)p(t)

where ag o

A(t) = ~ -

m (x(t)) , ;

~gJ ( x ( t ) )

uj(t) ax

j-1 (4.4)

B(~) = (g1(x(t)) I .....

I gm (x(t)))

aH

c(t) = [~_~i (x(t))) j=~ ..... m l=t, In

such

local

coordinates

v 0 = 0 is given by

the

,k

Input-output

map of the variational system for

48 (4.5)

YV(t) = I Wv(t,a,u) uV(a)d°,

0 ~ t < T

0 Where Wv(t,a,u) = C(t)¢u(t,o)B(a),with cu(t,c) the unique solution of

(4.6)

~--at cu(t, o):A(t) cu(t, e)

cu(a, o)=Ik

Similarly, locally the input-output map of the adjoint aystem for P0 " 0 is given by

(4.7)

Ya(t) " i We(t'~'u)ua(°)da' 0

0gb gT

with a W(t,a,u)=-BT(t)~u(t,o)cT(a) and lu(t,a) the unique solution of (4.8)

As

a

Bt matter

(¢u(e,t)) T

Tu(t,o ) = -AT(t)Tu(t,0) of

fact

it

follows

I(o,o)-I k

from

cu(t,o)@u(o,t):¢u(t,t)

and (4.6) that

satisfies (4.8), and so actually

Wa(t,o,U) - -BT(t)[¢U(o,t))TcT(o) - -WvT(o,t,u). We

shall

now

show how Wv(t,a,u) and Wa(t,o,u) can be defined for all t, a h O.

Let

(t,o,x) + ~u (x) denote the flow of the tlme-varylng vectorfleld t,a m g0(x) + 7 uj(t)gj(x), where u.(t),t e [0,-), j=1 .... m, are plecewiss constant.

j~1

J

m

Because

by

assumption

the

vectorflelds

gO(x)+j-I ~ ujgj(x)

are

for

every

(Ul,..,um) ~ ~ complete, ~u is defined for all t Z 0, o ~ O and x e M. Indeed each mapping

x ~ ~

o(x) Is

the

concatenation

of a finite number of dlffeomorphlsms

x ~ YsU(x), where (s,x) ~ 7sU(x) is the flow of some complete associated vectorm fleld go +j~1 ~ --ujgj, ~ = (~I' "''~m ) ~ ~" The global definition of Wv(t,o,u) for each plecewlse constant control u on [0,®) can now be given as

(4.9) where

Wv(t,=,u)i J . dHi[~,o(Xo) )(~t,o].gj(~,0(Xo)) u (@ ,a),: T@o,uo(Xo ) M ÷ T@t,otxo)U . .M

Wa(t,o,u) is globally defined as

Is

the

derivative

t,o ~ 0 of

u Slmllarly ~t,o"

49 u

*

Wa(t,o,o>lj"EI,o t) d"j(<,o%))J where that

[,u,t)*: T )¢u° ' o t.X o ).M

(4.1t)

"" T u

.

.M is

the

t,o a 0 u

codlrferentlal

o f ~e,t" I t

follows

VPt' ol'Xo)

t,0 ~ 0

Wa(t,o,u) - - Wv(o,t,u) T

Remark If u is defined on (--,®)

the same

argument

shows

that

Wa(t,o,u)

and

Wv(t,o,u) one defined for t,e e (-®,=).

We now come to the definition o£ self-adjolntness.

DEFINITION 4. I

A ~arlatlonal adjolnt

system

(4.2) along a plecewlse

if the input-output map

output map

(4.7) of

constant

control

u is called self-

(~.5) of (4.2) for v 0 - 0 is equal to the input-

the adjoint system

(4.3) for PO = O; l.e. If ue(o) - uv(o)

for 0 g o ( t, then ya(e) = yV(e), 0 K o ~ t, for any t > O.

Remark

Using

following v(t),

Lemma

2.1 we

condition:

resp. p ( t ) ,

for

see

any

that

self-adJolntness

variational

is

also

equivalent

to the

control uV(t)-ue(t) the tlme-evolutions

o f the v a r i a t i o n a l , resp, a d j o i n t ,

system f o r v 0 = PO = 0 should

satisfy < p(t), v(t) > = 0 for all t ~ O.

Free

(4.5),

(4.7)

self-adjolnt

if

and and

(4.11) only

it

follows

that

a variational

system along u i s

iC Wv(t,o,u) - Wa(t,o,u) = -Wv(o,t,u) T, t ~ ~ ~ O, and

hence if and only if

(4.12)

For

Wv(t,o,u) - -Wv(e,t,u) T,

later

use

we

remark

t,o a 0

that Wv(t,e,u) may

be. decomposed

Into a

product

of

e

matrix only depending on t and a matrix depending on 0. As a matter of fact

"vct o o>ij-d"i(< and so

if we

pick a local

coordinate

o).C chart around

XO, and l n p a r t l e u l a r

natural

coordinates for TxoM and TxoM we may write

(~.t3)

wv(t,(~,u) = G(t,u)it(o,u)

where O(t,u) is the m x k matrix whose l-th row represents the cove(tot In T~oM

5O

(4.14)

(~,0)*dHl(<,o(Xo))

and H(o,u) is the k x m matrix whose j-th c o l ~ n

represents the tangent ~ c t o r

In

TM x0

(4.15)

(~,), gj(~],o(Xo))

Then the self-adJolntness

(4.16)

condition can be equivalently stated as

G(t,u)H(0,U) = - HT(t,u)GT(o,U),

t, a ~ 0

We now come to the maln theorem of this chapter. that

if for a certain u the variational system

i.e.

A(t)=A,

if u-O

and

B(t)=B, x0

is

As a preliminary remark we note

(4.2) is a tlme-lnvarlant system,

C(t)-C are all constant matrices

an

equilibrium

point),

then

(bhls happens

self-adjolntness

for instance

amounts

to

the

e qua I i ty

ceAtB

(~.17)

If

we

-

furthermore

minimal,

then

It

Hamlltonlan, matrix J(cf.

-

BTe-ATtc T,

assum~ follows

that from

Vt the

llnearlz~d

Theorem

1.2 that

l.e. ATj ÷ JA - O, BTj - C, for [B2]).

system ~ = Av + Bu v, yV = Cv, Is this

some

linear

system

non-slngular

is actually antlsymmetrle

The following theorem shows how thls extends to the nonlinear

case.

THEOREM 4.2

Let only

(4.1) be

a mlnlmal

if every

nonlinear

variational

system

system. along

Then the system any

plecewlse

Is Hamlltonlan

constant

right

if and

continuous

control u is self-adjoint,

Before provlng Theorem 4.2 we note that the theorem can be equivalently stated in the

following way.

(2.15)

consists

of

The

Input-output

map (u,uV) -~ (y,yV) Of

the

input-output

map of the orlglnal

w!th the input-output map Input-output

the prolonged

system

(4.1),

system

together

(4.5) of the variational system along u. Similarly, the

map (u,ua)-~ (y,ya) of the Mamlltonlan

the Input-output map of the orlglnal system (4. I),

extenslon

(2.18)

consists

of

together with the Input-output

map (4.7) of the adjolnt system. Hence Theorem 4.2 can be rephrased as

51 THEOREM 4.2'

A minimal map

of

nonllnear

its

system

prolongation

input-output

map

of

(4.11 is Hamlltonlan with

its

initial

Hamlltonian

if and only

if the input-output

stat~ Xp(O) - (Xo,O) Coincides extension

with

with

the

initial state

xe(O) = (Xo,O). First we prove the easy "only if" direction of Theorem ~.2.

Proof of Theorem 4.2 (==>) Consider a Hamiltonlan system (not necessarily minimal) on (M,m), together wlth

its prolongation on TM and Hamlltonlan

extension on T*M.

The symplectic form ~ induces a natural bundle isomorphlsm ~: TM ~ T*M defined

by

~(X) = m(x)(X,-), X ~ TxM. We will show that ~ Is actually an isomorphism between the

prolongation

and

the

Hamiltonlan

extension,

l.e.

we

will

prove

that

~(Xp) = X e along solutions of (2.15) and (2.181 respectively since

~* XO m XpTXo --

(4.18)

°

~, XHj - XpT x

j - 1 ..... m

Hj

pT(-XH )o~ = Hj 3 (Notice the minus signs In the last two equatlons. sign

oonventlon

in the

definition

They appear as a result of the

of a Hamlltonlan

systen

(2.2).)

By DBrboux'8

theorem there exist local eoordlnates x - (q1,..,qn,Pl,..,pn) for M such t ~ t n m - [ dPlAdq I • We may cheek (4.181 on every such a Darboux nelghbourhood. In 1=I these eoordlnat°-s (and the corresponding natural coordinates. (x,v) for TM and (x,p) for

T*M

where

p is not to be confused wlth pl,...,pn partial coordinates

for M), the mapping ~ is given by 0

(~.19)

~(x,v) = (x,Jv) = (x,p),

-I

where J - [ I

On) n

Therefore

52 I I2n o

0

_-1 I xO t _-1

X0

o (~

axo

j

.

Lx°fix° ax0

aXo

o m

.

LJ ~-- v

v

•,X T

aXo T

P X0

aX0 )T aX0 since ( ~ J + J [ a T } ~ o because X 0 is locally Hamiltonlan. Similarly ~, XHj = XpTXH "

Furthermore

J

- ~ . { ~0 ° )J {0 XHj )-(j° 5)--x.~j

m, XHj and

(pTxHj)" ~ ~ (jv)TXHj = -vTjXHj = FOP the proof of the " i f "

)v - - H..j

0

direction off Theorem 4.2 we need some intermediate

steps. LEMMA 4.3 Consider

a minimal

Hamiltonian

nonlinear

extension

Xo' VO = P0 = O. Then

have there

system

(4.1).

Suppose

the

prolongation

and

the

the same input-output map for the initial conditions exists

a unique

dlffeomorphlsm

(which is even Peal

analytic) ¢ : TM ~ T*M satisfying @(x0,O) = (Xo,O) such that @* gi = XpTg I

i-0,1,...,m

@~ gj

J=1

m X H ~_

,' . . , m

3 (4.20)

gj £o

¢ =

Hj ~

Hgj° , - H i

j=1

...,m

j-I ..... m

Moreover @ Is a bundle isomorphism which is the Identity on the base manifold M, and so has in natural coordinates the form #(x,v)=(x,¢(x,v)) for a certain map ¢.

53 Proof: By Corollaries 3.3 and 3.7 are minimal. M,

complete.

and the Hamlltonlen extension

Furthermore it is easily seen that if g is a complete veotorfield on

then g and g£ are

wctorfield

the prolongation

on

complete

T*M. Moreover

Therefore

conclude to the

we

can

existence

of

@(Xo,O) = (Xo,O) and (4.20).

vectorflelds

on

TM,

and XpTg(x ) _ _ is

a

complete

a vectorfleld X £ on T*M with H:M * R is trivially H invoke the Sussmann theorem ([$I, Theorem 5]) to a

unique

dlffeomorpblsm

~: TM ~ T M

satisfying

(Note that for the Sussmann theorem we actually only

need accessibility instead of strong accessibility.) Clearly the input-output behavior

~havior

of the variational

of the original system is not influenced by the

or adjoint system.

By the mlnimallty of the original

system there exists a unique dlffeomorphlsm from M Into itself mapping x 0 onto x 0 and

mapping the system into itself, namely the identity mapping. By uniqueness of

¢ it therefore follows that @ Is of the form ¢(x,v) =(x,¢(x,v)).

[]

LEMMA 4.4

Under the same

assumptions

as

in Lemma

antl-symmetric matrix m(x) depending

4.3,

there

on x ~ M such

exists

a unique

non-slngular

that ¢(x,v) = m(x)v, for all

v E Rk.

Proof

It

follows @H

gjT(x)¢(x,v) = ~

from H gj° @- Hj

(x)v, V V ~ R k,

(see 4.20) and ¢ ( x , v ) - ( x , ¢ ( x , v ) ) that

j-1 ..... m.

(see (4.20)) It follows that (LX T

Furthermore

HgJ)° ¢ " L

P gl (x) 3.2,

3.6)

that

from

¢*gi " XpTgl(x)

Hj, or equivalently

(see Lemma

gl

[gl,gj ] H o @. g l ( ~ j ) , j - l , . . . . m,

i=0, I , . . , m.

Since

@(x,v) = (x,¢(X,V)) this yields 3 Lg i (Hj ) - v, ax

[gl,gj]T(x)@(x,v)

V e R k.

In general for all v e Rk

(4.21) with

is

the

[f1,[f2,[f3...[fs,gj]...]]]T(x)¢(x,v) fr'

linear

r

In

-

1,..,s

v

equal

to

some g l '

a (LfILf2Lf3"'LfsH j )v - ~-~

1 = 0,1,..,

m. S i n c e

the

right

hand side

and the system is minimal It follows that there exists a matrix

w(x) such that ¢(x,v)= ~(x)v.

Since ¢ Is a dlffeomorphlsm w(x) Is nonslngular for

every x. It follows from (4.21) that ~(x) satisfies

54

(4.22)

[f1'[f2'[f3"''[fs'gj]" '" ]]]T(x)m(x) " ~x (LcILC 2"'LfsHj)

with the fl as above. On the other hand in local coordinates @,gj with ¢(x,v)=(x,w(x)v) yields

[

I

0

XH £ together J

I[i°I 0

[

aHj T

or equlvalently aH. ~(x) gj(x) = - (-B-~)T(x)--

(4.23)

j-1 ..... m

Furthermore by Lemma 3.2 and 3.6, and (4.20)

¢,[gi,gj] £ = ¢,[gl,gj £] - [XpTgI, XHj£] " X(LglHj)£ for i=0,1,...,m, j=1, . . . . m. As In (4.20) tbIS yields in local coordinates aLga,~_HJT m(x) [gl,gj] (x) = - {----~_ } (x)

(4.24)

In general we obtain

(4.25)

w(x) [f]'[f2"''[fs'gj ]'']](x) = _[aax LfiLf 2"''Lrs Hj)T(x)

with the fr, r=1,..,s equal to gi' i=0,1,..,m. Comparing (4.25) with (4.22), and invoking

once more the minimallty we conclude that ~(x) satisfies w(x) = -~T(x).

Furthermore since ~(x) Is non-slngular

and antl-symmetric

It necessarily

follows

that k = dim M is even, say k = 2n. LEMMA 4.5 Let ~(x) be the ~ Lemma

4. ~n

x ~

Denote

matrix as in Lemma 4.4, under the same assumptions as in

the

(i ,j )-th

element

o£ re(x) by ~lj (x). Then

the

two

form ~: = [ mlj(x) dxi~dx j is closed (d~=O), and so is a symplectic form on M. l,j=1 Proof By mlnlmallty II contalns locally ~

IndeI~ndent functions. Hence we can take

local coordinates (x1,..,x~) for M such that every coordinate function xi is of the form Lfibf2...LfsH j for a certain j ~ ~1,..,m} and certain fr' r-1,..,s, equal to gl, i=O,1,..,m. It follows from (4.22) that there exist 2n independent

55

vectorfields

(4.26)

k I , .... k ~

Bx i

(kl)T(x)~(x)

= ~x

= efT

I=1,...,~

wlth e I the l-th basis vector of R 2n. Denote the k 11, .... k 12n, Now

such t h a t

in L 0 of the form [f1,[f2..[fs,gj]..]]

component

functions

define the 2n x 2n matrix K(x) with (l,j)-th element

of

ki

by

equal

to

klJ(x). It @ollows from (4.26) that

(4.27)

K(x)~(x)

= I2n

Furthermore we know from (4.20) that (4.28)

1=1, ...,2n

¢,~i . XpTk I

In local coordinates

I

LB ~

( x 1 , . . . , x ~ ) thls yields

°][kl ]ik

(~(x)v) ,,(x)

~-~-~x)v

kk 1

° @(x,v)

-(T~-) (xb

or equivalently

(4.29)

~k 1, ,

~k i T

B (w(x)v)ki(x) + w(x) ~-~--£x)v + [B-~--) (x)w(x)v-O Bx

for all v

W r i t i n g o u t f o r v = (v I . . . . v2~) T

a

(~(x)v)

2n B=ks

=

( I

and so the k-th component of

2n

2n

T;

" " "" "£~£x)v)ki£x) " Is equal to

B~ks

Furthermore

the

£-th component

2n ~ks

2n

~=I s=~ ~ T %)

- s=1 I

(Z 9.=1

Bk i of ~ - ( x ) v

"~"~-~

is glven

vs

by s~l

k-th component of ~(x) ~kl (x)v by

~k I £ ..... Vs, and ~Xs

so

the

56 2n akl£

Z

(Z

~I

"~T-%)'Z

~:I mk£ s=1

s

2n

aki£

[Z ~£-~--~-)vs • £=I

s=1

Finally the k-th component of (3~kT)T(x)m(x)v Is given by

~kl£ £:I

~Xk

2n

~n

s$I

s:1

2n

~kl£

£:I

~£s

Since v : (v1,..,V2n)T is arbitrary we therefore obtain from (4.29)

2n

( , . 30)

~mks

£-,

~

~ki£

£-I

s

2n

Bki£

£:I

~-x-~-."~o

Furthermore

by dlfferentlatlon of (4.27) we have

(4.31)

Y £:I 2n

ak£J m:!'£ ~ ÷£=I ~" akl£

"~-x-s--

2n amk£

£:I 2n

~kl£

2n

8kI£

~

,: 0

2n

£=I

and

k £j

=

:l.,J, s : I . . . . 2n

am£k

£:I

s

2n

~s£

[

k I£

Insertion in (4.30) yields

(~.32)

2n

~mks

£o, I~x~

k I£

8mlk kl~

+ 'a'xs

8~S~ ki£)

+ ~

= 0

By non-slngularlty of the matrix K(x) we therefore obtain (4.33)

Now

~

equatlons

a-%-o + ~-"-k-- " o

(4.33) form

closedness of the two-form

exactly

~,~,s

thB

local

- , .....

ooordlnate

expressions

for the

57 2n =: = Z mljdxiAdxj on M i,j=l (see for instance [A, Chapter 2]).

O

Finally we prove the "if" direction oF Theorem 4.2. Proof oF Theorem 4.2 ( < - - )

In Lemma 4.3, 4.4, 4.5 we have deduced the existence of

a symplectle form ~: = X it

follows

It remains

i,J-1 for j=1,...,

that to

~lj(x)dxIAdxj on M. Furthermore from (4.20) or (4.23)

~

proved

m, ~ T ( x ) ~ ( x )

that go

~HJ x)'~x - (

is a locally

Hence gj- -XHj , j-l, .. .,m.

Hamiltonian

~ctorFleld.

Now

by

Darboux's theorem we can take local coordinates x =(ql,..,qn,Pl..,pn) for M such n that locally m = ~ dPlAdqi. In such coordinates @*g0 " XpTg 0 amoUnts to i=I

0

~go (x)v J JL~-'~-

=

_[

)T(x)j v

or equivalently

(4.34)

J ~-

(X) + [

)T(x)J - 0

0 -I ~ T ~go T where of course J - [ I v n n ) " Hence the Jacoblan matrix ~ ( g o J)(x)'(B-~-) (x)j is n symmetric, so there exlsts on thls Darboux nelghbOUrhood a func$1on H 0

(4.35)

for

which

aH 0 go T(x)J = " 3-x- (x).

Hence on t h l s nelghbourhood gO " XHO"

0

Remark Note that in general it is hard to glv~ conditions In order that the system is globally Hamlltonian,

i.e, in order

that the

internal energy H 0

is globally

defined. (Except for the case that M is such that every closed one-form is exact, e.g.

if M Is simply connected. However since we insist on observability simple-

connectedness is not a natural assumption, cf.[C4].)

58 Let

us briefly return

("If"-directlon)

that

to Lemma if p(t)

2.1.

It follows

is a solution of

from

the

proof

of Theorem

4.2

the adjoint system for an input

ua(t), then p(t) equals ~(x(t))v(t), where v(t) is the solution of the variational system with

input uV(t) = ua(t) and initial

condition v 0 = (~(XO))-Ip O. Hence for

a Hamiltonlan system, (2.6) becomes

d

(4.36) where

~$~x(t)(v1(t),v2(t) vl(t)

is the solution

with output yiV(t), i=I,2. maps

of the

proved

variational

in [VII

of the

variational

system

for the

input uiV(t) and

(Recall that for a Hamiltonian system the input-output and adjolnt

by direct

characterization

)=Cu~ (t))TY~(t)-CulV(t))TY 2v(t)

methods,

of Hamiltonlan

system

coincide.)

and will

serve

systems completely

as

Thls formula was already a starting

point for the

in terms of their variational

input-output behavior in the next chapter. Finally, in previous work [VI,GI,V3,V5] it has been shown that minimal Hamiltonian systems with the same the Sussmann follows

uniqueness

immediately

minimal

(M2,m2). Then,

of

the

corresponding

the same

input-output theorem

?2" Clearly, as well

as

equivalence

[$I],

but also symplectomorphle.

Hamiltonlan as

Tnls fact also

systems

to

the symplectic

map for

there

with

state

spaces (M1~ml),

in Theorem 4.2, theme exist maps #i: TMI~ T M i,

form ¢i(x,v) = (x,mi(x)v), 1=1,2,

matrices

uniqueness

theorem

as implied by

from the proof of Theorem 4.2, as we will now briefly show.

Let Z I and Z 2 be two respectively I=1,2,

input-output map are not only dlffeomorphlc,

where mi(x),i=1,2 , are

forms mi, Ii=1,2. Now let Z I and Z 2 have

groundstates x I

~ M

and x~ ~ M 2. By the Sussmann

exists a dlffeomomphism @: MI*M 2 transforming Z I

also the prolongations

extensions

mapping

the two

of Z I and Z 2. It is easily

prolongations

between the two Hamiltonlan extensions

Is given

seen that the

by ~.:TM I ~ TM2, and

by ( *)-I : T *MI~T *M 2. By the uniqueness of

these equivalence mappings we therefore obtain the commutative diagram T I (4.3?)

r *

~I

T*M I ~

~*

~

2

"i

@2

T,M2

i .e.,

(4.38)

~

° ~2 ° ~* = ~I"

Since ¢i(x,v)=(x,~i(x)v),

into

o£ Z I and Z 2 have the same input-output maps,

the Hamiltonian between

the

I=I,2, we see that (4.38) is equivalent to

59 (4.39)

~ ~°2 = ~ I "

so ¢ IS a symplectomorphlsm. internal

Note furthermore that it follows from (4.39) that the

~nergles HOI, I=I,2, of Z I and Z 2 on

their

domains

of

definition

are

related by (cf.[V]]) (4.40)

H02 D ~ = HO I + constant.

i.e., the internal energy of a minimal Hamlltonian system is, up to a constant, uniquely determined by Its input-output map.

5.

THE VARIATIONAL

CRITERION

The main purpose of bhls chapter is to establish t h e

exact relationship between

self afljolntness of the variational systems corresponding to a minimal system, and the criterion conjectured also

collect

bog~ther

by Van der Schaft, as described in the chapter (I). We

the main results

of the monograph and present them in a

u n i f i e d way. F i n a l l y we demonstrate formally that the external behaviour set of a minimal

Hamlltonian

system

is a Lagranglan

submanifold

of

the manifold

of all

behavlours from minimal systems.

SECTION 5.1

In this chapter we deal exclusively with complete, affine and analytic systems m

(5.1)

"

go (x) + Z u l g l ( x ) , x ~ M, i=1

-xV- = Hi(x),_ _ We s h a l l

usc

the

I g i < m,

definitions

in

x(O)

u

chapter

•

~

=

c

x0 Rm.

(1.6)

of ZT (O)(Xo), £~ (0)

(XO),

variations of elements in these behaviour sets, and the corresponding variational fields. If (~, ~, ~) e ZT. (O)(Xo), and t---> 6u(t) is any plecewlse constant right continuous Rm valued function defined on [0,®) wlth support contained in (0,®) we define a variation of u by setting

u(t,c)

= J(t)

* E 6u(t).

It follows that t --> u(t,e) is plecewlse constant for each e, but it defines

an

admissible input to t5.1) only if

(5.2)

Supposing

Jtt)

this

that

nelghbourhood

+ c 6u(t)

V

of

E n,

condition 0,

we

t

~

(0,®)

is satisfied

obtain

for each E in

a

sufficiently

small

corresponding varlatlor~9 ytt,c) of ~(t) and

x(t,e) of ~(t). That for each £ E V we have t ~ (utt,g), y(t,E), x(t,e)) E Z~(0)(x0),

follows from the completeness assumption of (5.1), equation (5.2) and

the fact that the support of 6u is contained in t0,®). It follows from the smooth dependence

of

solutions

of

differential

equations

on

parameters

that

the

varlatlonal field along (u, y, x), t----> (6u(t), 6y(t),6x(t)) exists and satisfies the required regularity conditions.

It follows that we may realize any plecewlse

constant Rm valued

support

function

with

contained

in t0,-) as

the

control

61 of

component

the

variational field

along any

(~,~,~) • ZT _ (O)(Xo) , as long as

(5.2) is satisfied for lel sufficiently small. We shall always assume that this is the case, which admissible smallest

for

example

occurs

when ~ = Rm.

We now

give

variations. We shall write supp f, for the support closed

set

outside

which

f vanishes.

Also recall

the

definition

of

or f, whlcn Is the

that

if (6u,6y) is a

variational field along (u,y) • Z~(O)(x O) then 6u(O)=0.

DEFINITION 5.1

An admissible variation (u,y) of (~,~) ~ Z: (0)(x O) is a variation which satisfies (i)

6u is

pleeewlse

TI, T 2 > 0 such

constant

that

and

supp

6u is

compact.

Thus

supp (u(.,g) - ~ (.)) c iT1, T2] for

there

every ¢

exists

sufficiently

small. (ll) supp 6y c supp 6u. (lii) Suppose supp 6u c (O,T), and (J',~') ~ Z~ (O)(x 0) are such that 5'(t) = ~(t) (and hence

~'(t) = ~(t)) for t ~ [0, T]). Define a map (t,~)-+ u'(t,E) by

u'(t,e) = u(t,e), t ~ [0,T], u'(t,c) = ~'(t), t E (T,®). We variation

We

of ~', and

require

the

hence

obtain

corresponding

a corresponding

variational

deduce

variation

field

setting

that u'

Is

a

(u',y') of (~',~').

(6u', 6 y ' ) along ( ~ ' , ~ ' ) t o

satisfy condition (Ii) also. Clearly we have 6u = 6u'.

From now

on

variation

we

of

shall

abuse

and

(~,~) ~ E~_ (O)(x O) (wlth

variational field of an the

notation

condition

(II)

compact

(admissible) variation

may

6y = 0 on (0, T I) also. is automatically

simply

be

For

satisfied

written

as

say (6u,6y) is an support),

(admissible)

when (6u,6y) Is

(u,y) of (~,~). Note

follows: -

if

6u = 0 on (O,T I)

variations (6u,6y) of (~,~) e Z~(O)(x 0) this by

causality.

On

the

other

hand

the

the

that part of

other

then

condition part

of

condition (ii), expressed as: - If 6u = 0 on iT2,=) then 6y = 0 on iT2,®) also, Is a

definite

constraint.

Loosely

- (6u,6y) is an admissible support

and

stated

definition

(5.1)

may

varlatlon of (~,~) • Z~ (O)(x0),

thls is independent

of the values

be

expressed

as:

if (6u,6y) has compact

of (u,~) occurlng

to the right

of

the support of (6u,6y). Before proceeding with a discussion of the existence of admissible

variations we

need a technical result.

LEMMA 5.2

If D is a

dense subspaoe

of a Hllbert

space

H, and S

is a finite

dimensional

sub, pace of H, then O n S ± is a dense subspace of S 1, with respect to Its natural Hilbert space structure S.

Induced from H, where S ± is the orthogonal complement

of

62 Proof

If 0 # a 6 H, w e

codlmenslon

one

flrst

closed

prove

subsp~c8

that D n a ± is dense of H.

Let s e a i ~ n d

In a ±. Note let

that a ±

is a

Idn} c D b~ a s e q u e n c e

such

that d ---> s as n ---9 ~. We may w r l t e d = ~ a + s where a & R and n fl n n n ± e a , with s ~ 0 as n ~ -. Now there exists B # O, B e ~, such that n I% (Ba + a ±) n D d O, slnoe o t h e r w i s e D c a ±, w h i c h c o n t r a d i c t s the fact that D Is I dense in H. Thus there exists r e a and B # 0 such that BB + r = d e D. Now

s

i

since that

D

D h a ± is

write

S

result span of a I

is a subspace.

=

dense

span

above

In a ±. Now

since

e --+ 0 we have d -+ s as n ~ ®, w h i c h shows n n since S is a flnlte d i m e n s i o n a l s u b s p a c e we may

{a I .... anl w h e r e a1,..,a ~ is it

follows

that

D n aI

an

is

orthonormal±basis

dense

in

al

"

of

But

Induction a r g u m e n t we

is dense in span

the

= as s u b s p a c e s + ± S . By a

~

o b t a i n for any r,

lar+1,..,an~

But a l ± n. . . n

iS" From

al

{8 2 .... a_I ÷ S 1- Thus (D n a~ ±) n a 2 i ls dense In a 2 ± viewed ± n ± ± We deduce that D n (a I n a 2 ) is dense In span la-, ..,a

slmp~e

S±.

However

+ S±

~ < r ~ n - I, D n (a I n .... n ~r ) i ± and for r = n. D n (a I n-..n a n ) is dense in

a n ±= S ±, s o D n S 1 Is dense in S ± as roqulred.

[]

Referlng to the notatlo,, and d e f l n l t l o n s of chapters ( 2 ) and (3) we r e c a l l that a :~y,~Lem (5,1)

is

called

quasl-mlnlmal

If

It

Is

strongly

and

accessible

o b s e r v a b l e . We now have the r e [ l o w i n g e x i s t e n c e r e s u l t for a d m i s s i b l e

weakly

variations.

P R O P O S I T I O N 5.3

Consider (u.y)

a

quasl-mlnlmal,

e Z+ (0)(xn~,

functions

~m), w h l c h

(6u,6y)of

(u,y,x)

analytic

T

>

6u, in L2([O,T];

S c L 2 ([0,T]; tion

and

(u,y),

may

with

Rm)

n S z,

of(u,y,x)

complete

exists for

be r e a l i z e d compact

Is the c o r r e s p o n d i n g e l e m e n t

(6u,6y,6x)

and

0 there

as

a some

system

dense

set

finite

a component

(5.1). o£

dlmenslonal of

Given

plecewlse

any

constant subspace

an a d m i s s i b l e

varia-

s u p p o r t c o n t a i n e d in (O,T). F u r t h e r m o r e If + of Zl(0)(Xo), the c o r r e s p o n d i n g varlatlon

Is s u c h that supp 6x c (O,T) also.

Proof Referlng to chapters 2 and 4 we may w r l t e the v a r i a t i o n 6x due t o the u v a r l a t l o n 6u in the f o l l o w i n g manner, where ( t , ~ , x ) ~ ~t,o (x) ls th~ flow of the m (tlme-varylng) vector f i e l d go(t) * Z u l ( t ) g l ( X ) , t-I m

(5.:3)

:~

-,:,

i

.~x(t) ,=:~X(*.E;0)~ in!

('*~i,.o),

.

.

.

.

63 We

that

claim

if,

as

in t h e

(6u,6y) is an admissible

proposition,

of

varlatlon

(u,y) with oomI~Ct support contained in (0, T) then m

45.~) From

=

o (5,3)

and

moreover

modulo

(5.4)

since

varlatlonal

U ) ~u o(Xo))~ul (o)d~" I for(*o,o,gl (o,

i=I

we

outslde

equations

our

claim,

Furthermore,

that

the

that

45.1).

that

Mot-rover

if

oondltlon

modulo

(u,y) with constant

then

the

6x(T)

= O, and

structure

or

the

proposition

compact support

the

follows,

is indeed

oontalned

valid.

in

(0,T),

6x(t), ] < i < m,

(u',y') ~ Z~(O)(x O) (5.3) and Is

then 6x'(t) = 0 for t ~ iT,=) and

shown,

of

u with

pair 46u,6y) meets

if 6u' = 6u and (6u',6y',6x')

Thus

ease

varlatlon 6y of y has compact support

the

fOe t ~ [O,T], then equations

also.

th~

thah 6x(t) : 0 for t ~ T. It

final statement

variation of

is

(5.4), it follows from the equations

that the ooPrespondlng follows

thls

[O,T], 6u(t) - O ,

guarantees

6yi(t) = dHl(~t,0(x0))

It

if

interval

(2.5)

if 6u is a

which satlsfles

deduce

the

our

(lll) of

olalm,

that

compact support

functions 6u on

satlsfles

u'4t) = u(t), y'(t) = y(t)

so 6y' has

the existence

contained

[O,T],

compact

(5.1)

(ll)

unchanged

corrcspond!ng

definition

and

is

for t ~

variation support

definition

[O,T]. Thus

of

(u,,y,,x,),

eontained

also met.

of admlsslble

of

in (O,T).

41)

(5.4) are

the

oontalned

conditions

in

We hav~

varlatlons

(6u,~y) of

in (O,T) depends only on flndlng

with

compact

support

in

(O,T),

(O,T)

thsrefoPe

which

plecewise satisfy

equation (5.4). Fixing a local coordinate chart for M about X 0 we may write equations

(5.4) in the

form

45.5)

o = ~ H(o,u) 6u4o)d°.

wh~re H(o,u)Ij is th~ i ' t h in

local

coordinates.

component of the vector (kbO,o) Wgj(~uo(Xo))_ ~ TxoM

Tbls

may

be

expressed

in

terms

of the Hllbert space

L 2 ([O,T]; Rm) as th ~. orthogonallty of 6u with the subspac~ S deflned by

45.6) If

; ~ ~

the

dense

subspaoe

on

[0, T],

with

admissible

variations

of

is

{H(.,u)~

functions

set

D

s-

functions

Inparticular

~t.

of L2([O,T];R m) conslstlng

compact

46u,6y) of

contained

(u,y) are ,In one to one

in D fl S ±. ttowever

non e m p t y .

support

by

iemma

of in

pl~oewlse (0, T),

we

correspondeno~

constant see

th.at

w i t h the

(5.2) D N S ± is dense in S ±,

and

64 It remalns to verify our claim. Assume to the contrary that m

(5.7)

O~v=[ I=t

From

proposltlon

iOt (~;,a), gl(@u 0(go )) 6ui(~)da.

(3.8),

and

the

quasi-minlmallty

of

(5.1)

it follows

that

the

following truncated prolongation is weakly observable. m

~P= ~°(XP)+i~I ul gl(Xp)' (5.8)

yj = H~ (Xp), J V

'

I g j ~ m.

there exists a plecewlse constant control ~ on [0,®), for which yV(t)

is not identically

t,o

TM

I g j g m

yj = Hj(Xp), Inpartlcular

Xp

zero on [0,®), when the system Is initialized at time

state % °

zero at

Otherwise the initial states band

(¥;,0(x0),0)_ for system

(5.8) would

be indistinguishable,

Which as in section

3

would contradict the fact that system (5.8) is weakly observable. Now de~ine a control u' on [0,=)

for system

(5.1)

by

setting

u'(t) = u (t), +

t ~ [O,T),u'(t) = ~ (t-T) foe t ~ T, and hence o b t a i n a pair (u', y') e Ze(O)(× O) as

in

(ill)

Conclusion compact

Of Definition of

support

(ill) in

to

(O,T),

(5.1). see

Since (6u,~y) is a dmlsslble that the resulting v a r i a t i o n

where 6u' = 6u. Applying 6u'= 6u

we may apply

the

(6u',6y')

has

to

the

also

variational

system of (5.1) along u', at time T we reach the variational state (see equations ul (5.3)) (@ % 0 ) , v e T u' M. H o w e ~ r for t > T the output 6y(t) of the va-

*T,o(Xo ) riatlonal equatlon along u', now oolncides with the output yV(t) of system (5.8), initialized at ~ . By construction this is not identically zero, contradlcting the P fact that supp 6y' c (O,T). We conclude that v = O, establishing our claim.

SECTION 5.2

We now prove some intermediate results from which we deduce our main results.

LEMMA 5.4

Consider

a

quasl-mlnlmal,

Hamlltonlan; i . e .

analytic

and

complete system which is

given by equations (1.15),

(locally)

Given any (u,y) e Z~(O) (Xo), and

admissible variations (6iu,61y) Of (u,y) wlth compact support, i = 1,2, we have S; (62Y(t)T61 (t) - 61Y(t)T62 u(t)) dt = O.

65 Proof Suppose that the support of ( 6 u , 6 y ) I s contained in (5.3)

if (6u,6y,6x) IS the corresponding variation of

supp 6x c (O,T). By equation (4,36), or Van

d_ ~(x(t)) (~ix(t) dt

62x(t))

der

=

(O,T). By proposition

(u,y,x) ~ T; (0)(XO) , then

Schaft [ V | ] ,

62Y(t)T51u(t)-51Y(t)T62u(t)

where ~ Is the sympleotlo form assoclated with the Hamlltonlan system (1.15). Thus

;0 (~2Y(t)T61u(t) 61Y(t)T62u(t)) at -

= m(x(T))(alx(T),62x(T))

- ~(x 0)(alx(0),62x(O))

= O.

[]

We r e c a l l from chapter (4) the d e f i n i t i o n of the kernel function Wv(t,o,u) which defines the response of the variational system along u. Moreover assuming, as we do always, that the system (5.1) has a stabs space M of ~imenslon R, we may select a coordinate chart about xo and factor Wv(t,o,u) as G(t,u)H(~,u), where G(-,u) Is an m x k matrix valued function for each control u. In terms of thls factorlzatlon we

may define

=~G(t,u) H(a,u) + H(t,u)TG(a,u)T, t > a (5.9)

KA(t'a'u)

L-G(t,u) H(a,u) - H(t,u)TG(~,u) T,

C < a

PROPOSITION 5.5 Conelder a quasl-mlnlmal, analytic and complete system (5.1). Suppose that for any (u,y) e E~ (0)(Xo), all support, i

(5.10)

=

1,2,

admissible

variatlons (81u,~ly) of

with

cc~paot

S0 (62Y(t) T 61u(t) - 51Y(t)T62u(t)) dt - 0.

Then there exists a matrix valued function G (5.]I)

(u,y)

satisfy

such that for t,a h 0

G(t,u)H(o,u) - H(t,u) T G(a,u) T - KA(t,o,u).

Proof We f l x

T > O and show that (5.11) is true for any t , ~ ~ [O,T]. As in proposition

(5.3) the oonstralnts on admissible variations (6iu , 5iy) oC (u,y) with support

66 contained in (0, T) may be expressed by equation (5.5) or

T

f

0 =

H(t,u) 61u(t)dt ,

I

=

1,2.

0 By substituting the relationships between 61Y and 61u, namely

ely(t) = f(~ wv(t,o,u)~lu(o)do

I = I , 2,

into expression (5.10) we obtaln T

t

(5.12)

[61u(t)TWv(t,o,u)62u(0) 0

-

62u(t)TWv(t,e,u)61u(o)}dedt

0

However the constraints above may b~ expressed as t

T

f H(t,u)61u(t)dt - -~ H(t,u)~lu(t)dt, 0 which when

t

substitutedinto

((5.12),

remembering that Wv(t,o,u) = G(t,u) H(o,u),

ylelds after some manipulation the following identities

.fT t o

f 0

61u(t)T(Wv(t,o,u)

T

62(o)Tdodt

T

Jf 0

* Wv(O,t,u)T)

0 61u(t)T(Wv(t,e,u) + Wv(o,t,u) T) 62u(o)Tdodt t

These identities in tUrn yield the following expression

(~3>

0o

To prove

the

situation. functions subspace

fo~ fo~ ~lu ~2u(o~dodt

identity

Let

H

(5.11)

be

the

of Rm valued

T f(t)Tg(t) dr, and let S be the on [0, T], with inner product = - fO 2 {H(-,u)Te; e e Rn}. Let H 1 be the Hilbert space L2([O,T]x[O,T];~"-- ) con-

sisting of m × m

matrix valued functions T 0

Note H I may linear

on [O,T] × [0, T]

with inner product

T

- f f

flnlte

from thls, we give a Hilbert space setting to OUr Hllbe~rt space L 2 ([0, T];R m) consisting

trace (f(t,e)g(t,o)T)dodt.

0

be viewed as oomblnatlons

the closUre of elements

(in H I) of the subspace

consisting of all

in H ~ H. (If f,g e H then f ~ g ~ H ® H

is the function in H I glven by (t,a) ~ f(t) g(t)T.) Let D be the dense subspace of

67 H, conslstlng of all plecewlse constant functions In H with support contained In (0, T). Similarly we let D I be the dense subspace

of H I consisting of all finite

linear combinations of elements in D 8 D. Let

S1

be the

elements

subspace of

H I, consls~Ing of all

in S ® H and H ® S. S 1 may

finite

be identified

wlth

llnear

combinations

of

the space

of all matrix

where K I and K 2 are m x k matrix valued functions with components

in L 2 ([0, T]).

valued functions of the form

H(t,u)TK1(a) T+ K2(a)H(a,u)

Now the ortbogonal complement of S I in H I, denoted $I" ,

is just the closure in H I

of the subspace conslstlng of all flnlte linear comblnatlo~ S±® S ±. Thus DIA $I ± is

the

tions

D f] S±® D r% S I. But

of

elements

in

subspace

consisting

S 1, so D I f] S11 is dense in $II. Noting m vectors trace (A a b T) T

o - ;01 o The constraints

=

that

by

of

all

lemma

of elements in finite

(5.2)

linear

comblna-

D f] S ~ is dense in

if A is an m x m matrix, and a,b are

bTA a, we may rewrite (5.13) as

T trace (KA(t,a,u)(61u(t)

(5.5) and the fact that

~2u(a)T)T)da dt

61U are

plecewise

constant

imply

that

61u ® 62u e Din $I i. It follows that (5.13) is equivalent to

= 0.

However

since D A

decomposed

Into

$1 ± is dense in $I i we conclude the

direct

s~

that K A e S I. New

S 1 may be

S I = sIA8 S| S where sIA(] $I S - 0, and sIA is

the

space of matrix valued functions.

(5.14)

for

some

KI(t)H(o,U ) - H(t,u)TKt(u) T matrix

valued

function

S K I ; and S I is

the

space

of

matrix

of

those

elements

valued

funetlons

K2(t) H(s,U) + H (t,u)TK2(c) T,

for

some

matrix

valued

function K2. sIA

consists

satlsfylng 0 = K(t,u) + K(o,t) T, whereas S] S satisfying 0 = K(t,o) - K(o,t) T. Clearly, so the repcesentatlon

consists

K

of

SI

of those elements K of S I

since K A ~ S1 we also have K A ~ sIA, and

(5.11) follows from (5.14).

[]

68 Before

continuing

our

serles

of

results

we

more

some

introduce

notation.

Let P~ (u) = P~ be the projeotlon onto the range of t f 0 By

standard

H(c,u) H(~,u)Tdc

arguments,

Brockett

n-] lUl (ti,ti+1] u (tn,®) Such dealingwlth with

analytic

of

we

may

partition

[0,-) into

that Pt is constant on each subinterval.

systems,

a discontinuity

[B3],

u.

and H(-,u) is plecewlse (HOWever

we shall

analytic,

not make

a union

Since we are

each t I colncldes

use of

this

fact.)

Note

that P~ P~ = PS P't : PS for S < t. We may also write Pt - HiRe, where Rt : "t(u) is

an

k x

kl(t)

matrix

satisfying

HtT~t : Ik1(t), the

kl(t)

x

k1(t)

identity

matrix, and k1(t) = dim r'ange P~. s Similarly we may define Pt' s > t, as the projection onto the range of fs H(~,U) H(a,u)Td~. t s ---> PtS is piecewlse

constant

projection

t.

+

+

for

each

as before so + Pt (u) = Pt= * sup Pts is a well

Moreover

+

t ----> Pt is

+

PS Pt = PS for s ~ t. We write Pt = Zt z T satisfying

defined ÷ p+ piecewise oonstan~ with Pfi s =

zT Zt = ik2(fi) P the

k2(t)

×

where Z t = Xt(u) is an k x k2(t) matrix k2(t)

identity

matrix,

÷

and k2(t)

= dim

+

range Pt' Note also that Pt H(a,u) = H(a,u) Cot t > a and Pt H(a,u) = H(a,u)

for

t < a. Moreover both of the following matrices are Invertlble

(5.15)

nt T

wheres(t) is

any finite time satisfying

{~tH(a,u)H(e.u)Tda) n t,

EtT

0

s(t) (~ H(a,u)H(~,u)%~) St, t

Pt ( t ) . Pt"

LEMMA 5 . 6

Consider and

a quasl-mlnimal,

analytic and complete system

(5.1).

Then for any T > 0

plecewlse constant control u, there exists another plecewise constant control

SUCh

that

Moreover

~(t) = U(t), t ~ [O,T] and PT(U)

given

any

T > 0 then

there

exists

is

the

a plecewlse

k × k

identity

constant

matrix.

control u such

that P~(~) is the k x k identity matrix.

Proof

only

Tne

proof

consider

of

f i e l d o f t h e form distribution

the

first

the second.

Lo(X)

assertion

Recall

(%,t]~_ gj(~,0(x0)). eolncldes

with

is

that each

almost

Identical

column

off H(fi,u) represents

to

the

second

so

we

a vector

By the ~ t r o n g a o c e s s l b i l l t y a s s ~ p ~ l o n the

TxM f o r

all

x E M. Suppose

that

we

are

given

69 s > 0 and proper over

a

control

subspace all

u on

[O,s]

of R k, as

pleoewlse

differentiation,

t

constant

and the

Lo(~U o(Xo )) contained

[O,t 2)

=

of

[tl,t 2)

in a

independent

set

[O,T) such

of

that

range H(t,~) is

over

an

interval

controls

contained

with ~(t) = u(t) for t e [O,s].

proper

subspace

make

Into

use

T Csu o ( X o ) M .

of as follows.

such

on

that

[O,t2).

(a:)TH(a,ul)

This

Choose

let

VI

there exists

and let V 2 : ~ i 61 ker H(a,u2) T V2 we see V 2 ¢ V I. oe[t2,t3 ) = l If VIfl V 2 = {01 we finish. By repeating the argument above with V I replaced VIi + V2,l

and

again

(al)TH(a,u2)

repeating

the

- 0

on

argument

It 2, t3),

by

we eventually obtain

Rk = VIi + V21 + .... + VNI and a control ~ on [O, tN+ I) satisfying ker H(o,5) T

-

t ~ [O,t2), such that range I r2 Let a 2 .... a 2 be a maximal Independent

for some ~ ~ It 2, t3). that

Contradlctlonk

m 0 on [O,t 2) and

for

vectors such r2 span {al.--a2 ~.Since

have

Let a11,..,alrl be a maximal

a control u 2 on [O,b 3) with u2(t)=u1(t)

of

then

u [tl,tl, 1) I=I a control u I on

If not, by the above result

Set

By

Let [O,T) -

k non empty sub-lntervals.

that H(a,u I) # 0

vectors

of

If V I = {Of we finish. span [al,..,a rl}. 1

H(o,u 2) ~ V~,

in a fixed

[s,~), ¢ > s, and ~ ranges

definition o£ L O, it is clear that we would

yields a result which we n o w be a partition

and

ranges

O er][tl,tl+ I)

: V i. If N # k we define ~ on [O,T), by arbitrarily extending ~ with

a plecewlse

constant control.

T Finally, if v e R k satisfies v T ( f H(a,u) H(a,~)Tdo) 0 on [O,T], so v g V i, I = I .... N. Since

v

= 0 then v T H(o,~) ~ 0

N nV i i=I

= (V I +

-.-.

=

c.">

.

iol. T

it

follows

that

nonslngular.

PROPOSITION

Consider

exists

=

Inpartlcular

0

and

hence

the

mm

symmetric

matrix J H(~,~) H(o,~)~do is 0 P~ (~)" Is the k x k identity matrix as claimed. []

5.7

the

a

v

situation

plecewlse

described

in

constant control

proposition

~ on

[O,T]

(5.5).

such

Given

that

constant control u satisfying u(t) : ~(t), t ~ [O,T] we have

any

for

T any

>

0

there

plecewlse

70 (5.16) Proof (5.11)

Wv(t,o,u) + WV(o,t,u)T ~ 0 Applying for

som~

proposltion matrix

(5.5)

valued

we

t,o 2 T.

see

that KA(.,.,u) satisfies

function G(.,u), and

every

the

equation

plecewlse

constant

control u. From the definition of K A we may assume that G(t,u) is independent of the values of the control u(s), for s > t. Moreover it is clear that we may write

(5.17)

G(t,u) = G(t,u) + RI(U,U)

(5.181

6(t,u) = -O(t,u) + R2(t,u)

where (5.19)

t > ~

R1(t,u)H(o,u) - H(U,u)TR2(o,u) T - 0,

Cleary Ri(t,u), i - 1,2, are also

independent

of the values

of the control

u(s)

for s > t. From (5.171 and (5. 181 we obtain

(5.20)

I

G(t,u) - ~ (R2(t,u) - R1(t,u)).

We may rewrite (5.19) as

(5.21)

Writing C

R~(t,u)P~

H(a,u)

- H(t,u)~Ps + R2(o,u) T o o,

= ]IrgrT' p+ s = )~s •s' T by using

(5.151,

t ~ ~ > r ~ a.

integration,

and

a

technique

Brockett [B3], we obtain

(5.22)

Rl(t,u)H r

= H(t,u)Tx s Kl(S,r),

t ~ s > r,

~T R2(o,u)T = K2(s,r ) ~rT H(a,u)

s > r > o.

However substituting these equations into (5.21) we obtaln for t ~ s > r ~ o

H(t ,u )TEs K I (s ,r )~rTH( o,u ) = H(t ,u )TZsK 2(s ,r )nrTH( o, u).

By integration we deduce that K1(s,r) - K2(s,r). Thus

(5.231

R2(o,u)Z s = H(e,u)T[IrK1(s,r) T,

From (5.221 we again deduce by integratlon

s > r >

a.

of

71 KI(s'r)

" AI(°)

~r

f o r some m a t r i x valued f u n c t i o n

A 1. S u b s t i t u t i n g

into

(5.23) we o b t a i n

R 2 ( a ' u ) Zs " H ( o , u ) T E r ][r T A I ( s ) T , Thus again by i n t e g r a t i o n (5.24)

s > r ~ a

we deduce t h a t

lit T A1 ( s ) T = A 2 ( r ) Ys

, s > r

for some matrix valued function A 2. In fact r r A2(r) = [ ! ~r T H(a,u) H(e,u)TnrdO)-1 f0 ErTH(a'u)R2(a'u)d~ Hence it is clear that A2(r) is independent of the values of the control u(s) for s > r. Recall that Ps(U) = Fs ZTs, and that by lemma (5.6) we may change u to a control ~ which is identical to u on [O,s], but for which P+(h-)s, and hence Zs

is

the identity matrix. We may therefore write (5.24) as

(5.25)

]]rT At(s) T ~ A2(r),

s > r

since A2(r) has not been altered by this change in control. We now deduce from (5.25) that there Is a matrix Q - Q(~) such that

A2(r ) u ErTQ Now

Ts K1(s,r) -

ZsA1(s) I[r so

by

(5.25)

we

get ~s K1(s'r) " A2(r)T " QT~r'

Applying this result to (5.22) we obtain

(5.26)

R1(t,u)n r - H(t,u)TQTnr,

t > r.

On the other hand Rr KI (s'r)T ~ IIr~rT AI (s)T so by

(5.24)

Er A2(r) Es = q Zs, Applying this to (5.23) we obtaln R2(o'U)Zs

= H(e'u)TQ Es'

s > a.

In this case an argument as above gives

(5.27)

R2(t,u) - H(t,u)TQ.

In any case we may use (5.26) and (5.27) in (5.20) to obtain

we

get Nr K1(s,r) T -

72 1

(5.28)

G(t,U)~r = H(t,u)T ~ (Q.QT) ~r'

t > r.

We now apply the l a s t part of lemma (5.6), to y i e l d the existence of a control on any interval [O,T] such that ~P~(u) = ~TRT T is th~ Identity matrix. Thus for any control u, which coincides on [0, T] wlth ~,

(5.28) yields

G(t,u) = H(t,u) T ~I (Q_QT),

t ~ T.

Since

Wv(t,a,U) + W(a,t,U) T - G(t,U) H(O,U) + H(t,U) T G(o,u) T = H(t,u)T ~I (Q.QT) H(a,u) + H(t,u) T ~I (Q_QT)T H(c,u) = 0

as long as t,e > T, we have proved (5.16).

We note that proposition of

proposition

result

below

(5.5), shows,

45.7) does not quite guarantee that under the conditions

every it

[]

does

variational guarantee

system that

is self adjolnt.

the

variational

However

systems

as the

are

self

adjolnt along "periodic" trajectories.

COROLLARY 5.8

Under

the

condltlons

o~

proposition

(5.5),

for any plecewlse

constant

control

u

satisfying

H(g,u) = H(o÷T,u), G(t,u) = G(t+T,u), t, o a 0

where T is a posltlve constant depending on u,

W(t,o,u) + W(o,t,u) T = O, t,o ~ 0.

Proof

Clearly

we need

only

prove

the

desired

identity

for t,o 6 [O,T]. Now

such t and o

W(t,o,u) + W(o,t,u) T = G(t,u) H(O,u) + H(t,u) T G(o,u) T

G(t+T,u) H(o,u) + H(t,U) T G(o+Tju) T

- G(t+T,u)~ 0 hO T H(0,u) + H ( t , u ) T ~

nt T C(~+T,u) T.

for

73 Since t + T > e and o + T > t, 45.28) yields H(t+T,u)T

=

1

(Q_QT) H(O,U) ÷ H(t,u) T ~I (Q_QT)T H(o+T,u)

H(t,u)T ~I (Q_QT) H(o,u) + H(t,u) T I (Q_QT)TH( o, U)

where we have used "periodicity" again. The result now follows trivially.

O

SECTION 5.3

In this section we present our main results by combining the results o~ this and the previous sections.

THEOREM 5.9

Consider

a minimal,

analytic and complete system

(5.1).

The system is (locally)

Hamilton[an if and only if glven any (u,y) E Z~ (O)(Xo), and admissible variations (61u,61y) of (u,y) with compact support,

S 0 Proof

= I,2, we have

(62Y(t) T 61u(t) - 61Y(tlT62u(t)ldt = 0

Necessity

proposition

i

follows directly from lemma

(5.7)

to obtain

(5.4). To prove sufficlency we apply

T > 0 and a control ~ on

[O,T] such

that for any

control u coinciding with ~ on [O,T] we have (5.16) i.e.

Wv(t,g,u) + Wv(g,t,u) T = O,

t,o ~ T.

J It therefore variational

follows

that

if system

45.1) is now

initialized

at ~T,o(Xo) every

system is self adjolnt. We may therefore apply theorem

(4.2), to see

that the system is Indeed Hamlltonlan.

Before

we

give

the

maln

result

U

stated

in

chapter

(I) we

prove

the

following

corollary of theorem (4.2):

COROLLARY 5.10

If Z is an analytlc

complete system

(5.1) with g0(x0) e

Lo(Xo), such that every

+

variational

system

is

self

adjolnt,

mlnlmal (locally) Hamlltonlan system.

then Ee(O)(Xo)may

also

be

realized

by a

74 Proof

Take

minimal

a

÷

realization

of ~. ( O ) ( x O) as

guaranb~ed

in

[SI],

Sussmann

denoted ~. Since the condition that every varlatlonal system is self adjoint is a

p r o p e r t y o n l y of I;e+ ( O ) ( x 0) i t gO(Xo) e Lo(X0), ~ Is

also

Is

also

true

strongly

;

of

Z (O)(Xo).

accessible.

Thus

by

Since

~ satisfies

theorem

(4.2), Z is

(locally) Hamlltonlan.

The

significance

of

D

this

result

minimal, not j~st quasi-minlmal. and

Goncal~s

[GI]

only

lles

in

the

fact

~hab

the system ~ Is indeed

In the previous works by Van der Schaft [VI,V4]

quasl-mlnlmal

Hamiltonlan

systems

are

constructed.

It

should be pointed out however that the system ~ wlll not be globally Hamiltonlan in

general,

only

(locally)

Hamiltonian

(i.e.,

the

Internal

energy

H O Is only

loCally defined). Using a method In Crouch [C4] it Is easy to establlsh that any minimal

(locally)

Hamtltonlan

realization

of

an

Input-output

map has a

qussl-

minimal globally Hamiltonian realization. Havlng

establlshed

the

exlstence

of

minimal

Hamiltonlan

realizations

of

input-

output maps we may now give our maln reallzablllby result.

THEOREM 5.11

If CZ (Xo) is

the

Input-output

map

Of

an

analyblc complete system (5.1) with

g0(x0) e L0(x0), then @z(x 0) has a minimal, analytic and complete Hamiltonlan realization, if and only if for any (u,y) e Ze(0)(Xo), and any two admlsslble

variations

(61u,61y)

of compact support, i = 1,2,

; (62Y(t) T 61u(t) - 61Y(t) T 62u(b)) dt - 0.

0 Proof Necessity follows from lemma (5.4) as in theorem (5.9). To prove sufficiency we co~truct

a minimal strongly accessible realization ~ of Cz(x0) as In corollary

(5.10).

conditions

The

on v%~(O)(Xo) a l s o

hold

f o r --~[(O)(z O) where

z0 e M is

the

initial state Of the minimal realization ~ on a state space M. As in theorem (5.9) we conclude that ~ is Hamiltonlan.

[]

SECTION 5.4

In thls sectlon we

briefly review

the foregoing results

of thls chapter In the

context of non-initlailz~d systems, and show that the external bchavlour s~ts o£ mlnlm&l Hamlltonian Systems a~e characterlzed formally as Lagrangian submanifolds of the manlfold oonsistlng of. all external bg~havlours. We refer to the terminology of

subsection

(1.6).

Our

flrst

task

is

to

glve

a

definition

of

admissible

variations which generallz~s definition (5.1). W8 shall for the present, revert to the

dtstlnction

between

a

variation of a control

(behavlour) and a variational

75 field along a control (behaviour). We consider the followlng class of non-lnltialized, analytic and complete systems m

(5.28)

~ = go (x) + Z u I g l ( x ) ,

xeM

Yl

U e ~ c~ m .

=

HI(x)'

1

~

I ~ m,

Recalling the definitions ~i' re' ZI(T)(XT)' ~ (T)(XT) and variations of elements of these sets, we Introduce the sets ~7(T)(XT), Z:(T)(XT), by first defining Z;(T) as the restriction to (-=.T] of all elements in rip and then deflnlng ~;(T)(x T) as the subset of Z;(T) corresponding to elements (u,y,x) satlsfying x(t) ~ x T. T=;(T) and £;(T)(x T) are defined by just projecting ou~ the state trajectory coenponent of each element. Now

we

define

the

sets

ZI(TI.T 2) (XTl,XT2)(~.~,~)

for

any

(~.y'.~) ~ XI,

-® < T I ~ T 2 < ®, and states XTl= x(T1), XT2 = ~(T 2) as follows; (u,y,x) e ZI(TI,T2)(XTI,XT2)(u,y,x)

if

(1)

(U(t),y('&),X(t))

(il)

there exlsts (u, y, x ) e Zi(T I)

=

(J(t),~(t),X-(t)),

t e [T 1, T2),

such that (u,y,x) ~'estrlcted to (-®,T I)

coincides wlth (u~ y? x-). ÷

+

÷

(Ill) there eXiSts (U ~, y , x ) ~ Z I (T 2) such that (u,y,x) restricted to IT2,®) coincldBs wlth (u+,y+,x+).

Note that by construction we have (u,y,x) ~ ~i also. We re(T I,T2)(xTI,xT2)(~,~) by

projection

once

more.

may

However

define for

our

purpose

this

deflnitlon is not satisfactory since It proposes to describe external behavlour using internal structure.

(Thls Is also true for Z;(T)(Xo) , but for initialized

systems this Is not so serious.) We therefore define sets Ze (TI,T2)(~,~), for.any (~,F) e £e' -~ < T1 ~ T2 < ~ as follows; (U,Y) e ~e(TI,T2)(~,F) if (u,y) e Ze and (u(t), y(t)) - (u(t),Y(t)), t ~ [T,I,T2). We defin~ an equivalence relation on ~e(TI,T2)(~,~) as follows. (u~y) - (u ,y ) If given any control V on [Tt,=), the palrs (ul,y]), (u2,Y 2) e Xe defined by

ul(t)

= u(t),

u2(t) ~ u (t)

u1(t) = v(t), u2(t) = v(t) satisfy yl(t) = Y2(t), t e IT],=).

,

t e ( - - , w 1) t e [TI,®),

76 It is easily verified that If Ze is the external behavlour of an observable system (5.28), then

each

equlvalence

Ze(TI,T2)(XTI,XT2)

class

in

Ze(TI,T2)(U,~) coincides wlth

a

set

(~,y) for some cholce of XTl,XT2 e M. We are now at liberty to

~ge Ze(TI,T2) (x T ,x T )(~,y) In definitions concerning external ~ehavlour, at 1 2 least when applle~ to observable systems. We shall therefore only consider minimal systems In the sequel, easlly

be extended,

systems.

It

proving an analogue of theorem (5.91 but the results could

as

is not

in theorem

clear

(5.11),

how to extend

to strong

the results

accessible

non observable

to non strongly accessible

systems however, so a true analogue of theorem (5.11) can not be given.

DEFINITION 5.12

An admissible

variation

(u,y)

of (~,~) E Ze is a

~rlablon

whlch

satlsfles

the

following conditions 8ssumlng (u,y) is the pcojeetlon of (~,~,~I e Zi: (i)

6U is plecewlse constant and supp 6u is compact

(ll)

supp 6y c supp 6u

(ill) SUppose supp 6u c [T I,T2], ~(Tll ~ XT1,X(T 2) - XT2 and (J',~') Ze(XTI,XT2)(u,~I.

Define a map (t,c1 + u'(t,£) by setting

u'(t,a) = u(t,e) for b e [TI,T 2) u'(t,e) = u'(t) for t e (--,T I) u IT2,®). We

deduce

(u',y')

that

of

u'

(~',~').

(6u,,6y,) along

is

a

We

(~',F')

varlatlon require to

o£ ~', and hence obtain a verlatlon the

corresponding

variational

field

satisfy condition (ii) also. Clearly we have

6U' = 6U.

As

before

we

now

of (~,~) ~ Ze wlth

abuse compact

notation support.

and

say (6u,6y) Is

Again

part

of

an

admissible

condition

(ll)

variation is

just

causality. The analogue of proposltlon 45.31 is now stated.

PROPOSITION 5.13

Consider where

a minimal,

(u,y)

is

the

analytic and complete projection

of

system

(u,y,x)

(5.28).

~ Z I,

and

Given any any

(U,y) E Ze,

TI,T 2 wlth

-® < T| < T 2 < ®, there exists a finite dimensional subspace S c L 2 ([TI,T2] ; Rm) and a dense set of plecewlse constant functlons 6u in L2([TI,T2]; Rm) n S ±, which may be reailzed compact

support

as a component contained

In

of an admissible

variation (6u,6y) of (u,y), with

(T l,T 2). Furthermore

(~u,6y,~x) is such that supp 6x c (TI,T2).

the correspondlng variation

77 The proof o£ this result follows in exactly the same way as proposition may be given as the space of functions {H(.,u)T~; ~ e Rnl k x m matrix whose j-th column Is

a local

coordinate

(5.3). S

where H(t,u)

is the ( u of .~0,t),

representation

gj(~tU, O (x(0)) and the variations 6u are characterized by

0 = ; T2 H(o,u)6u(~)dc T1

(5.29)

As In subsections

(5.3) we define Pt' s ~ t, as the projection onto the rang~, of

the matrix f H(o,u) H(o,u)Td~. We also define the related projeotlons t +

s Ps(u) - He]isT = sup Pt' t

(5.30)

Given a minimal T,

by

an

Pt(u) - Z t Zt T - sup Pts s

system Z described

analogue

lemma

Of

by equations

(5.6)

J(t) = u(t), t e

(--,T)

identity

(respectively PT(~)

matrix

we

may

,(u,y,x) e £1' and a time

(5.28) find

a

control ~

such

that

(respectively J(t) = u(t), t e iT,-)) and P~(J) is is

the

Identlty

matrlx),

the

and (~,~,~)

~I(T,T) (x(T), x(T)) (u,y,x). Using

an

analogue

situation,

of

proposltlon

In an analoguous

result for non l n l t l a l l z e d

(5.5),

which

proof o£ proposltlon

Is

almost

Identical

in

this

(5.7) we obbaln a mope pleasing

systems:

PROPOSITION 5.14

Consider a minimal, all

admissible

analytic and complete system

variations

(61u, 61y) of

(UW)

(5.28).

wlth

If for any (u,y) e Ze'

compact

support,

i

=

1,2,

satisfy

f

(62Y(t)T61u(t)

-

61y(t)T62u(t))dt

= O,

then e%,ery variational system is self adjolnt.

The p r e c i s e

reason

proposition

(5.7),

for the the

better

result

Improved r e s u l t

e q u a t i o n s ( 5 . 2 6 ) and ( 5 . 2 7 ) may be w r i t t e n

Rl ( t , u )

in

this

case

ls

i n o u r analogue o f

that

!n

the

lemma ( 5 . 6 )

as

- H ( t , u ) T QT

R 2 ( t , u ) = H ( t , u ) T Q. It follows from (5.20) that if Wv(t,a,u) - G(t,u) H(u,u) as usual, then

Hoof

of

shows tha~.

78 I (Q.QT). G(t,u) = H(t,u) T ~

Self adjolntness of each variational system, defined simply as

(5.31)

Wv(t,0,u) + Wv(e,t,u) T = 0,

t,e e R

is now a trivial conclusion.

COROLLARY 5.15

A mlnimal, analytic and complete system 45.28) i s Hsmiltonian i f

and only i f

the

condltlox~ of proposition (5.14) hold.

The proof of this result follows just as its analogue, theorem (5.9), but we need +

only

consider

therefore

a

"subset" ~e (0) (x O) of

clear that the assumptio~

~e in

of proposition

the

sufficiency

prooF.

It

is

(5.14) are far stronger than

required to merely guarantee that the system Is Hamlltonian. We now demonstrate a correspondingly stronger result for non-inltlallzed systems:

PROPOSITION 5.16

Suppose that

(5.26)

system Z. Let

(u,y) e Ze, and suppose

represents a mJnlmal, a n a l y t l c ,

mlnlmal

but not necessarily

control

coi~traint set R.

compact

support

where

variation (6u,6y) of

(u,y) (u,y)

further

Hamiltonian Let

that

system with the same state space M and

(Du,Dy) be an admissible

is viewed as an element wlth

complete and Hamllton!an

(u,y) ~ ~e where ~ is another

compact

support,

variation of

of ~ .

where

(u,y) wlth

If every 8dmlssible

(u,y)

is

viewed

as

element of Ze, satisfies

(5.32)

f

(6u(t)TDy(t) - 6y(t)TDu(t)) dt = O,

then (Du,Dy) is also an admissible variation of (u,y) viewed as an element of Ze.

PROOF

With (U,y) e Z

e

we may write t

45.33)

6y(t) = f

Wv(t,a,u)6u(a}do

an

79 where Wv(t,e,u) satisfies

the

self

adjolntness

conditions

(5.31)o

Substltutlng

(5.33) into (5.32) we obtain after some manipulation w

(5.34)

f

6U(t)T[Dy(t) - ;

WV(O,t,u)TDu(a)ds] dt = 0

As in proposition (5.13) given any interval [TI,T 2] we may suppose 6u takes values in a dense subset of L2([TI,T2]; Rm) n S ± where the constraints 6u ~ S i are expressed

by

equations

(5.29),

or 0 = fT2 H(a,u) ~u(o)do. Thus

(5.34) we deduce tha~ for any t

(5.35)

Dy(t) - ;

from

equation

that

supp Du

TI

Wv(e,t,u)T Du(~)do =

H(t,u)T(u)

t

where

a(u) is

c [TI,T2] , so (5.36)

a suitable k vector. We claim a(u) is by

definition

H(t,u)T(u)

Moreover by part

- 0,

supp D y c

zero.

Suppose

[TI,T2] also. It follows from (5.35) that

t > T2

(II!) of definition (5.12) of an admissible

variation,

(5.36)

remains valid if we replace u by any control ~ which coincides with u on (-=,T2). Now we may rewrlte (5.36) using (5.30) as

0 - H(t,~) T PT2(~)+ a(~) ~ H(t,~) T ZT2 Z~2a(~). Hence by lntegratlon we see that 0 - ZT2a(~). However we may

choose ~ so

that

P~2(~) and hence ZT2 is the identity matrix. Thus for this ~, a(~) = 0, and so the left hand side of (5.35) vanishes for this u. But the left hand slde of (5.35) does not depend on u(t) for t > T 2 so we see that it vanishes identically,

(5.37)

Dy(t) ~ ; t

Wv(O,t,u)TDu(o)d0.

We now use the fact that supp D y c supp Du c [TI,T2] again to deduce that 0 = f

Wv(C,t,u)T Du(o)do for t < T I. Writing Wv(O,t,u) T - H(t,u)TG(a,u) T we

T1 obtain m

0 - H(t,U) T S G(o,U) T Du(0)do, TI But this may be rewritten using (5.30) as

t
I.

giving

80 0 = H(t,u) T PTI--(U) TII G(o,u) T Du(o)do,

t < TI

0 = H(t,u) T ~TInTTI ~T~G(a,u)TDu(a)do,

t < TI

Or

(5.38)

By definition (5.12) both (5.37) and (5.38) ace unchanged when u Is replaced by any

control ~ which

el~ment (u,~,~)

Colnoldes

with

u

on [T I,®), and

the

correspondlng

belongs to Zl(Ti,T2)(x(T1),x(T2)(u,y,x). We now integrate (5.38)

to obtain ~TTI f® TI G(°'u)TDu(°)da = O. But we may choose ~, as In our analogue of lemma (5.6), so

that

NTI are the identity matrices. We conclude that

- -

PT1

(u) and

hence

f Wv(o,t,u)TDu(e)do ~ O. TI Thus by (5.37) and the fact that supp Du a [TI,T2]

t

-m

t = - f

Wv(O,t,u)TDu(o)do

and by the self adjolntness condition (5.31) by(t) This

demonstrates

Wv (t,o,u)Du(o)do. that

(Du,Dy)

is

an

admissible

wrlatlon

of

(u,y) g Ze as

claimed. As in section

(1.6) of chapter

(I) we let NM,~, m be the union Of all behavlour

sebs Z@ as E ranges over all mlnlmal,

analytic and complete systems

(5.28),

and

designates a submanlfold of NM,~, m consisting of a behavlour set described by a Hamiltonlan system E. Using termlnology deflned in sectlon last result of thls chapter.

(1.6) we ha~e our

81 THEOREM 5.17

Each submanlfold ~Z is a Lagranglan submanlfold of IM,D, m. Proof ~

By corollary (5.15) ~

is an Isotroplo submanlfold. By proposition (5.16)

is eo-lsotroplc. Thus ~Z is a Lagrangian submanifold.

0

SECTION 5.5 In this section we shall lll~strate the results obtained In thls chapter by means of some very simple examples. First,

as

deduced

in section

5.1

the admlsslbl~

varlatlons (6u,6y) of (u,y) o£

compact support on [O,T] are chacaoterlzad by the condition (5.5) T f H(s,u) 6u(s)ds = 0

(5.39) 0

Let us see what thls amounts to in the case of the harmonic oscillator

(mass-

spring system)

)

u

q

i.e. the linear Hamlltonlan input-output system ~ - Ax + bu, y - cx giver, by

(5.40) BecaUse

the

system

Is

linear

we

only

have to consider variations (6u,6y) of

(u,y) = (0,0), which are equal be the aotual Input-output pairs (u,y). The ]mpulse response matrix equals ceA(t-S)b

(5.~1)

=

ceAt.e-ASb, and so H(s,u) is given as

m ~ S l n mS] H(s.u)

= e-ASb

=

[- cos ~s

j

82 where m =

¢' _k is

the fundamental frequency of the system. Lct us investigate m 27 (5.39) for T equal to the period of the system, l.c., T = --~,

conditions

f

(5.42)

sln

0

the

u(s)ds ~ 0.

OOS uS

3

This yields

2~lu (5.43)

f

2~Im

0

0

Or, equlval~ntly,

th~ Fourier s ~ l ~ s

a0

(5.44)

u(s) cos us ds ~ 0

u(s) sin ws ds o

®

U(S) = --~ +

of u(s) on [O,T] can be wrltt~n as

(a n Cos n~s + bn sin n ~ )

[

n-2

Concluding, (O,2~lu) arc

the

admissible

contain the fundamental

As

a

second

example

proportional to q 3

(5.45)

wrlations

all those Input-output

of

pales

(u,y) - (0,0) of compact (u,y)

support

for which the input u

does

on not

~requency w = / ~ m

let

us

consider

the

mass-sprlng

system

with a spring

force

i.e.

mq = - kq 3 + u 2

Defining

p ~ mq we obtain,

H0

with

.

L_ + 4k q 4 H1

system 8H 0 =

8H I

p

~p

m

u ~p

(5.46)

y = q

-~ The v a r i a t i o n a l

= ~~H-0

~ ~aH I ° k q 3 -

equatioRs

are g l w n

as

- q, the nonlinear

Hamiltonian

83

[][

(5.47)

y

Y

II[

-3k~I 2

42

= ( I 0)

0

UV

q2

lql] q2

and the a d j o l n t variational equations as

211 3:2::

(5.48)

Let

us

consider

the

(a~j o l n t )

varlatlonal

equations

along

the

e q u i l l b r i t~n

trajectory q ~ p ~ u ~ y ~ O. Then, since q a 0, (5.47) and (5.48) easily yield

q2 (t) "

ftuV(e)de'

ql (t)

I m it

0

0

]euV(s)dsd° " YV(t) 0

(5.~9) t

pl(t)

Thus

= - f ua(o)do, 0

p2(t)

I

it

= ~ 0

if u v m ua we obtain yV ~ ya, and so this

(as i t s h o u l d

be).

A variation

(5.50)

T ] H(o,0)

= (- m

uV(o) do = 0

I) T. T h i s y i e l d s

T

] uV(o)do - 0

(5.51a) 0

a

f u (s)dsdo

= ya(t)

Variational

system

0

Is self-adjoint

(u v = 6u, yV = 6y) h a s c o m p a c t s u p p o r t

0

where H ( o , 0 )

o

j

on (O,T) i f

84 T

(5.51b)

euV(e)de = 0 0

Using (5.51a) we immediately deduce that (5.51b) is equivalent to

(5.52)

which

f O

o v

T

j[

U (s)dsda - 0

O

is by (5.49)

just the

condition yV(T) = O.

Now consld~r

two

variations

(6ui, 6yI), (6u2, 6y2) of compact support on (0, T). Then

[

[6y2(t)6u1(t)

- 6y 1 ( t } 6 u 2 ( t ) ] d t

=

0

it

sT[~u1(t) 0

0

J"e ~u2 (s)d~do - ~ua(t) 0

T

t

I d 0

it 0

0

0

o

f oeu I (s)dsdo]

dt

=

0

2

0

t ~u2 (o)do it f~U1(s)dsda]dt0

=[ 0

f 0

t

0 a

2

0

= O, by (5.51a) or (5.52), and so Theorem 5.11 is confirmed.

0 ft 0

0

o 0

I

=

6. G E N E R A L

In

this

chapter

crlterlon

for

we

NONLINEAR

shall

briefly

show

Hamiltonian

systems

as

how

SYSTEMS

the

developed

self-adjointness in

chapters

and

4 and

variational

5 for

systems

(2.1) can be extended to general nonlinear systems

= f(x,u)

,

x(O)

D xO e M

(6.1) yj = hj(x,u),

j = 1 .... m, u = (u I .... u m) ~ n

Here as before M denotes the k-dlmenslonal s t a t e space manifold a n d ~ I s assumed to

be (an open subset o f )

admissible

controls

oontlnuous

functions.

ning

vectorflelds

the

u(t),

as

(outputs (6.1)

can

product

and be

our discussions

with

interpreted

manifold

system

manifold,

inputs),

they

Include

the

pieoewlse

constant

TM×W,

we

shall

right concer-

again

assume

(real-)analytic..

of a nonlinear

analytic

as

be very preelse about the class o~

f(',u) for every U s 9. Tbroughout

In order to facilitate interpretation

long

not

Later on we shall add some 99mplete__ness_ assumptions

that all data in (6.1) are

dimensional

Rm. We s h a l l

(6.1)

as introduced

representing

local as

we first recall the more general,

the

coordinates

describing

which

is

a

spoace

in [WI]. o£

Let W be a 2m-

external

(yl,-.,Ym,U1,...,Um).

(k+m)-dlmenslonal

locally

parametcizad

x = (Xl,-.-,x k) for M and m of the coordinates

for W, namely

Variables

Then equations

submanlfold by

geometric

the

L of

the

coordinates

(ul...,Um):

DEFINITION 6.1 ( [ W I ] , [ V I ] , [ V 6 ] )

A ( g e n e r a l ) n o n l i n e a r system with s t a t e space M and space o f e x t e r n a l v a r i a b l e s W Is given

by a submanlfold

and some coordinates

R~mark that

Usually there

mapping G : B ~ TM

parametrlmad by the coordlnatas

a

fibre

that L Is "globally"

bundle

B

over

M

parametrlzed

(with

in the sense

projection

W, such that G(B) = L, making the dlagram

G

of M

of W.

it is also required exists

B

L c TM x W locally

.> TM × W

M

commutative. For a dlscussion o f these Issues we r e f e r t o [BI,WI,V6].

w) and

a

86 A

general

nonlinear,

(cf.[VI,V2]).

M

has

system to

be

Is

a

form ~, and W also has to

called

Hamiltonlan

b-dlmenslonal

be a sympleotlo

If

sympleotlo manifold

the

manifold

following with

with symplectic

holds

symplectlc

form m e . The

symplectic

form ~ induces a symplectic form ~ on TM (iT, V2]). Let ~ be locally n n given by ~ - [. dPi a dql , then locally ~ - ~ (dPiAdqi + dPi^dqi). Now we require i=I 1=I that L c TM x W, describing the system equations (6. I), is a L a ~ a n g l a n

submanifold

of

(Of.[VI,V2])

locally n

f o r W, l . e . e coordinates exists

can

Its

take

product

a

.g_e_n_e~atlng function

of TM x W Is locally

y~ ~

symplectlc coordinates

semi-canonical

x of M and u = (ul,,,,,Um).

Furthermore

form ~ @ (_,,e). Then (y1,..,Ym,U1,..,Um)

H(x,u)

for

x(O)

- - %~ ~aH

because L.

Thls

L is Lagranglan

there

implies

that

as

cj

I

L

a

given by the equatlons

XH(x,u)

=

(6.2)

we

with

=I=I ~" eldUi ^ dy i with c I = _+ I , such t h a t L i s parametrized by the

locally

submanlfold

TM x W

(x,u),

j

~

-

x0 ~ M

1,...,m,

-+

J

where for

every

Hamlltonlan

function H(.,u).

u, XH(-,u) Is

the

locally

Hamiltonlan v e c t o r f l e l d on M with

Comparing (6.2) w i t h (6.1) we conclude t h a t a l o c a l coordinate expresslon (6.1) i s Hamlltonlan

if for every

function H(',U),

u, f(.,u) is a Hamlltonlan vectorfleld aH and if h.(x,u) equals + ~ (x,u), J - ],...,m. J J

wlth

Hamiltonlan

m

B~mark If the generating function H(x,u) is of the form H^(x) - ~ U. H.(x), then u "-I J J we r~cover t h e definition of an (afflne) Hamlltonlan system as ~iven in (2.3).

Now we

proceed

to the definition

a general nonlinear

system.

In

of (6.1) along a certain control

(6.3)

~(t)

of the prolongation local coordinates u(t)as

af (x(t),u(t))v(t) - ~-~ ah

YV(t) - ~-; ( x ( t ) , u ( t ) ) v ( t )

and Hamlltonian

extension

we define the variational

the tlme-varylng

system

linear system

+ af (x(t),u(t))uV(t), Bh + ~- ( x ( t ) , u ( t ) ) u V ( t )

of

v(0) - v 0

87 with

h

(h],-..,hm)T , and the a d~olnt system

=

(6.4)

as

~Bh~T, + ~-~-~J ~x(t),u(t))ua(t),

-p(t) = [~x)T(x(t),u(t))p(t)

p(0) = P0

@f T ya(t) = [-~J (x(t),u(t)lp(t) + r~h~T, k'~) tx(t),u(t))ua(t)

It is easily checked that Lemma 2.1 goes through for this more general definition off the variational and adjolnt system. AS before we call the orlglnal system (6.1) together wit h (6.3), extension

of (6.1).

Definition

respectively

(6.4),

The coordinate-free

6.1 follow.

Since

the

.prolong@tlon,

definitions

for

(x,v,x,v) + (x,x,v,v),

Hamiltonlan

L c TM × W we have TL c T(TM) × TW.

canonical involution on T(TM) (in local natt~al coordinates

with

resp.

nonlinear system as In

a

cf.[T, Vl])

we

therefore

obtain

Applying the

(x,v) for TM given by

a

new

nonlinear

system

state space TM and space of external variables TW. This system is called the

prolongation, since if we denote natural coordinates for TM by (x,v) and for TW by (y,u,yV,u v) then

we

recover

loo811y

the

equations

Furthermore It can be seen that in the afflne case

(6.1)

together

wlth

(6.3).

(2.1) - (3. I) this deflnltlon

of the prolongation reduces to the global definition of the prolongation given in equation 2.17 of chapter (2). In general,

if N

is a submanifold

of a product manifold ~I x Q2', then we can

associate to N, in a canonical way, a submanlfold Nlift of T QI x T Q2' which is a La~anglan submanlfold of T QI x T Q2 with the product symplectic form Ul @ (-~2) (~I' resP'~2' denote

the natural

symplectlc forms on

T QI' resp. T Q2.) Namely, #

let ~QI and ~Q2 denote the natural projectlons of T Q1 and T Q2 on QI and Q2" Then

Nllft = {(al,a 2) E T*Q1 x T*Q21(-~QI(al), .,-Q2(a2)) - (ql,q2) E N, and for all (XI,X 2) ~ T(ql,q2)N, (~I(XI) = o~2(X2)} Therefore,

since

T*(TM) x T*M. bundle natural

L c TM x W

Because

T*(TM)

is symplectomorphle symplectlc

to

form ~ on

we

can define the Lagranglan submanlfold Llift of

with

Its

T(T*M)

natural

with

T*M([T]),

symplectlc

the symplectlo

we also have

form

as

a

cotangent

form ~ induced

that Lllft

by the

is a Lagranglan

submanlfold of T(T*M) x T*W with its product symplectlo form 6 % (.~e) (where ~e is

the natural

symplectlo

form

on

T'W).

Hence

we

system with state space T*M and space of external function of Lllft is given by (6.5)

H (x,p,u,u a) = pTf(x,u) + (ua)Th(x,u)

have

obtained

variables

a Hamlltonlan

T*W. The generating

88 where

(x,p)

are

natural

coordinates

for

TWM

and (y,u,-ua,y a) are

natural

ooordlnates for TWw. (Notice the order and the signs of ua and ya.) This system is called the Hamlltonlan

extension,

since

In local coordinates

as above we recover

the local expressions BH

f(x,u)

= ~-= aM

--~

r af~T. ~.~j ~ x , u ) p - [ Bh ~ ) T (x,u)

=-

ua

(6.6) BH

y = ~ = h(x,u) ~u a

y

a

For a suitable take recourse coordinates

aH

- ~-j -

af

[~)

T

(x,u)

p

¢ ah~T,

+ ~j

definition of mlnlmallty to the notion

of extended

~x,u)

u

a

of a general analytic nonlinear system we system

as deflnBd

In [VI,V6].

In local

the extended system Is simply obtained by adding an integrator to all

the input chan~%sls of (6.1)

(6.7)

x - f(x,u)

x(o)

= xo

6 - v

u(o)

= o

y = h(x,u)

and regarding

(6.7) as a system with (k + m)-dimenslonal

state

(x,u) and m inputs

v. Note that the extended system Is a system of the form (3.1) and hence we can apply the mlnlmality notlons

of Section 3. The accessibility algebra Lax t of the a extended system Is the Lie algebra generated by the vectorfield~ f(x,u) ~ and a ~-~, j = 1,..,,m, on (x,u), J ! eal in ge er tod accessible

If

Its

extended

and

the

strong

accesslbl-llty algebra

Now system

de i e

L 0 ext

to

is

the

st,o gi

Is (strongly)accesslble, or equivalently If

dim Laxt (x,u) ~ k + m or dim L 0 ext(X,u) = k + m for all (x,u). (In order ~o show the

consistency of

already

of the form

bility of

has

to show

(3.1) strong accesslbillty

its extended system.

vation space the

thls deClnltlon one

that for a nonlinear system

Is equivalent

Thls Is done in [VI,V6].)

to strong accessl-

Furthermore

the obser-

Hex t of (6.7) Is given as the linear space of functions on (x,u) of a to f(x,u) -~-~ or ~-~-, I=1 ..... m,

for[, Lf ...Lf hj, wlth fr' r = ] ..... s, equal

I and 3• = I ,...,m.

InSorder

to define

observabillty

of a nonlinear

have to add an extra set of outputs wj to the extended system (6.7)

system

(6.1) we

89 (6.8)

wj = u 3 ,

j = 1 ---,m

Clearly the observation space oC (6,7) together with (6.8) is Hex t plus the functions Ul,...,u m. Now we define

(6.1) to be (weakly)observable

Ul,...,u m

has

to distinguish

if (6.7) together

Equivalently, Hex t together with the functions

with (6.8) Is (weakly) observable.

(nearby)

points

in

(x,u).(The

consistency

o£ this

definition is checked In [VI].) Because of analytlclty it again ([H]) follows that if

(6.1) is accessible

constant

for each

dxHex t = dlm M.

then the dimension of the codlstrlbutlon

(x,u),

so that

(6.1)

is weakly observable

(d x denotes differentiations

dxllext (x,u) is

if and only if dim

with respect to the x-coordlnates.)

Finally (6.1) is called (quasi-) minimal if it is strongly accessible and (weakly) ob3ervable. Since the prolongation of an extended system is equal to the extended system taken of the prolongation o£ the system, it immediately follows that Corollary 3.3. holds for general nonlinear systems as well:

(6.1) is (quasi-) minimal

if and only if its prolongation is (quasi-) minimal. With respect to the generallzatlon of Proposition The observation the

3.4 and Corollary

3.7 we make the following observations.

space Hex t for a general Hamlltonian system

BH 3H functions ~ ' ' ' " @ - ~ m '

together

with

(repeated)

all

(6.2) is spanned by

Polsson brackets

(with

respect to the Polsson bracket on M) of these functions with H(x,u) and all

(re-

peated) differentiations to u1,...,u m ([VI]). The general Hamlltonian system (6.2) is quasl-mlnlmal equal

if and only

if

the

dimension

to the dimension of M, and minimal

functions

Ul,...,u m distinguishes

the quasl-mlnlmal

case).

points

of

the

codls~rlbutlon

dxHext

Is

if furthermore flext together with the in

(x,u)

(not only nearby

The observation space H ex t

points as in

of the Hamlltonlan

extension

of a general nonlinear system (6.1) is given as follows. Since the Hamiltonian is of the form H(x,p,u,u a) = pTf(x,u)+(ua)Th(x,u)

we haVe

~_HH__ ~uja " h j ( x , u ) (6.9)

j - I ..... m

BH

..... Buj

-

(BC T

tBh ~T.

~-~j~ (x.u)p + ~F~j j ~x,u)u a - yj

Furthermore s i m i l a r l y t o Lemma 3.6

a

90

{.(x,p,u.ua), ~-~_ ~u. a ) ~ (6.10) {H(x,p,u,ua),

As

Lf5

3

in

Theorem

Corollary

3,5

3,7 w8

we

u ~) pT[f(x ' ) ~ ,

@H

therefore

have

that

is

@

.e

obtain

(6.1)

@f

that-ext

= pTLo ext ÷ Ii~,xt' and as in £ (quasi-) minimal If and only if its Hamll-

tonlan extension is (quasi-) minimal.

As in Definition 3.1 we call a variational system (6.31 se_l[f-adjolnt if Its inputoutput

map

for

(6.4)

for

PO = O.

f(.,u)

for

v0 =

0 is equal (As

all

before

constant

to we

the

Input-output

map of the adjolnt

system

wlll assume throughout that the ~ c ~ o c f l e l d s

u e n are ~ l e t ~ . )

In local coordinates the Input-

output map of the variational system IS Of the form t yV(t) - J Wv(t,o,u)uV(~)do + ~ 0

(6.11)

(x(t),u(t))uV(t)

whereas the Input-output map of the adjoint system is glv~.n by t @h T ya(t) = f Wa(t,~,u)ua(e)de * [ ~ ) (x(t),u(t))ua(t) 0

(6.12)

where W a (t,~,u) = -wvT (o,t,u) (0f.(4.11)).

Hence

a

wrlatlonal

system

along

a

control

u

is

self-adjoint

If

and

only

if

(compare with (4.12))

(6.13)

Before systems

~[v(t,c,,u) = - wvT(c,,t,u)

t,a ~ 0

a_h (x(t},u(t)) - (~uh)T<x(t),u(t)) ~u

t ~ o

proceeding we

to

the

generalisation

of

have to remark on the Sussmann

Theorem

uniqueness

4.2

for

general

theorem which

nonlinear

is crucially

t~ged in the proof or that theorem (cf. Lemma 4.3). In the Sussmann theorem it is assumed that the input and output space of the system are globally defined. In the terminology output, M x U

resp. (so

Furthermore avoid systems

Of

Deflnition

input manifold,

that ft-,u) are

veetorfie!d

and

that

globally

that W = y x U, with Y, reap. L c TM x W is

defined

globally

veetorfields

for

such

we

that

shall

thereform

W = Y × U -

for each u ~ ~ .

Since

~m

assume ×

in the

~m

and

throughout

that

that f ( , , u ) i s

U, the

parametrlzed each

it is assumed that Y = Rm and that U is (a part of) ~ m

complications are

6.1 this means

u

e

by U).

In order to

the

nonlinear

a well-defined

Sussmann theorem tha outputs y do not

91 directly Recall

depend on the inputs u we still need a generallsatlon

that a system

is called .complete if the associated

of that theorem.

vectorflelds

for con-

stant Input are complete,

PROPOSITION 6.2

Consider two minimal systems Z I , Z 2

~I

f1(xl,ul )

=

Z1: y 1 = h I (x I ,u I)

~2= , x e M1

Z2: Y 2 = h2(x2,u 2) , x e M 2

such that hhe extended systems are complete. the

systems

initialized

f2(x2 U2 )

Assume that the Input-output maps of

in x I ~ M I and x 2 e M 2 are

equal.

Then

there

exists

a

unique dlffeomorphlsm ¢: M I + M 2 with ¢(X 1 ) - X 2 such that

i)

¢,f1(.,U) = f2(-,U), for all U e ~m

ll)

h2(¢(x),u)

= h1(x,u), for all u e Rm, x ~ M]

Proof The trick is to consider the extended systems o f the

Input-output

initialized output

maps

Sussmann

in

maps

of

the

extended

(Xl,0), respectively

for

systems

Z I and Z2. It is clear that (6.7)

(x2,0), are equal

of Z I and Z 2 i n i t i a l i ~ d

theorem

systems

(6.7)

in

x I , resp.

together

with

@,fl = f2

ll)

a ~* auj

ill)

h 2° ~ = h I

j

auj

where (u I , • -.,um) are the

with

(6.8)

If and only if the Input-

x 2, are (6.8)

dlffeomorphism ~: M I x Rm + M 2 x K~ with ~(Xl,0) = (x2,0)

i)

together

equal.

there

Hence

exists

by

a

such that

":' I , , , , , , m

standard coordinates for U = Rm. By 11) it follows that

if of the form ¢(x,u) - (¢(x), ¢(x,u)) for certaln maps ¢: M I * M 2 and ~: M|

U ÷ U. Using i) and i!) we hav~

and in general ~ . L ~ e x t

the

unique

: L~ext.

Using strong accessibility, the struct~'e of

LO ext" and the fact that ~(x1,O) = (x2,0) it follows that

e(x,u)

= u for all

x,u, so that ~(x,u) = (¢(x),u). The rest of the proof follows easily.

92 The generallsatlon of Theorem 4.2 is now straightforward. THEOREM 6.3

Let

(6.1)

complete.

be

a

minimal

nonlinear

system,

such

that

the

extended

system

Then the system is a Hamiltonlan system (6.2) if and only

is

if every

variational system is self-adjolnt.

Proof For the easy Suppose every

"only if" direction we refer to the proof of Theorem 4.2.

variational system is self-adjolnt.

Then equivalently the input-

output map of th~ prolongation and Hamlltonlan extension inltializ~d in (x0,0) are equal. Hence by Proposition 6.2 there exists a u n l q ~

dlfFeomorphlsm ¢: TM + T*M

transforming the prolongation into the Hamlltonian extension. As in Lemma 4.4 and 4.5

it

follows

that ¢(x,v) = (x,m(x)v), with re(x) a non-slngular

antl-symmetric

2n matrix representing a symplectlc form ~ "i~l ~ ~lj(x)dxiAdxj on M. Furthermore every locally

u the vectorflelds a

function

f(-,u) are

locally

H: M x ~m ~ R such

Hamiitonian.

that

Hence

fT(x,u)m(x) = - ~

there (x,u).

for

exists Since

(_~)afjT(X,U) ~(x) ~ ~"ax-(X,U), j = 1,.-.,m, we also have that - "~Ja ( =~ a) l l _,~a h

j = I, ....m, or equlvalently

(6.14) Hence i t (6.15)

!

axi ( ~ ;

follows hj

that

there

j = 1,-..,m,

exist

functions

: - au--; + k j ( u )

By t h e s e c o n d l l n e

(6.16)

) = ---)-3 ax I

of

(6.13)

i = I,...,~

kj (u) such t h a t

j : 1,...,m it

"~-ui = au I * "~ui-~u j

follows

auj +

that

aujauI

.j-'~. .:.

auj

auj

So there exists locally a function K(u) with kj =

and hence

i,j

= 1,-..,m

aK

~-~7' 3

Then H(x,u) = H(x,u) + K(u) Is the Hamlltonlan of the system, i,e. for this H the system is given by (6.2).

[7

93 Remark Note that the condltlon of completeness of the extended system means that for

every u e Rm the

tlme-~rarylng

vectorfleld x(t) =

f(x(t),ut),

x(O)

- Xo,

is

complete.

Flnally we shall bmlcfly show that also the variational criterion for Hamiltonlan systems

as developed

in chapter

(5) goes through for general nonllnear systems.

Recall that the varlatlonal system along u is glven by

(6.17)

v(t)

= A(t)v(t) + B(t)uV(t)

v(0) - v 0

yV(t) : C(t)v(t) + D(t)uV(t)

where

A(t) - ~ x ( X ( t ) , U ( t ) ) ,

D(t)

=

ah (x(t),u(t)).

aF ( x ( t ) , u ( t ) ) , B(t) - ~-~

C(t) - ~ah ( x ( t ) , u ( t ) )

and

Hence the input-output map for v 0 - 0 Is given by

t yV(t) , f Wv(t,a,u) uV(o)da + D(t)uv(t) 0

(6.~8)

where W v (t,o,u) = C(t) cU(t,e) B(o) and Cu(t,e) Is the unique solution of

a-~--~u(t,o) = A(t) at

(6.19)

Because

we

~u(t,o>

~(o,o) -

z

assume that the vectorflelds f(.,u) for constant

u

are

complete,

Wv(t,a,u) is actually deflned for a l l t ~ o ~ 0 and is In local coordinates

given

by

Wv(t,o,u)

(6.20)

=

G(t,u) H(o,u)

wlth

(6,21) Remark:

O(t,u) = C(t) ¢ ( t , 0 ) , Let ~ be a constant

H(e,u) - ¢ ( a , 0 )

control

B(o)

and l e t ~t,o(~,u) denote

the

v~ctor£1eld

= T(X,U) o

G -

X(e) = u(o}

-

on the extended state space M x Rm. Then analogously to (4.9)

(6.22)

Wv(t'a'U)ij

= dhi(@t,o(Xo ' u ) ) ( ~ t , a ) * I f '

a~j ](~o,O(xO ''~) )

flow

of

the

94 where dh i is a one-form on M x Km,

and i f , 8 ] denotes the Lie

vectorflelds f(x,u)

and -~. on M × ~n. For ~tecewise J slmilar expression is obtained by coneatenation.

As in deflnltlon 5.1 a variation

bracket of the

constant

(6u,6y) of (u,y) ~ Ee(0) (x0) is

controls

u

admlsslble

a

if

6u Is plecewlse constant, supp 6y c supp 6u c (O,T) for so~e finite T, and also condition

(Ill) of definition 5.1 holds. As in proposition 5.3 these variations

are constralned only by the conditions

46.23)

T f H(a,u)6u(s)do - 0 0

Moreover as in proposition (3.8) we may prove that checking (weak) observabillty of the prolonged system of a general nonlinear system may be

done

by

setting u v

4=6u1 equal to zero. It is easily seen ([VI]) that equation 44.36) remains true for general Hamiltonian systems. All this yields ~he Immediate generallzatlon of Lemms 5.4 LEMMA 6.4 consider a quasi-minimal, Given any

analytic,

complete,

general Hamlltonlan system

(u,y) ~ Z~ (0)(x0), and admlsslbl~ variations (61u,61Y)

of

(6.2).

(u,y) with

compact support, i = 1,2, we have

(~2Ty(t) 61u(t) - ~lTy(t) 62u(t)) ~t = 0

(6.24) 0

For the generallzatlon of Proposition 5.7 we proceed as follows. With the aid of Lemma 5.2 it again follows that there are a great may non-zero admissible controls 6u. Let us take two admlsslble variations 61u, 62u satisfying (6.23) for some T and

their

corresponding

vaclatlons 61y, ~2y. Suppose

that

the

variational

condition (6.241 holds. Then, using (6.181 we obtain T 0 = ; (61L(t) ( f WV(t,o,U) 62U(~)do * DCh) 6#<~)) 0 0 t 62Tu(t) [ f Wv(t,a,U) 6lU(O)dO * D(t) 61u(t)) ) dt 0 T t = f f [61Tu(t) Wv(t,e,u) 62u(e) - 62Tu(t) Wv(t,~,u) 61u(o)ldodt 0 O T • f 61Tu(t) {D(t) -DT(t)) 62U(t)dt 0 -

(6.251

95 Using

variations 61u, 62u with support

concentrated around a certain time t It

follows by contlnulty from (6.25) that (6.26)

D(t) = DT(t),

~ ~ 0

Hence (6.25) reduces to Tt 0 - f ; [61Tu(t) Wv(t,a,U) 62U(o) - 62Tu(t) Wv(t,~,.U) 61U(o) ) dadt 0 0

(6.27)

which

is

the same

equation

as

equation

(5.12)

in section

5.2.

The proof

of

propositlons (5.5) and (5.7) now proceeds as before. Thus we obtain

PROPOSITION 6.5 Consider a quasi-mlnlmal, analytle and complete system any (u,y) e Z~(O)(x0) , _

all

admissible

(6.1).

variations (61u,61y) of

Suppose that for (u,y)

or

compact

support, 1 = 1,2, satisfy (6.24). Then, given any T > 0 there exists a plecewlse constant

control ~ on

[O,T]

such

that

for

any

plecewlse

constant

control

u

satisfying u(t) = ~(t), t e [0,T] we have

(6.28)

W v (t,o,u) = - WvT(e,t,u),

It therefore follows from

t,0 ~ T

(6.13) and

(6.26) together with

(6.28),

varlatlonal criterlon (6.24) is satisfied then for a system (6.1) ~T,o(Xo,~), wlth ~ as adjolnt.

Combining

in

this

Proposition rasult with

6 . 5 , every Lemma

varlstional

6.4 and Theorem

that if the

Initlallzed system

is

at

self-

6.3 we obtaln

the

following gene~811zatlon of Theorem 5.9.

THEOREM 6.6

Consider a minimal, analytic and complete system system

Is

complete.

Then

the

system

is

(6.1), such that the extended

Hamiltonlan

if and

only

if for

any

(u,y) ~ Ze(0) (x0) and admissible variations (~iu,61y), I = 1,2, of compact support of (u,y) we have

(6.29)

T S [61Tu(t)62y(t) - 62Tu(t) 61Y(t)] db = 0 0

7. FINAL REMARKS AND SOME OPEN PROBLEMS

In this final chapter we shall touch upon some extensions to the theory presented in this monograph.

Main

emphasis

is on the

generalization of

the mathematical

tools used so far, although section 7.2 is Concerned with the physical Interpretatlon of our wrlatlonal characterization.

7. I

SECTION

EQuivalence

of

the

self-adj ointness

and

condition

Jakuhezyk'a

algebraic condition.

In chapters

2 through 6 we have presented a characterization of Hamiltonlan con-

trol systems from th~ ~ h a v l o r

of variations in their input-output and state tra-

jectories. On the other hand in chapter z~d Hamlltonlan systems; rectly. We give

theorems

for

more

general

(1.4) and (1.5),

via the input-output map dl-

below a connection between the two approaches. Our sel f-adjolnt-

ness criterion, theorems (4.2),

stated

I we showed how Jakubczyk had characterl-

systems

by J a k u b c z y k ,

(1,2)

theorem

(4.2)' for afflne systems (1.1), and theorem (6.3) bears

(1.6),

the

which

closest is

resemblance

expressed

in

to

terms

the of

criterion

the

general

nonlinear system (1.2). Notice that Jakubezyk's conditions comprise two parts, one of which gives conditions under whlch a realization of an input-output

map exists

at all, and the other part provides extra alg~bralc oondltlons which ensure that the

(minimal) realization is Hamlltonlan.

bralc

condltfons

to

be equivalent

One therefore expects the extra alge-

to the

condltlons

expressed

in our

theorem

(6.3). However the algebraic condl~!on given by Jakubc~yk in theorem (1.6) is not particularly elegant; his characterlzatlon

seems to be more suited to identlfylng

the Hamfltonlan structure expressed in equations (1.21), as in Jakubczyk's theorem (1.4).

(Contrast the simplicity of equations

(1.23) with the algebraic conditions

given in theorem (1.6)). For these reasons we prefer to demonstrate here that our self-adjolntness criterion for afflne nonlinear systems, theorems (4.2),

(4.2),, characterizing Hamilto-

nlan systems (1.15) as ones for which the variational systems are self adjoint, is equivalent to algebraic conditions on the input-output map of the system expressed in terms of the Volterra series, extended to the multlvarlable case and infinite Volterra series. We shall not give a direct demonstration that these conditions on the Volterra kernels; which define the input-output map of a system (1.1), imply that the system is indeed Hamiltonian, since this is essentially what Jakubozyk did.

Let us first remark that it is far from trivial to show directly that these oondltlons on the Volterra

kernels are equivalent to our variational characterization

of Hamiltonlan systems stated in Theorem (5.11), namely

97 Im

(7.1) In

f (62y(~,)T61u(t) - 61y(t)T62u(t)]dt - 0

0

Van der Schaft [V2] tnls is done In the case of

linear systems.

Systems with

input-output maps defined by the equation t

y(t)

S [al w2(~,OlO2)U(al)U(%)do1~o2

=

o

0

are considered in Crouch [C3]. Here the conditions amount to the follc~Ing

W2(t,e I,~2 ) + W2(oI,t,o 2) m 0 W2(t,Ol,a2) + W2(o1,o2,t) + W2(o2,t,o I) m O, and the equivalence

with

(7.1)

is demonstrated

by explicit

computation,

modular

certain technical details which are dealt with in chapter 5. The g~neral case must involve a heavily combinatorial typ~ arglanent, such as in [La]. Consider now a system (1.1) m

(7.z)

~ = go(X) +

X ulgi(x), I,.I

Yl = H I ( x ) '

x E M, x(O) - x 0

I g i g p.

The Volterra series expansion of the input-output map of (7.2), generalizing that given in (1.6) to multiple inputs and multiple outputs, may be written as

ft

i0 Yi(t) '= Wo

ioi I (t,o 1) Ul1(Ol)do I + ... + 1~ W 1

(t'Xo) + 0

I

0

ft/I /kl 0 "'0 +

If

tol I i k 1 1 . ! . i k Wk

(t'°1"'°k'xo)U11(°1)ui2

(a2)''ulk(°k)doId°2"'d%

....,

(t,x) * X(t)(x) denotes

the

flow

of

dependent vector flelds Gi(o) as follows,

GiCa)Cx) = y ( - o ) , g l C y ( o ) ( x ) ) .

go'

as

In

chapter

I, deflne

the

tlme

98 We may now write the Volterra kernels in the form

~/k (7.3)

l o i I •..i k

(t, o.1.. • o.k,xO)

- Gik(Ok)(xo)(Gik

i(Ok_I)(.)(

..... (Gil(o.1)(.)(HioOX(t)(.))

..... )

(We ~ e

the notation G(H) for the Lie derivative of a function H by a vector field

G.)

also

We

given

in

generalize

(1.19)

to

the

the

definition

case

of

of multiple

the

bracket

inputs

and

operation outputs,

on the by

kernels

applying

the

bracket simultaneously to the indices, for example

Wk

10i I--- [I r,ir+ I ]..

Wk

:

"ik(t,ool...[ee,ar+]].

lOil-..leir+l.-.l

k

• .o.k,X O)

(t,ol'..o'r,o'r+l..-ek,xO)

_WkiO11'''le+11r'''Ik(t,o.1...o.r+1,~r...Ok,xO

).

For slmpliclty of notation we Shall often drop the tlme p~rameters, for e ~ m p l e

iO...[ir...is]...i k Wk

i O. • .[i r. • .Is]...i k for

Wk

(t, o.l... [Or... Os]... Ok,X0).

We now state the main result.

THEOREM 7.1

Every variational system of (7.2) is self adjoint if and only if the Volterra kernels satisfy [i 0. ..Ir]ir+ I ...i k (7.4)

Wk

i 0...Irir+ I...I k = (r+l)W k

Before we prove this we give a simple lemma.

for k > r > I.

(See also [La] for similar

computa-

tlons and [J7] for related results.)

LEMMA 7.2

The

conditions

(7.4) hold if and only if the foll(~ing oondltions

hold simulta-

neously

(7.5)

[io'''Ir-1]irir÷1"''ik Wk

= rWk 10'''Irlr+l'''Ik

,

k > r > I

99 (7.6)

Proof

Wk

ir[iO'''Ir_l]Ir+1'''i k

For fixed k identltles

= -Wk

i0 . . . . "'irlr*1

ik

,

k Z r ~ I.

are equal to the identities (7.5) except

(7.4)

for

k = r in (7.41 [i0...i k]

I0..-I k

Wk

=

(k+1) W k

However by definition [ ! O - . . i k]

[lO...ik_1]l k

Wk

which

by

= Wk

(7.51 with

k = r and

(7.61 wlth

(7.5) and (7.6) together

imply (7.4).

Conversely

(7.4)

for

fixed

k

ik[iO...Ik_ I ] - Wk

for

k

k ~ r is equal

> r ~ I yields

(7.5)

to (r+1)Wk iO'''Ir . Thus

for k ~ r > I. Also

by

definition [lO-.-Ir]Ir+]...i k -W k

[iO...Ir_1]Irlr+1---! k + Wk

ir[lo...Ir_1]lr+l...1 k -

Wk

Thus (7.4). and (7.4) with r replaced by r-1 yield

(7.6).

Proof of Theorem 7.1

We prove this result by using the Input-output characterization of self-adJolnthess,

namely

that

same input-output (2.20)

that

the

sets of equations

the map

prolongation (theorem

prolongation

4.2').

and

Hamlltonlan

extension

(7.1)

have

the

from (2.171 and

aide given by the following

respeetlvely. m

Yl = HI~(XP) yl v

of

In the ease m = p, we recall

and Hamlltonlan

m

~p =~0(Xpl + I=I Z ul&i(Xp)+ j=1 Z (7,71

extension

= }{l(Xp)

v

uj gj (Xpl I < i < m, Xp(O) = (xo.O) ~ TxoM I ~ i ~ m

100 m

xe = X T (xe) + P go

m

[

i=l

ulX T (xe) + P gl

~

UJ aX

J=l

(xe) Hj £

(7.8) Yl

Hl£(xe)

=

Yla = Clearly

pTgi(xe )

for each Input the out~ts

coincide

with Hi(x) of

sufficient

to compare

(7.1)

1 S

I

I<

i <m

~

m, xe(0)

(Xo,0) e Tx 0 *M

=

HI£(x p) and Hi£(Xe) coincide

subject

to

the input-output

the

maps

ylV = ~l(Xp) and yl a - pTgl(x e) respectively.

same

control.

of (7.7) and

since they also It

(7.8)

Is

therefore

with

outputs

We do thls by computing the Volterra

kernels for each system explicitly. We denote the Volterra kernels for the p~olong~tlon (7.7) by

Wv,l ok l 1 " " I r - l J r l r ÷ 1 " ' ' ( t , o l . . . e r _ l , e r , e r + l

(7.9)

. . . . . (Xo,O))

1011'''Ir_lJrlr÷1 "-" or simply Wv, k , and those of the H~mlltonian extension 47.8) as Wa,1011"''Ir-lJrlr+1""(t k ,Ol...Or.l,ar, ar+ I .. ,(x0,0))

(7.10)

or

slmply

Wa,kl O i 1 " ' ' i r ' l j r l r + 1 " ' "

These kernels are the coefficients of the products (7.11)

ul1(al)U i 2 (02)... ulr-1(°r-I )UVjr(a r )Ulr+l (Or+l)-..

and

(7.12)

ui1(gl)ui2(a2)'''Ulr-1(°r-1)u~r(ar)uir+l

cespectlv~ly.

(°r+1)'''

Note that we reserve the index j for the adjolnt or wrlational

con-

trols, and the index I for normal controls and the outputs. Computations systems Lions

of (7.9) and (7.10) can be done by applying the formula 47.3) to the

(7.7) and (7.8) respectively. which we

now describe.

(a,xe) + ~H(o)(x e) . [go~]

denote

the

However there are a great many slmpllflca-

Let (O,Xp) flow

of

+

"~(a)(Xp) denote

X T " From P go

so

%(o)~i({,(o).) = ,~(-a)~i('r(o).) - 61(o).

lemma

the flow or (3.2),

go' and [go,gi ] =

101 Moreover [go,gj £] = [go,gj] £ so Y(-a),gj£(~(a).) It follows

by applying

= [~(-o),gj(Y(a).)) ~

(7.3) to (7.7),

and

= Gj(0).£

using gi(Hr) = ~ - ~ ,

that

If no j

indices are present in (7.9)

1011 • . . i k Wv, k =

~0.

(xo,O)

If only one j Index is present in (7.9), the USe of the identities (Learns 3.2)

gj~(Hr)

: g j ( H r ) ~, g i ( H e ~) = g i ( H r ) ~

yields ioi I ...ir.lJrlr+ I • ..i k WV ,k = 1011 " " " I r - l J r l r + 1 " " "ik = wk

(7.13)

.)£ (t,a1...q

k,

(x0,O)

= wklO i1"''Ir-ljrlr+1'''Ik = Gik(0k)(X0)(...(Gir+1(Or÷1)(.)

(

Ojr(ar) (.) ( .... (Gi1(aI) (.) (HIoOY(t)(.)) ...). If more than one j index is present in (7.9) the identity gj£(Hr£) = 0 shows that the Volterra

kernel Is also identically

zero,

as in the case of no j indexes.

(Note that for Volterra kernels of a system 47.2), j ~nd i indexes play an identical role.) From lemma (3.6) X T (PTgl) = pT[g0'gi] so P gO T P gi OYH(°) = PTY(-a)*gi(Y(°)')

" pTGi(o)(')

"fH(-°)*X T (YH (~) ") = X T = X T P gl P gi°~H (°) P

and

"

Gi(a)

It follows by applying (7.3) to 47.8) that If no j indexes ace present in (7. I0) 10t t - . . I k Wa

"

PT[clk(%) (')'[Gik-l%-1) (')'['''[Gi I (al) ()' GI o(t)(') 3''']("0'0) " 0

102

j index

If only one

present

Is

in

(7.10), the i d e n t i t i e s

(Lemma

XT (Hi ~) :g0(Hj) ~, X ~(p~r ) :-gr(Hj) ~, P go

3.6)

and hence

Hj

XH(-O),X H £(XH(O).) = X

j

£

: X

~,

Hj OXH(O) H.OY(o)3

yield

W

i01 1 "" "ir-lJrlr+1""ik a,k

-Glk(Ok)(xo)(...Gir+

=

l(°r*l)(')(

[Glr_l(Or_l) (.), [ . . . . . .

(7.14)

, [G12(o2)(.),

[GII (al)(') 'Gi0(t)(')]'''](tr°X(Ok)')''')" If

more than one j index is present In (7.101, we use the fact that X H.£(Hr ~) : O

to deduce that the Volberra kernel is also identically

j

zero, as in th~ case of no

index.

In conclusion we see

that

tonlan

equal

extension

coincide.

are

If condition

the

input-output

if and only

(7.4)

maps

is satisfied

then

(7.6) is also satisfied. Using (7.6) in (7.131

(7.15) we

of

if all the

the prolongation

and Hamll-

kernels in (7.13) and

by lemma

vla the

(7.2) we

(7.14)

know condition

ldentlty

g1(g2(H1) - g2(g1(H)) = [gl,g2](H),

deduce

from

(7.14)

iOil...irJrlr.1.-.i k 1011...irJrlr+1--.l k that Wv, k = Wa, k , which

is sufficient to conclude that the input-output maps of (7.7) and (7.81 coincide. Conversely if the input-output maps of (7.7) and (7.8) coincide, we can equate the expressions

in

(7.1 3)

Dynkln-Specht-Wever

be

seen

by

formula for

-W

(7.15)

Jr[10...Ir_1]ir+

•..ir.lJrlr+1

•..I k

10I 1 • . - i r _ l J r l r + v,k

1 •..i k

1011 a,k

-W

We deduce

that (7.5) holds

formula to the Lie brackets In

applying

Wk

and (7.14).

,

repeatedly

I ..-I k

as

given

(7.141. That (7.6) holds, may

Jrl0..-ir_lir+]--.i k to W k to

which

In

by applying the

is

(7.141.

seen

This

to

however

obtain a

coincide

is

equal

wlth

to

because (7.131 and (7.14) coincide. Thus since (7.5) and

103 O

(7.6) hold, by lemma (7.2), (7.4) also holds, which complehes the proof.

SECTION 7.2 A physical interpretation of the variational criterion

The

basic equality which holds

for

(afflne

or

general) Hamiltonlan

which forms the starting point for their variational

systems, and

characterization

in chapters

(5) and (6) iS equation (4.36), suggestively rewritten as

I T (% T 61y 61T~ 6~)dt u

0

(7.16) ~x(T)(61x(T)'62x(T) ) - Wx(O)(61x(O)'62x(O)) Here (61u,61y,8]x) and (62u,82Y,82x) denote

arbitrary

(infinitesimal)

variations

to any solution (u,y,x) of the Hamlltonian system under consideration. Although this formula is very appealing, a direct physical Interpretation of it is hard to find. We shall now deduce a consequence of (7.16), which has some physical meaning. Let

us

consider

[0, T] for which

arbitrary

solutions (u,y,x) on a time-interval

the input u(t) is identically

(a,b} containing

zero. We define two variations

to

such a solution in the following way. First take the one-parameter variational family

(u(t+~), {(t+E), x(t+~:)), t

(7.17)

By

stationariby these are

e

[O,T]

solutions of the

Hamiltonian system for any small

e.

This results in the first (infinitesimal) variation

61u(t):= lira u(t+a) - u(t) = u(t)

~(t__+~D__-_~_(_t_! x(t.c) 61^(t):== l l m c-~O

For

* ~(t)

the second varlatlon we take

proPerty condition

that the solution x(O)

tion ~(T) of

= ~(t)

E

x(t}

any input function

F(t),

t ~ [0, T], with the

of the Hamiltonlan system starting from initial

= x(O} for this input fdnctlon coincides at time T with the solu-

the system

without

input variation 62u, and let

input (u = 0). Take such a

function

F(t) as the

62y and 62x denote the resulting output, respective-

ly state, variation. Now apply formula (7.16).

Since

G(t) ~ 0 we have

81G m O.

104 Moreover by d e f i n i t i o n of

F(t) we have 62x(0) ~ 62x(T) = 0. Hence (7.16) s I ~ c i a -

llzes t o

ST,(t>

(7. ~8)

,(t)dt o o

0 Thls equation admits ternal

the following interpretation.

forces, and y(t) as displacements

If u(t) can be regarded as ex-

(as Is commonly

the case, see the Intro-

" f TFT(t ) y(t)dt is the external work performed by the force F(t) on 0 the system along the solution (u = O , y , x ) . ductlon),

Hence

then

(7.18) expresses

that

the

"virtual

the fact that for any force function F(t), t ~ [0, T], such

motion" ~_2x(t) sa~isfles ~ ( 0 )

~ 62x(T) = O, the external

work

performed by F(t) on the time interval [O,T] is zero.

The above analysis

can be immediately

generalized to the case that u is not iden-

tlcally

zero, but a conservative

force. Let us restrict ourselves to afflne Hamll-

tonlan

systems x = XH0(x) - j=IZ UjXHj(X), Y~j = JH~(x)' j = I ..... m.

m

conservative

If it is of the form

(7.19)

uj(t) = a~jas (ytt)),

for

a

added

certain to the

"potential external

Internal

force

forces

S(y)

is now

u(t) is

j = I, .... m

function S : Rm + R. In this

energy"

Then

by adding

= S(HI,...,Hm).

case

to the For

the

external

internal

be

Hamiltonlan H 0 the extra

this modified

zero, and so it follows

force ~(t) can

Hamlltonlan

system

the

that for any F(t), t e [O,T], as

before T

S (F(t) - u(t)) T {(t)dt = 0

(7.20) 0 Slnce

fTuT(t)y(t)dt" . 0 this

fT ~aS (y(t))y(t)dt = S(~(T)) - S(~(O)) 0

yields fTFT(t)~(t)dt - S(y(T))

(7.21)

- S({(O))

0 l.e. the external work equals the increase in potential energy. In case u(t) is arbltraPy ted.

Consider again an

(non-oonservatlv~),

the sltuatlon

becc~es more complica-

input variation 62u(t) - F(t) - ~(t) as ~ f o r e ,

resulting

105 in

an

output

variation

62y(t)

and

state

62x(t).

variation

Since

62x(0) = 62x(T) = 0 we obtain from (7.16) T . ; (F(t) - u(t)) T ~(t)dt 0

;

T • uT(t)6~(t)dt

0

The quantity on the rlght-hand side can be rewritten as

T'T

T

; u (t)62Y(t)dt = uT(t)62Y(t)]0 -

fT~T(t)6~(t)d t

0

0

. - jTGT(t)62~(t)dt 0 since 62Y(0) = 62Y(T) = O. Hence we obtain T

(7.22)

; FT(t)y(t)dt + fTuT(t)62Y(t)dt = 0 0

Here t h e while

quantity on the r i g h t

;

0

is the increase o f

both terms on the left-hand side

tual) external

T. T

; u (t)y(t) dt

internal energy of the system,

can be interpreted

as some

kind of

(vlr-

work.

SECTION 7.3 Poisson control systems and moment @a._ps_s

In this final section we make some connections between the results presented a b o v e and

other

work

Hamlltonlan

on

mechanlcs/systems;

connections may either help in understanding

indicating

where

these

issues not resolved by us or further

the scope of the results presented. The main mathematical this may Polsson

construct we used here was the symplectlc structure. However

be generalized structure

on

quite significantly

a

smooth

manifold

by

M,

the

is

a

of Poisson structures.

use

billnear

form

on

the

A

smooth

functlons, also denoted {.,.], satisfying

(~)

{f,{g,hll + {g,{h,fl! + {h,{f,gll - o

(Jacobl)

(b)

{f,gh{ - ~[f,h{ + h[f,g{

(,.elbniz)

(o)

{f,g{ = -[g,f{

(anti -symmetry)

for smooth

functions

f,g and h on M, We call a manifold endowed with a Poisson

structure a Poisson manifold, see Welnsteln

[WBI],

[We2], Van der Sehaft [VT], and

the hook by Oiv~r [0], for details whloh we sketch now. Given a Polsson manifold we may

also

define

the notion of a Hamlltonlan

field; X h denotes the Hamiltonlan vector field defined by

Xh(f)

- {h,f}

vector

106 f on M, and h is the corresponding Hamiltonlan function.

for every smooth function

Clearly the Polsson structure defines a linear mapping

Bp :T*M P ~ TpM,

whose rank clearly trlc.

determlnes

even

the

"rank"

dimensional

A Polsson

in fact

df(p)

structure

as in Welnsteln

a symplectlc

manifold

is

span

of

In general

manifolds,

on

a

more

at p •

structure

= dg(Xf(pl)

M. The rank

= {f,g~(p) is

skew

complex than a symplectlc

manifold

is locally about

off rank

Polsson

the

vary but the

flelds

see O l v e r

generalize

Polsson

zero

at p.

is

symme-

structure;

p a product of Thus

a Polsson

to the dimension off M everywhere determines a symplectic

the

symplectlo structure, We may now

Polsson

vector

which

therefore

the rank may

Hamiltonlan

the

[We2], a Polsson manifold

and

structure with rank equal structure.

of

since dg(Bp(dfll

* Xf(p)

is

distribution

integrable

with

structure restricts

on M defined

leaves

being

define

to

the

by the

symplectlc

corresponding

[0].

definition of a Hamlltonlan system

at least in the case where the control enters afflnely,

tu a Polsson system,

(see Van dec Sehaft iV7]),

by startlng with a Polsson manifold and writing the system i n the usual form m

XHo

=

(x)

-

Y (X) i~l UiXHi

(7.231

Yi where XHI are Polsson

the Hamlltonlan

structure.

fact

that many

than

Hamiltonlan

slonal

reason

systems. would

torques)

vector fields defined by the functions H i , via the foe

introduclng

systems may For

he

leaves.

as

([KI]).

class

of systems

case

input

and

output

systems systems

see Olver

discussed

spaces

£ c Rm, we

generall~tlon

are

can define

globally

- ~j.-

(x,u/,

J

j

-

1 .....

m

first,

on odd

and

dlmen-

subsequently

(with o r

with-

complicated

(6) is more subtle. dlstlngulshed,

In

e.g.

a global Polsson system on a Polsson

u e £ c Rm y~., =

in the

of the notion of Poisson systems in chapter

= XH(X,U) ~H

evolving

[0]. See also more

manl£old M as

(7.251

lles

as Polsson systems

velocity equations

The extension

general

W - y x U = Rp x R,

Polsson

The angular

to the more the

mechanical

is a good example,

in Krlshnaprasad

such a

be expressed more readlly

example

expressed

to the symplectlc

out external examples

The

physical

manifolds

"reduced"

| ~ i ~ m

Hi(x)__

-

107

with XH(.,u) the Hamiltonian vectorfields defined by the functions H(.,u) via the Poisson Structure.

In general

the definition requires

a development similar to

that in Snlatyckl and Tulczyjew [Sn], relating (locally) Hamlltonlan vector fields and a suitable

form of Lagranglan submanifolds

of

manifolds.

idea

structure

The

of

prolonging

the

Polsson

tangent bundle is dealt with In Hermann [H2].

(tangent bundles

to) Polsson

of a manlfold

to

its

Whereas an infinitesimal symplectlc

automorphism is necessarily a locally Hamlltonlan vector field on a symplectio manifold, an infinitesimal autc~orphlsm of the Polsson structure on a Polsson manifold Is not necessarily a locally }{amiltonlan vector field, see Welnsteln [We3]. This combined with the variable rank of a Polsson structure will probably make the required

generalization

of the

afflne

Hamlltonlan system

(7. I) a more delicate

matter than in the symplectic ease (see [All). A peculiar

feature

of Polsson control systems

is that in case the rank of the

Poisson structure in a point p, i.e. the rank of Bp, is less than the dimension of the

manifold

(or

equivalently,

the

if

Polsson

structure

does

not

define

a

symplectle structure), then the system cannot be accessible. This follows from the fact

that

Hamiltonlan

the

dimension

vectorflelds

at

a

equals

point

p

of

the

the rank Of Bp.

dlstrlbutlon

spanned

by

all

(In fact, the ma'xlmal integral

manifold through p of thls distribution is the maximal sympleotic leaf through p as

alluded

to

before.

)

Since

the

acoesslbllllty

distribution

trivially

is

contained in this dlstributlon, its dimension is always less than the rank of the Poisson

structure.

This

implies,

inparticular,

that

we

may

always

restrict

a

Polsson control system wlth initial state to the maximal sympleatlo leaf through this point to obtain a Hamiltonian

control

system on this symplectlo leaf

(see

[VT]). A Polsson s t r u c t u r e

formulation of

the r e a l l z 3 t l o n problem may a l s o provide a

better setting In which to deal wlth another aspect of our work. Our main result

fo~

afflne

systems,

characterizing

Hamiltonian

systems

from

their

input-output

data, theorem (5.11), relates to initialized systems. In a practical sense this is not

partlcularly

useful,

since

a

physical

obssrvation

of

s

system

probably

consists of Input-output records from many initial states. The realization theory for non initialized systems, where one is given Input-output records from sOme (or all) possible initial states has not been worked out; ture

all

exists

Input-output

at

records

on the

subject.

corresponding

The main

to

systems

indeed

very little lltera-

difficulty arises which

are

not

in considering

orbit-mlnlmal

(or

accessible in the analytic case), since then one has to compose input-output maps arlslnE from initial states in different orbits.

The extent to which these input-

output maps may be viewed as arising from different systems, Or subsystems of the same system presents a significant problem. Only when the original system yleldlng the

Input-output

data

is orbit minimal

does

this problem

disappear.

Since, as

already mentioned, many physical systems may be modelled more naturally as Polsson

108 systems,

whlch

ture is less sider

than the

dimension

the r e a l l ~ t l o n

cusslon

Is

be orbit

can never

theory

even more

already

forced

de,loped The

problems in

initialized

space,

to our

results,

non

initialized

apply

ease,

systems.

and output

in this case.

corollary

5.15 and

to con-

This dis-

theorem

5.17,

than [O,-), since there we were

systems.

equally

even

to the

more

In particular

spaces

general

fundamental

OUr

results

COuld

only

be

ere only locally

distinguished.

in chapter

6.

of this

of our results,

as discussed

in Van der Sohaft

observations.

We

encountered

a

problem

problem would yield a m o r e

We repeat

t>3~gln with

The

is,

[V5], where

the most

it here modified

nonlinear

symplectlc

form m e . We use the notation

trajectory

w(t)

arising

from

e yen

as pointed

general statement general

ver-

in light of our fore-

system Z as

In definltion

space W, a n d state space M. Assume W is a sympleotlc

with external

for

theory for these systems is not fully de-

A resolution

is stated.

considered

were

we would like to deal with systems where the In

(6), that the realization

sion of his conjecture

systems

problems

either.

going

It seems appropriate

systems

records on (--,®) rather

consider

above this

out in chapter veloped

of the state

for orbit minimal systems.

However

put

to

in case the rank of the Polsson struc-

for non initialized

relevant

dealing with Input-output

minimal

(6.|),

manifold

with

6w(t) to denote a variation of an external

the system,

and

consider

trajectories

on ~ for slm-

pllclty.

CONJECTURE

The

complete

complete,

mlnlmal

and

is Hamlltonian

analytic

system Z, such

that

the

extended

system

ls

If and only if for every external trajectory w

f me(~lw(t),62w(t))dt - o for all admissible

variations

61w and 62w of w, wlth compact support.

Of

also

to

course

we

may

wlsh

generalize

space W which is a Poisson manifold,

the

situation

to

a n d deduce the existence

include

an

external

of a Polsson system;

but an exact focmulatlon of this problem is yet to be made.

A particular

Polsson

Lie algebra G, example

Olver

sflruct~e

and Is usually [O].

exists

on

the

refiered to as

dual

space

of

tlve to a basis Vl,...,v k. Let G* denote bracket

dimensional

the Lie-Polsson structure,

r l,j,r = I,... k be the structure Let eli

basis. Define the Lie-Poisson

any f i n i t e

constants

see for of G rela-

the dual space Of O and wl,...,w k a dual

between two functions

F,H:G* ~ R by

109 (7.26)

where

IF, H) (x) = x([DF(x),

DF(x),

DH(x)

DH(X)])

x e G

~ (G*)* - G, and [.,.] Is the LIe bracket

defined by the Fcechet derivative

DF(x)(y)

= llm F(x÷ny) ÷ F(x) h

k I

c ~ aF lj ~-~'l (x)

i ,j ,r-1 k that

bracket

Polsson structure

o£

linear

also

important

is Important

becauge

[J5], mentioned

It

in chapter

and his general realization consider

:3 k and [vl,V j] "

functlenals

(7.26) which

ry of Kostant-Klrillov-Sourlau

to

~?.(x>~

l-I ~ I

the space

Lle-Polsson

(7.26) reduces

k

r-1 Note

r-l[

on G* becomes

is of co~rse

eli

a LIe algebra

Isomorphlc

(see for example Wallach

leads

to

the

[We]).

introduced

(I), to the case of infinite

dimensional

theory for Hamlltonlan

a Poisson

bracket

on M denoted

when C=(M) is viewed as a Lie algebra

viewed as a Lie algebra

In this

case

to

~ctor

Hamiltonlan

xV(m)

We

obtain

by

the

a Lie

Lie

subalgebra

under Lie bracket

if each mapping each

v ~

m f G,

under

Lie

algebra

of

manlFold

Polsson

a Lie bracket

(I).

algebra

homo-

and K=(M) ls

~ Is called a symplec-

diffeomorphlsm

G, we

wlth sym-

a is a C ® function on

~ is

of vector fields.

> g.m is a sympleotic the

Lie algebras,

{.,.} as in chapter

~(a) - X , where

Hamiltonlan

is

Jakubczyk

group on a smooth mani-

vectorfleld.

corresponding

by

systems.

M and X a is the

tlc action

Thls development

generalization

We then have a linear map T: C®(M) + X=(M),

morphlsm

the of a

since it arises as we now briefly show in the theo-

a Lie group G acting as a C ® Lie transfo~matlon

form m, we obtain

under

to G. This example

fold M, ¢: G x M ~ M, ¢(g,m) - g.m. If M is also a symplectlc plectlc

is

x,y ~ G*

With respect to the bases wl, v i the definition

{F,H! (x) -

on G . Here DF(x)

of F

can

of M, g e O.

associate

a locally

Field X v In the usual way

°1

" ~

algebra

algebra

L.

t-O exp tv.m

homomorphlsm We

say

F: G * X®(M), v ÷ X v, whose

that ~ is

a

Polsson

action

I£

Imag~ there

F of C=(M) ~nder Polsson bracket and a Lie algebra homomorphlsm

makes the following diagram commutative

we

denote

exists I which

a

110

£

G

~-

1.,

(7.27)

In ease I existed

only as

a linear

mapping

then

each X v • is a globally

Hamilto-

nlan vector field r(v) = X v : X~(v). In case of e Polsson action we have also

(7.28)

A([Vl,V2])

Let G* denote

=

{A(Vl),A(v2) }

the dual of G as before,

define the moment map p: M * G

(7.29)

Vl, v 2 e G .

then

given

a Polsson

action ¢ as above we

by setting

p(x)v = A(v)(x)

for x e M, v e G. ^

G

also

has

a natural

(coadjolnt)

action

on

G*. ¢ : G x G* ~

G*

defined

by the

generators

of

following when £ ~ G* , g e G and v e G,

;(g,£)(v)

(7.30)

where Ad g : G + G is

- £(Ad g-1(v))

the

edjoint

~ Ad*g(£)(v),

action

of G

on G, defined

on

a

connected G by

Ad exp v = e~(p a d v

and ad v(w) - [v,w] is Just the Lie bracket

on G. In fact the moment map is equl-

^

variant for the actions ¢ and ¢ of G on M and G* respectively;

u(~(~,m))

(7.31)

Moreover denoted

since M m,

manlfold,

Is

- ~(g,~(m)).

G is a Lie transformation a

that is

manifold

and

the

group of M, the orbit of m e M under G,

orbit

of p(m) - p under

G

in

G

is also

a

denoted Gp.

Clearly p maps

Mm

onto Gp. The

famous

theory

of

Kostant-Kirillov-Souriau

shows

,

that G~ is

in fact

a symplectlo

manifold

,

symplectlo form on %

,

is given a t

x e %

and p : M

÷ Gp is a m

by

eoveclng

map.

The

111

(7.32)

nx(XV,X w) = x([v,w])

where ~ ( x )

=

dd-~ ; (exp tV, x) It

action of G on i t .

=

0 iS

the vector

field on G p induced

by the

Indeed Xv is a Hamlltonlan vector f i e l d on G~ wlth Hamlltonlan

function Hv defined by the l l n e a r funetlonal

(7.33)

Hv(~) - &Cv)

for v e G and & e G . Now the Lie-Polsson

structure

of G* into symplectlo

on G* defined in equation

leaves, which are the orbits

(7.26) yields a partition

of the corresponding

Hamilto-

nlan vector fields determined from

XH(F)(x)

In

~se

H = H v is a

- IF, H}(x) -xC£DF(x),

linear f ~ c t l o n a l

DHCx)]).

on G* ~

In (7.33),

we obtain DHv(x) = v.

Now

XV(F)Cx} = d'td F(¢Cexp tv,x))It - 0 " dd_t_~(exp tv,x)CDFCx))It

d-E d x (Ad exp-tv,DF(x)) It = 0 - x([DF(x),v])

=

Thus

the

Hamiltonlan

vector

fields X v and X H

ooinolde,

- O "

= XHv(F)(x). and

we

deduce

that

the

sympleotic leaves of the Poisson structure areVjust the oo-adjolnt orbits.

Assume now that we are given a general Hamiltonlan manifold

control system, on a symplectlc

(M,~)

(7.3~)

~ - XH(x,u)

x e (M,~) , u e n ~ Rm, x(O) - xO

,

with outputs either

(7.35)

J

yj = -~uH.Cx,u) J

~ 1,...pm

p

or

(7.36)

The

y = H(x,u)

observation

chapter

(6);

space

namely

Hex t the

of

(7.34)

smallest

Lie

with

outputs

algebra

of

47.35) functions

is

constructed on M

under

as

in

Polsson

112 BH {~-~T(.,u), j = 1,...,m, u • R~ and is closed under Polsson

bracket, which contains bracket

with

J functions

all

{H(.,u); u • ~)

and

B B-~?.' 3 Analogously, the

differentiations

j = I, .... ,m. For simplicity we shall denote flext herb as II.

by

observation space H of (7.34) wlth outputs (7.36) is the smallest Lie Alg~bra that contains {H(.,u); u • ~} and tions {H(.,u)I u g ~} and fact we have

is

closed

under

Polsson

bracket

wlth

all

func-

by ~-~--~, J - I ..... m. As a matter of 3 lemma relating II and H (see for a proof [G2,

differentiations

the following

little

Proposition 2. I])

LEMMA

Let q : Rm ~ Rn be a smooth map. Then the following linear spaces are equal

V ~, span

Buj q ( u ) , j

= 1. . . . ,m, u ~ Rm}

V' = span { q ( u ' ) - q ( u " ) ,

Inpartioular the

smallest

that

it follows Poisson

H (and hence H)

¢I funotlons

on M.

We

u',

u" ~ Rm}

that H is a subalg~bra

Lie

alKebra

is

£inite-dlmenslonal,

can also

containing

D

of H and that H is also given as

{H(.,u); u • ~). Now say

view @I' " ' " ¢ N

H

let

us

= span {¢I,...,¢N I,

as linear

assume with

coordinate functions on

the dual Lie algebra II , which is, as we discussed above a Poisson manifold. Calculating the time-evolution of ¢i along (7.34) we obtain

(7.37)

~d¢l - IH(x,u),¢il dt

N ~ aik(U)¢k, k-1

i - I ..... N

for

some functions aik(U), since [H(x,u),¢il is contained in 2. BH because ~-~(x,u) ~ H it follows that the outputs are of the form J N (7.38) •, ! Cjk(U)¢k J - I ..... S k I

Furthermore

5

Equations that

(7.37) with

the

(7.38) define a control system on S . It immediately follows

input-output

map

state (@1(Xo) ..... @N(xO)) equals system (7.34) with outputs For general to Hljab

this

the

input-output

affine

system

map

of

the

for

the

Hamltonlan

initial control

affine control systems the above constrUOtlon is due

[Hi] and Fliess and Kupka

control

control

(7.36) we started with.

(non-Hamiltonlan)

functions alk(U) ace bilinear

of

iF3].

~n u and

In the aBfine case it follows

that the

the JC~k are constants, so that we obtaln a

system on H . In the Hamiltonlan

case H

is a Po!sson manifold

113 and

the

natural

([Wel]), which

mapping

gives

x e M --> (el(X) .... ,¢N(X)) e H

that

(7.37) together with

is a

Poisson mapping

(7.38) defines a Poisson control

system on H • In fact, since H(x,u) e H foe any u e ~, H(x,u) is of the form N hk(U)¢k, and serves as the generating function for the Polsson control system k=l_w on B . Concluding, under the assumption that H is finite-dlmensional, we can immerse

a Hamiltonian

control

system

(7.34) with outputs

(7.35) into a Poisson

control system on B . Of course by taking

(7.39)

y =

N [ hk(U)¢k (= H(x,u)) k=l

to be output of (7.37) we immerse the modified Hamlltonian control system (7.34) with outputs (7.36) into the modified Polsson control System (7.37) with (7.39). Furthermore reCall that a Poisson control system with initial state zO e H always restricted to the symplectie leaf through

may t~

Zo, which Is exactly the oo-

adjolnt orbit through z0 in H • The construction sketched above was affine

Hamiltonlan

control

systems

upon an idea employed by Goncalves

used by Van der Schaft [V7] in the case of in case H

is finite-dimensional,

and

builds

[O2] for the case that the Lie algebra (under

Lie bracket) generated by the veetorflelds XH(,,u), u e ~, is isomorphic to R. In this latter case there is Immediately given a Polsson action of the Lie group G corresponding to L on M, and a corresponding momentum mapping ~ : M + L

defined

by

(7.40)

where

~(x)v = F(x) d such that XV( - _d-~xP tv{t©O) equals X F. In eoordlnates this mo-

F • H is

ment map is given as follows. Let v 1,...,v N be a basis of G, N of G*. If v = ~ via i e G then l(v) ~ H is given by l=l Pl = X ( v i ) e H, a i e R. Thus

and Wl,...,w k a basis N ~ aiP i where I=I

N

(7.41)

U(x) "

X wiPi(X) i=l

Hence we obtain the commutative diagram (7.27) with G a Lie algebra isomorphic to L and H

given

by H. However,

in

general

L

is only

isomorphic

to H modulo

the

oor~tant funotlons (see chapters (3) and (6)), and so we do not immediately obtain a

Poisson

completely

action

on

resolved

discussed below,

M Oy

and

corresponding

J~kubczyk

in

the

momentum infinite

mopping.

This

dimensional

problem

was

generalization

114 The

interest

control

in

the

systems

structure

Idea

lles

of

immersing

partly

in

the

of minimal realizations

see Van der Schaft

[V7],

study realization

fact

that

control the

of Hamlltonlan

Goncalves

theory

Hamlltonian

existence,

[JS].

G*

into

the

coadjolnt

view

partition leaYes

of

of G

of

the

orbits Cp(m). Thls

considering

Polsson

Hamiltonian

Into co-adjoint

orbits

Lie

Polsson

structure

on

Poisson

structures,

dimensional"

systems,

as in Krishnaprasad

The geometry

of the co-adjolnt

We mention

is solvable, in tUrn [C2]

details

details

of

and Bloch

Finally

we

Souriau

theory

arbits

how

realization

is infinite

dimensional.

control

set £ is thought

indeed a Polsson manifold) may

form

the

Poisson

ponding Hamlltonian It

follows

that

dimensional by £ (free formal

Lie Lle

(Lie)

identified

we

Lle

cannot

complete Rather

G

over R generated elements

of formal

is now described

< ~ awW, ~ bwW> W

which

makes

taken

over ~

the diagram

sense

W

because

~

form

the

case.

in

are

the Konstant-Klrlllovorder

to

deal

with

the

when the Lie algebra

in chapter variables

a symplectlc

(I), in which and ~

manifold

is the M,

(or

{he; e e ~} on M then we

generate,

along

with

with

F

algebra

the

corres-

dimensional.

= H vla

a

A over R with

Let A*(G*) denote

power series

Irving

series

application.

by ~), identified

of £.

This work and

Volterca

(7.36),

(7.27)

in speclal

see Crouch

extend

free

more

"infinite

the

(formal Lie series)

finite

generated

the dual

sets

of

spaces

in elements

by

awb w

W

the sum on the right

and a w, b w are

(7.27).

we

other

from

[Au] in the case where G

finite

G,

given

diagram

G.

with the algebra

are

very intensively

described

they

algebra

in

the

into symplectic

We do not assume H is finite

algebra

of ~. The duality

to

(7.34),

we are

algebra H that

more

there

of as a set of non commuting

fields X h

polynomials

of

groups

to

[K2].

and a set of smooth functions

~ctor

the

demonstrated,

arising

structure;

manages

the situation

set of words XlX2...x k, xj e £. Assume

relate we

that some

dimensional"

theory of systems

to

[Ki] in the nilpotent

system

[JS]

since the par-

partition

and Kostant

realizations

dimensional"

Recall

Note

and

thls way,

into the partition of

as

the

has been studied

control

Jakubczyk

"infinite

seen

and Marsden

[B1] for an "infinite

indicate to

of the

Hamiltonlan

corresponding

the

G*.

here the work by Auslander

provides

Is

inparticular

and of course that of Kirillov

where

discussed,

and

uniqueness

systems,

since,

is precisely

complicated

cases.

then

systems

Poisson

Also it may enable us to

tition of the state manifold M into orbits M m, is transfered

general

into

systems may be attacked

[GI], Jakubczyk

for non initialized

systems

coefficients

is finite.

in ~). It Is now

(Here the summation clear

how to

Is

complete

The map A is defined on £ by A(e) = h~, ~ e £, and extended to

115 Lie polynomials in G by Inslstln8 that ~ IS a Lie algebra homomorphlsm from G onto H. For example A(~la2-~2(,1 ) = lh 1,h 2J for ~I,e2 e ~]. Thls Is possible since G is a free Lie algebra. P is defined in the obvious way as r(w) = XA(w), w ~ G.

The {formel) momentum mapping

~: M ÷ G

is now easily defined as bef(~e, equation

(7.5), by setting

(7.42)

= X(w)(x),

. where as in for

the

x e M, w e a

c h a p t e r ( 1 ) , g ( x ) = Z < g(x),w > w. (Compare t h i s expression w i t h t h a t

finite

dimensional

w

case,

equation

(7.41).)

Moreover a

skew-symmetric

bilinear form on G* can be defined as in equation (7.32) by setting

(7.43)

gp(W,W') = < p,[w,w']>

w e G, p e G

where [w,w'] = ww'-w'w. Jakubczyk now shows t h a t the formal momentum p - g ( x ) defines the i n p u t - o u t p u t map

of

an analytlc

system

and <~(x),w> define

(7.34),

coefficients

(7.36)

if and only

if the rank

of ~nalytlc maps as explained

of gp Is finite

in section

(1.2).

1~qe state space of the system is clearly the co-adJolnt orbit of p under G, whloJa is finite dimensional because of the rank condition on ~]p.

Referents

[A]

R.A.

Abraham,

and

J.E.

Marsden,

edition), BenJamln/Cummings,

[AI]

G. Alvarez-Sanehez,

"Foundations

of

Mechanics',

(Second

Reading (1978).

"Geometric Methods of Classical Mechanles applied to

Control Theory", Thesis, Univ. of Calif., Berkeley (1986).

[Au]

L. Auslander

and B. Kostant,

"Polarizations

and Unitary Representations

of Solvable Lie Groups", Inventlones Math., Vol 14 (1977) pp 255-354.

[BI ]

R.W.

Brockett,

"Control

Theory

and

AnalytiCal

Mechanics"

in Geometric

Control Theory, eds. C. Martin and R. Hermann, in Vol VII of Lle Groups, History, Frontiers and Applications, Math Sci Press, Brookllne (1977).

[B2]

R.W.

Brookett

Equations,,,

and

A.

Hahlml,

in Ordinary

"Lie

Aleebras

Differential

and

Equations,

Linear

ed.

Dlfferentlal

L. Weiss,

Aoademlo

Press, New York (1972).

[B3]

R.W. Brockett, Finite dimensional linear systems, Wiley, New York (1970).

[sl]

J.M. Bismut, Mee~nlque Al6atolre, Lect. Not. Math., 866, Springer, Berlin

(1981). [BI]

A.M.

Bloch,

"Left

Invariant

Homogeneous Spares",

Control

Systems

on

Infinite

Dimensional

Proceedings 24 th IEEE CDC, Fort Lauderdale (1985),

pp 1027-1030.

[Bul]

A.G.

Butkovskll

and

Yu.

I.

Samoilenko,

"Control

of

Quantum

Systems",

Automat. Rem. Contr., Vol 40 (1979), pp 485-502.

[Bu2]

A.G. Butkovskll

and Yu. I. Samoilenko,

"Control

of Quantum Systems II",

Automat. Rein. Control., Vol 40 (1979), pp 629-645.

[ci]

P.E.

Crouch

Hamlltonlan pp 293-318.

and

M.

Irving,

Reallzatlons",

"On Math.

F1nlte

Volterra

Systems

Theory,

Series Vol

Which 17

Admit (1984),

117

[c2]

P.E.

Crouch

and

M.

Irving,

"Dynamical

Realizations

of

Homogeneous

Hamlltonlan Systems", SIAM J. Control and Optlmlzation, Vol 24 (1986), pp 374-395.

[C3]

P.E.

Crouch,

"Hamiltonlan

Realizatlons

of

Finite

"Theory and Applications of Nonlinear Control

Series",

in

Systems", C.I. Byrnes,

Volterra

A.

Llndquist, eds., North-Holland (1986), pp 247-259.

[c4]

P.E. Crouch,

"Dynamical Realizations of Finite Volterra Series,,, SIAM J.

Control and Optimization, Vol 19 (1981), pp 177-201.

[FI]

M. Flless, "Realizations of Nonlinear Systems and Abstract Translhive Lle

Algebras", Bull.

[ F2]

M.

Flless,

Amer.Math. See. ( N . S . ) ,

Vol 2 ( 1 9 8 0 ) , pp 444-446.

"R6alisatlon Locale des Syst~mes Non L!n6aires,

Alg~bres de

Lie Flltr~es Transitives, et S4ries Gdn4ratrlces Non Commutatlv~s",

Inv.

Math, Vol 71 (1983), pp 52]-533.

iF3]

M. Flies and I. Kupka, "A Finiteness Criterion for Nonlinear Input-Output Differential

Systems",

SIAM J. Contr. and Opt., Vol 21 (1983), pp 721-

728.

[01]

J.B.

Gonoalws,

"Realization

Theory

For

Hamlltonlan

Systems",

SIAM J.

Control and Optimization, Vol 24 (19B6).

[G2]

J.B.

Goncalves,

"Nonlinear

Observablllty

and

Duality",

Syst.

Contr.

Letters, Vol 4 (1984), pp 91-101.

[Ga]

J.P.

Gauthler

RealiZations

and

G.

in

the C ®

Bernard, case",

"Existence

and

Syst. Contr.

Uniqueness Letters,

of

Vol

Minimal

I (1982),

pp 395-398.

[HI

R. Hermann and A.J. Krener, "Nonlinear Controllability and Observabillty" IEEE Trans. on Automatic Control, Vol AC-22 (1977), pp 728-740.

[Hi]

[J1]

O.B. Hljab, "Minimum Energy Estimation", Ph. D. Thesis, Berkeley, (1980).

B. Jakubczyk, and

"Hamiltonlan Realizations of Nonlinear Systems" in "Theory

Appl icatlons

of

Nonlinear

Control

Systems",

Llndquist, eds., North-Holland (1986), pp 261-271.

C .I.

Byrnes,

A.

118

[J2]

B.

Jakubczyk,

',Existence and

Uniqueness

of

Realizations

of

Nonlinear

Systems", SIAM J. Control and Op~imlza~ion, Vol 18 (1980), pp 455-47].

[J3]

B. Jakubczyk, "Local Realizations of Nonlinear Causal Operators", SIAM J. Control and Optimization, Vol 24 (1986), pp 230-2~2.

[J4]

B.

Jakubczyk,

"Construction

off

Formal

and

Analytic

Realizations

of

Nonlinear Systems" in "Feedback Control of Linear and Nonlinear Systems", D. Hinrichsen and

A. Isidorl, ads.,

Springer, New York (1982), pp 147-

156.

[us]

B. Jakubczyk,

"Existence o£ Hamlltonian realizations of nonlinear causal

operators", Bull. Pol. Acad. Sol., Ser. Math., to appear (1986).

[J6 ]

B.

Jakubozyk,"Realizatlons

"Proceedings Conference linear

Control

Theory",

of

nonlinear systems:

Three approaches",

on the Algebraic and Geometric Methods Paris

(1985),

M.

Fliess,

M.

in

in Non-

Ha Zawinkel,

eds.

Reidel, Dordrecht (]987), pp 3-31.

[JT]

B. Jakubozyk,

"Existence of Hamlltonlan structure of nonlinear systems",

Institute of Mathematics, Polish Academy of Sciences, Warsaw (1986).

[J8]

B.

Jakubczyk,

"Poisson

structures

and relations

on

vector

fields

and

their Hamlltonlans", Bull. Pol. Acad. Sci., Set. Math., to appear (1986).

[KI]

P.S.

Krishnaprasad,

Asymptotic

"Lie-Poisson

Stability",

Nonlinear

Structures, Analysis,

Dual-Spin Theoretical

Spacecraft Methods

and and

Applications, Vol 7 (]984), pp 101]-1035.

[K2]

P.S. Krishnaprasad and J. Marsden, "Hamiltonlan Structures and Stability for Rigid Bodies with Flexible Attachments", to appear Arch. Rat. Mech. (1986)

[K1]

A.A. Kirillov, "Unitary Representations of Nilpotent Lie Groups", Russian Math. Surveys, Vol 17 (1982), pp 53-104.

[L]

C.M.

Lesiak and A.J. Krener,

Series

for Nonlinear

(1978), pp ]090-1095.

Systems",

,'The Existence

and Uniqueness of Volterra

IEEE Trans Automatic Control,

Vol AC-23

119 [La]

and P.E. Crouch,

g. Lamnabhl-Lagarrlgue

Identities",

Arizona

State University,

"Algebraic end Multlple Integral Dept. of Electrical

and Computer

Engineering, Tempe (1985).

[M]

R. Merino,

"Hemiltonlan

Techniques

in Control

Arms and Power

of Robot

Systems", in "Theory and Applications of Nonlinear Control Systems", C.I. Byrnes, A. Lindquist, eds., North-Holland (1986), pp 65-74.

[Mr]

P.M.

Morse

McGraw-Hill

[o]

P.J.

and

Feshbach,

"Methods

of

Theoretical

Physics,

Vol

I,

(1953).

Olver,

Graduate

H.

"Applications

Texts

in

of

Lie

Mathematics

,

Groups Vol

to

Differential

107, Springer

Equations",

Verlag,

New

York

(1986).

[P]

R.S.

Paials,

"A

global

Formulation

of

the

Lie

Theory

of

Transitive

Groups", Memolres o£ the A.M.S., No 22, (1957).

JR]

R. Ree,

"Lie Elements and an Algebra Associated with Shuffles,,, Ann. of

Math., Vol 68 (1958), pp 211-220.

[si ]

H.J.

Sussmann,

"Existence

and

Uniqueness

of

Minimal

Realizations

of

Nonlinear Systems", Math. Systems Theory, Vol 10 (1977), pp 263-284.

IS2]

H.J.

Sussmann

and V.

Jurdjevlc,

"Controllability

of nonlinear systems",

J. Differential Equations, Vol 12 (1972), pp 95-116.

[Sal]

R.M.

Santllll,

"Foundations

0£

Theoretical

Mechanics

I",

"Foundations

of

Theoretical

Mechanics

II",

S~rlnger,

New-York (1978).

[sa2]

R.M.

Santilll,

Springer

New-York (1983).

lear]

W.

Sarlet,

inverse

"The

~oblem

Helmholtz

conditions

of Lagranglan

revisited.

dynamics",

Journal

A new

approach

Physics

to the

A: Math.Gen.,

Vol 15 (1982), pp 1503-1517.

[sn]

J. Snlatycki and W.M. TulczyJew,

"Generating Forms of Lagranglan Submani-

folds", Indiana Univ. Math. J. Vol 22 (1972), pp 267-275.

120 [st]

J.

Stelgenberger,

"On

B1rkhofflan

Representation

of

Control

Systems",

Wlss. Z. TH Ilmanau, 33 (1987), Heft 1.

IT]

F. Takens,

"Varlatlonal and conservatlve Systems", Report ZW 7603, Univ.

of Gronlngen (1976).

[Ta ]

T.J.

Tarn,

G.

Huang and

J.W.

Clark,

"Modelling

of

Quantum Mechanical

Control Systems", Math. Modelling, Vol I (1980), pp 109-121.

[Tu]

W.M. Tulczyjew, "Hamiltonlan systems, Lagranglan systems and t h e

Legendre

transformation", Sympo Math. Vol 14 (1974), pp 247-258.

[Vl]

A.J. van der Schaft, "System theoretlc descriptions of physical systems", Door. Dissertation Unlverslty of Gronlngen,

1983, also CWI Tract No. 3,

CWI, Amsterdam (1984).

Iv2]

A.J-

van

der

Sehaft,

"Hamlltonlan

dynamics

wlth

external

forces

and

observations", Mathematical Systems Theory, Vol 15 (1982), pp 145-168.

[V3]

A.J.

van

der

Sohaft,

"Symmetries,

conservation

laws

and

time-

reversibility for Haml!tonlan systems wlth external forces", Journal of Mathematical Physics, Vol 24 (1983), pp 2095-2~O1.

IV4]

A.J.

van

der

nonlinear

Schaft,

Hamiltonlan

"Controllability systems",

IEEE

and

observabillty

Trans.

Automatic

for

Control,

afflne AC-27

(1982), pp ~90-492.

Iv5 ]

A.J. van der Schaft, "System theoretic properties of Hamlltonlan systems with

external

forces",

Proceedings

Dynamical

Systems

and

Mlorophyslos

(Control Theory and Mechanics) eds. A. Blaqui~re, G. Leltmann, Academlo, New York (1984), pp 379-400.

IV6]

A.J.

van

der

nonllnear

Schaft,

systems",

"Observabiilty

SIAM

J.

and

Control

and

"Hamiltonlan

and

controllability Optlmlzatlon

for

smooth

20

(1982),

Vol

pp 338-354.

[v7]

A.J.

van

Systems", Theory

of

der Udln~

Schaft, (1985),

Dynamical

to appear

Systems

Quantum

In Proc.

and Mlcrophyslcs

Mechanlcal

4th Int.

Seminar

(Information,

Complexity,

and Control in Quantum Physics), Udlne 1985, eds. A. Blaqui~e, G. Lochak, Academic, New York (1987).

Control on Math.

S. Diner,

121 [ V8]

A.J.

van der Schaft,

,,Stablllzatlon

of Hamlltonlan

Systems",

Nonl.

An.

Tn. Math. Appl., Vol 10 (1986), pp 1021-1035.

[V93

A.J.

van

der

Schaft,

"Controlled

Invarlanoe

for

Hamiltonlan

Systems",

Math. Systems Theory, Vol 18 (1985), pp 257-291.

[rIo]

A.J.

van

der

Schaft,

"On Feedback

"Theory and Applications

Control

of Nonlinear

of

Control

Hamiltonlan

Systems",

Systems"

In

Byrnes,

A.

C.I.

Lindqulst, ads., North-Holland (1986}, pp 273-290.

[Wl]

J.C.

Wlllems,

"System

theoretic

models

for

the

analysis

of

physical

systems", Ricerche dl Automatlca, Vol 10 (1979), pp 71-106.

[w2]

J.C.

Willems

using

and

External

A.J.

van

der

and Internal

Schaft,

Varlables

"Modellng wlth

Systems", in Dynamical Systems and M1crophysles EdSo A. Avez,

of

Dynamlcal

Appllcatlons

Systems

to Hamiltonlan

(Geometry and Mechanlcs),

A. Blaqul~re,

A. Marzollo, ACademic Press, New York (1982),

"Dlsslpatlve

Dynamlcal

pp 233-264.

[W3]

J.C. Wlllems,

Systems",

Part I & Part II,

Arch.

Rat. Mech. Anal., Vol 45 (1972), pp 321-392.

[wa ]

N.R, Wallach,

"Symplectio Geometry and Fourier Analysis", Math Soi Press,

Brookllne Mass, (1977).

[Wel]

A. Welnsteln,

,Symplectie

Manifolds

and their Lagranglan

Submanlfolds",

Advances in Math., Vol 6 (1971), pp 329-343.

[we2]

A.

Welnsteln,

"Lagranglan

Submanlfolds

and Hamlltonian

Systems",

Amer.

Math. Soc. Vol 98 (1973), pp 377-4]0.

[We3]

A.

Welnsteln,

"The

Local

Structure

of

Polsson

Manifolds",

J.

Diff.

Geometry, Vol 18 (1983), pp 523-557.

[Y]

K.

Yano

and

S.

New York (1973).

Ishlhara,

"Tangent

and

cotangent

bundles",

Dekker,

Lecture Notes in Control and Information Sciences Edited by M.Thoma and A.Wyner Vol. 62: Analysis and Optimization of Systems Proceedings of the Sixth International Conference on Analysis and Optimization of Systems Nice, June 19-22, 1984 Edited by A. Bensoussan, J. L. Lions XlX, 591 pages. 1984. Vol. 63: Analysis and Optimization of Systems Proceedings of the Sixth International Conference on Analysis and Optimization of Systems Nice, June 19-22, 1984 Edited by A. Bensoussan, J. L. Lions XlX, 700 pages. 1984. Vol. 64: Arunabha Bagchi Stackelberg Differential Games in Economic Models VIII, 203 pages, 1984 Vol. 65: YaakovYavin Numerical Studies in Nonlinear Filtering VIII, 2?3 pages, 1985.

Vol. 71: N. Baba NewTopics in Learning Automata Theory and Applications VII, 231 pages, 1985. Vol. 72: A. Isidori Nonlinear Control Systems: An Introduction VI, 297 pages, 1985. Vol. 73: J. Zarzycki Nonlinear Prediction Ladder-Filters for Higher-Order Stochastic Sequences V, 132 pages, 1985. Vol. ?4: K. Ichikawa Control System Design based on Exact Model Matching Techniques VII, 129 pages, 1985. Vol. 75: Distributed Parameter Systems Proceedings of the 2nd International Conference, Vorau, Austria 1984 Edited by E Kappel, K. Kunisch, W Schappacher VIII, 460 pages, 1985.

Vol. 66: Systems and Optimization Proceedings of the Twente Workshop Enschede, The Netherlands, April 16-18, 1984 Edited by A. Bagchi, H.Th.Jongen X, 206 pages, 1985.

Vol. ?6: Stochastic Programming Edited by E Archetti, G. Di Pillo, M. Lucertini V, 285 pages, 1986.

Vol. 6?: Real Time Control of Large Scale Systems Proceedings of the First European Workshop University of Patras, Greece, Juli 9-12, 1984 Edited by G. Schmidt, M. Singh, A. Titli, S. Tzafestas XI, 650 pages, 1985.

Vol. ?7: Detection of Abrupt Changes in Signals and Dynamical Systems Edited by M. Basseville, A. Benveniste X, 3?3 pages, 1986.

Vol. 68: T. Kaczorek Two-Dimensional Linear Systems IX, 39? pages, 1985.

Vol. ?8: Stochastic Differential Systems Proceedings of the 3rd Bad Honnef Conference, June 3-7, 1985 Edited by N. Christopeit, K. Helmes, M. Kohlmann V, 3?2 pages, 1986.

Vol. 69: Stochastic Differential Systems Filtering and Control Proceedings of the IFIP-WG ?/1 Working Conference Marseille-Luminy, France, March 12-17,1984 Edited by M. Metivier, E. Pardoux X, 310 pages, 1985. Vol. 70: Uncertainty and Control Proceedings of a DFVLR International Colloquium Bonn, Germany, March, 1985 Edited by J. Ackermann IV, 236 pages, 1985.

Vol. ?9: Signal Processing for Control Edited by K. Godfrey, R Jones XVIII, 413 pages, 1986. Vol. 80: Artificial Intelligence and Man-Machine Systems Edited by H. Winter IV, 211 pages, 1986.