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

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,.,)(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) --> I, ak, bj~j, and

# = sup rank {~---~tÀ 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



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" .... (~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, 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--~ -

24 k

- 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

= 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 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- +

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



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-

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( 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. 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
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 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.

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