Nonlinear - Local Controllability

8/11/2019 Nonlinear - Local Controllability

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Proceedings

of the International Congress of

Mathematicians

August 16-24, 1983, Warszawa

H . W.

KNOBLOOH

Nonlinear Systems: Local Controllability and Higher

Order Necessary Conditions for Optimal Solutions

1 .

Introduction

We consider control systems

which

are defined in terms of an ordinary-

differential equation

x

=f(t;x,u).

(1.1)

u is the control variable and may he subject to a constraint of

the

form

u eTJ. We allow specialization of u to an admissible control function

u(-),

th a t is, a fun ction w hich is piecewise of class0onR and has a range

whose closure is contained in TJ. The function / on the right-hand side of

(1.1) is assum ed to be sufficiently sm ooth . H enc e, if a n adm issible con

trol function is subst i tuted for u in (1.1), we obta in a differential eq ua tio n

which allows

application

of al l standard results concerning the existence,

uniqueness and continuous dependence of solutions (see e.g. [1], Sections

2- 4 ) .

Any one of these solutions will be denoted by x(-) and cal led an

admissible trajectory. We also refer to the pair

(u(-),x(-))

as a solu tion

of (1.1). If we speak of an optim al solution, we m ean a s olution wh ich m ini

mizes th e function al with in th e class of all admissible trajec torie s satis

fying boundary condit ions of the usual type. I t is always taci t ly assumed

that the value of the functional can be identif ied with the terminal value

of a component of the state vector.

We are concerned in this lecture with two types of problems which can

be studied

independent

from each other. However, i t is clear from the

beginning that one can expect some kind of duali ty between statements

pertain ing to each of these problems. Am ong other things we will u nd er tak e

in this lecture an at tempt to put this duali ty into more concrete forms.

Problems of the first type deal with necessary condit ions which have

[1369]


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13 70 Section 14: H . W . Knobloch

to hold along singular ares. A singular arc is a portion of an optimal sol

ution which is such th a t th e control variable is specialized to interior

values of the control set

TJ.

We restrict our attention to conditions which

have to hold pointwise along a singular arc and which assume the form

of a multiplier rule, i.e., a rule which can be expressed as an inequality

of the form y(t)

T

a(t)

^0,

where y(-) is the usual adjoint state vector.

Problems of th e second typ e carry the label local controllability .

The precise definition goes as follows. Let there be given, for all

t

in some

interval tt

0

,t], an arbitrary solution u(-)

9

x(') of (1.1) (called reference

solution from now on). Local controllability along this solution and for

t

=

t

means

:

There exists,

for

every

sufficiently

small

e

> 0, a

full

neigh

borhood of

x(t)

which can be reached at time

t

=

t

by travelling along

admissible trajectories starting at time

t

==?e from

x(t

e). In other

words:x(t) is an interior point of the set of all states to which the system

can be steered from

x(t

s)within time s. We remark th a t our notion of

local controllability coincides with Sussmann's small tim e local control

lability (cf. [2],Sec. 2.3), if the system equation is autonomous and the

reference solution stationary.

It is somehow clear from the above definitions that problems of both

types are concerned with local properties of solutions and that these

properties, in a certain sense, exclude each other. If a solution is optimal

(in the senseas explained above), then the set of all states into which the

system can be steered from

x(t

e)within timeeis situated on one side of

a certain hyperplane through the terminal point

x(t)

and we have no

local controllability

for

t = t.Thus one can expect some kin d ofcorrespond

ence between results concerning singular extremals and those concerning

local controllability which roughly speaking, amounts to reversing

conclusions in a suitable way. We will demonstrate in Section 2 how this

kind of reasoning can be put on more solid grounds by presenting two

theorem s one giving necessary conditions for singular arcs and th e

other giving sufficient conditions for local controllability in which all

statements are expressed in terms of one and the same object, namely the

local cone of attainability. This is a set of elements of the state space which

is associated w ith each poin t of th e reference solution. Since we may th ink

of a solution as a curve parametrized by th time t, we denote this set

by

Jff.

The precise definition is given in Section2 ; it will turn out to be

a modification of the definition of the set

II

t

which was introduced in

[1],

Section 9. Infact,jf

t

is a subset of

II

t

.

The reason that we dispense

here with some elements of

II

t

is the gain in mathematical structure.

jT

t

enjoys certain properties which cannot be inferred from the definition


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Nonlinear Syste m s: Local Controllability and Optimal Solutions 1 3 7 1

of

II

t

:

It is a convex cone and its maximal linear subspace is invariant

under a certain operator

J

7

(Theorem

2.1). The operator.Twillbe described

in detail in Section 2. In contrast to the philosophy

adojvted

in [1] we

prefer here a definition which depends on the choice of a special reference

solution since it helps to bring out the system theoretic aspect of this

operator. One can look uponJ

7

fromth e viewpoint of linear systems the ory;

it then appears as a generalization of the process which leads to the con

struction of the controllability matrix. Indeed, if the system equation

is linear and is given as

x

=A(t)x +

B(t)u,

(1.2)

then the columns of this matrix can be generated from the columns of

B(t) by repeated application of J

7

. One can also look onit from the differ

ential geometric viewpoint. If the system equation is autonomous and

the reference solution stationary then the simplest way to explain the

application of r is in terms of a Lie-bracket involving / ( = the function

which appears on the right-hand side of (1.1)). This, by the way, explains

why the forming of the Lie-bracket with/ is a nonlinear analogue of the

linear mapping is defined in terms of the matrix

A

(t) of the linear system

(1.2).

Eegardless of which view one prefers, what counts for our purposes

are the following two facts, (i) We can define r without any restrictive

assumptions, as linearity of the equation or time independence of the

reference solution, (ii) One can use J

7

in order to generate new elemen ts

ofdC

i

out of givenones :From the previously m entioned invariance prop erty

ofc/C

i

one infers that the following statement holds true:

i p e j r , implies F

tt

(p)ei

t

,

^ = 1 , 2 , . . .

(1.3)

To get an impression of the scope of this result it might be helpful to

consider a special case. Let us assume that the system is linear in u, and

hence defined by a differential equation of the form

m

A^U{t;x) +^]u

v

g

r

{t;x), u

=

(it

1

,...,

m

).

(1.4)

=

Furthermore, let us assume that the reference control satisfies the condi

tion

u(t)

eintZJ for all

t e p

0

, * ] .

Using standard variational techniques

it is then not difficult to see that

g

v

(t-,x(t)) eX

t

for all

t e

[t

0

, ] . Hence,

it follows from (1.3) that the linear space spanned by (r^g^it, x(t)),


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13 72 Section 14: H. W. Knobloch

p = 0,1, ...,v =1, ...,m,is contained in

ct

t

.

This space was intro du ced

in

[1]

and den oted th er e b y 33(2) (cf. Section 1 , in pa rtic ula r p . 5), i t can

b e

identified

with the columns of the control labil i ty matrix

for

the l inear

ized sy stem (i.e., for the lin ear syste m (1.2) with

A(t)

: =(dfldx)

[t;x(t),

u(t))

a n d B(t)

: =

(dfldu)(t; x(t),

u(t)).

It is therefore no t surprising to redis

cover

33 (J) as

a part of

#

%

and to establish i ts invariance with respect

to the opera tor

B\

this could be verif ied also by standard arguments.

The importance of the statement (1.3) rests upon the fact that i t a l lows to

extend the space

93

(t) by adjoining

further

e lements p without losing i ts

tw o

basic

propert ies, namely, that of being a subspace of

jf

t

and being

inva r iant w i th respect to r. In other words

:

One can add to the genera tors

g

v

of th e spa ce

93

(t)all elem ents

p ec/T

t

which

satisfy

t h e condit ion +p

e i

t

and then treat the enlarged set as i f i t would be the set of generators

for

33(2).

W he ther th is is a useful insight or no t , depen ds on the concrete

possibilities of constructing vectorsp wi th the proper ty +p e

cf

t

and w hich

are not already elements of

93(2).

W ha t is kno wn in thi s respect is ve ry

l i t t le,

nevertheless i t seems worthwhile to review carefully the material

exist ing up to now.

The first examples of non-trivial elements

p

which are contained in

X

t

tog ethe r w ith the ir nega tives are amo ng w ha t we will call second

order elem ents an d discuss in detai l in

Section

3. The name stems from

th e fact th a t th e necessary condit ions which can be /expressed in te rm s

of thes e elements are comm only cal led second ord er . W e will pres ent

in Section 3 a gen eral definition of th e or de r of a n elem ent of tf

t

and

give a complete description of the set of all

second

order elements. Special

emphasis is put on thosep which appear together wi th

p in this set and

which therefore must be orthogonal to the adjoint state variable along

an optimal solut ion. I t has been known since long that

for

a system of the

form (1.4) th e m ixe d Lie-brackets

Pv,t*'-=

C^jffJ

(1-5)

enjoy this property along a singular arc. But the background of this was

not recognized unti l recently when Vrsan [4] announced the fol lowing

result: Local controllability along a reference solution of the system (1.4)

can beinferred from th efollowing two conditions :

(i) the reference control assumes values in the interior of TJ for allt

9

(ii) the controllable subspace 33(2) and the elements

r

v

(p

v

J

9

v,ii = l , . . . , m , y = 0 , 1 , . . .

genera te together the whole s ta te space .


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Nonlinear Systems: Local Controllability and Optimal Solutions 1 3 7 3

As we will see in Section 3, bo th t he second order equality t y pe condi

tions and Vrsan's result are true since the hypothesis (i) implies

p

VffX

e

jf

/

for any system of the form (1.4). It is also possible

under suitable

extra hypotheses to add further second order elements to the

p

V t f

in

such a way that one arrives at a similar type of controllability criterion.

However, all second order elements

p

whichare such that +p

eX

t

reduce

to zero if the dimension

m

of the control variable is equal to one (note th a t

Pv

>f

A = 0, in view of (1.5)). The absence of those elements can be under

stood from the n atu re of th e corresponding necessary co ndition s: t he y

can be compared to the standard second order tests in calculus. Basically,

these tests are inequalities (semi-definiteness of a quadratic form), which

eventually m ay lead to equality ty pe statements

;

namely, if th e

form fails

to be definite. All these sta tem ents, however, are trivial if the re is no t more

than one variable.

I t should be pointed outthatVrsan's result reflects a typical non-linear

system prop erty: There exists a kind of crosswise interaction between

the components of

u

which is exercised through the state vector (note that

p

V t f

= 0 if the

g

v

do not depend upon

x)

and which cannot be recognized

by means of linearization since for a linear system the action ofu

=

(u

1

, ...

...,

u

m

)

is just the superposition of the action of the components

u

v

.

In

precise mathematical terms this interaction is expressed by the fact that

one can simply adjoin the

p

V i f t

to the generators ofSB(2) w ithout destroying

the controllability properties of this space.

Next, we wish to say a few words about possible extensions of58(2)

in case of a scalar control variable

u.

It is clear from what was said above

that one has to search for possible candidates among higher-than-second-

order elements, but it is presently not obvious how this search can be

carried out in a systematic way. What one expects to find is some kind

of hierarchy among the subspaces of

tf

, which corresponds to the hier

archy among higher order tests in optimization. Of course the controllable

subspace SB (2) of the linearized system equation should be the member

of lowest rank.

The first attempt to put this idea into a more concrete form has been

undertaken by H. Hermes and completed by H. Sussmann [3]. It led

to a controllability criterion for a system of the form

x =f

0

(x)

+ ug(x), u

scalar, (1.6)

with a stationary point (u

09

x

Q

)playing th e role of th e reference solution.

The crucial condition which enters this criterion concerns the Lie-

brackets associated with system (1.6) and evaluated at

(u, x)

=

(u

0

,

x

0

).


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Tobe m ore specific, itisassumed that all brackets which involve the quan

tity

g

an even num ber of times can be expressed as a linear combination

of Lie-brackets which are of odd order with respect to

g.

This even-odd

relationship resembles the one which is well

known,

from elementary

calculus: If all derivatives up to order 2Jc vanish at an extremal point

of a function, then the (2ft+l)th derivative must also vanish there. Now,

vanishing of some derivative at an extremal point is an equality-type

necessary condition. In optimal control theory conditions of this kind

appear in the form of an orthogonality relation y(t)

T

p = 0. As we have

seen before, they arise from elements

p

of the state space which satisfy

the condition +p eX

t

. We wish therefore to pose th e following question

which is a natural modification of the Hermes conjecture: Assume that

the above stated condition holds for all Lie-brackets which are of order

at most2Jfcwith respect to

g,

Jcbeing a fixed positive integer. Is it the n true

that the linear space spanned by all Lie-brackets which are of order at

most2Jc +1with respect togbelong to #

t

?

In this generality the question probably cannot be answered along

the lines of existing methods;in particular, it is unlikely th at Sussmann's

proof of the original conjecture could be carried over. Note that it is

required to establish the existence of specific elements in

jf

t9

regardless

of whether we have local controllability or not. It seems, however, con

ceivable that special cases can be treated e.g. with methods taken from

[1] and th at one w ould the n be able to examine from case to case how

much of the assumptions underlying the Hermes-Sussmann result is

actually required. From the viewpoint of applications one would anyhow

welcome results which are more restricted in its scope in return for more

flexibility with respect to the hypotheses. Some steps in this direction

have been undertaken and will be discussed in the lecture. In particular,

it seems very likely though not all details have been cleared tha t for

systems of the form (1.6) one can extend the space

SB (2)

by adjoining

third order elements (i.e. vectors which can be written as third order

polynomials in the components of

g, g

x

, g

xx

,

etc.) under the assumption

that the Lie-bracket

[g,

[ss/o]] (1-7)

evaluated at the reference trajectory

x

=

x(t)

is contained in 93(2) for

t e

[2

,*].

The reference solution need not be stationary; however,

u(

)

has to assume values in th e interior of th e control set

TJ.

To compare

,

this result with the Hermes conjecture, one has to take into account that


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Nonlinear Sys tem s: Local Controllability and Optimal Solutions 13 75

in case of a stationary reference solution the space 93(2) is independent

from

2

and coincides with the linear span of those Lie-brackets which are

first order with respect to

g.

Hence i t is required

in case of a stationary

solution that (1.7) is a linear combination of first-order Lie-brackets

in order to ensure the existence of certain third-order elements in

X

t

.

The

conclusion is certainly much weaker than what would follow from the

Hermes conjecture (in case of Jc 1). On the other hand, one is relieved

from the necessity of checking

all

Lie-brackets which are of order 2 with

respect to

g.

In fact, there are some examples in the engineering literature

(e.g. Lawden's spiral) where (1.7) is the only one among these brackets

which is easy to compute.

The results which have been outlined so far (one more will be added

in Section 3) can all be proved by a combination of methods, which could

be summarized as th e ana lytic approach to control theo ry. A consider

able portion of it has been developed in [1] and used the re to establish

higher order necessary conditions for singular arcs. The starting point

is the notion of control variations. These are parameter-dependent local

modifications of the control function and the trajectory around a given

reference solution. Later, in order to handle formal problems, one finds

it convenient not to relate all results with the reference solution but to

work directly with the right-han d side of th e system equation. The ana lytic

approach leads thereby straight into an ad-hoc-made algebraic theory of

non-linear systems, which appears at first glance to be a rather natural

generalization of linear system theory. The connection with the differential

geometric approach is less obvious; the comparison of these two basic

methods in control theory will play a major role in the lecture. At present

it is safe to say th a t the analytic techniques seem to be rathe r efficient

if one w ants torefineexisting results and, in particu lar, get rid of restr ict

ive assumptions concerning the system equation or the reference solution.

Furthermore, they seem to be well suited for a better exploitation of the

specificna ture of a given problem. This also can be of an advantage if one

has to compute from the right-hand side of equation (1.1) those quantities

which one has to know in order to apply the general results. An illustrativ e

example is th e econom ic version of th e generalized Olebsch-Legendre

condition which was given in [1] (Theorem 20.2).

The following two sections constitute a short account of the essential

definitions and facts on which th e analytic approach to non-linear systems

theory is based. Except for occasional remarks we will not enter into a dis

cussion of the proofs. All details as far as the y cannot be found in


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13 76 Section 14: H. W . Knobloch

exist ing l i terature will be giv en in a dissertat ion, which is presen tly

prepared a t the Depar tment of Mathemat ics in Wrzburg.

2 . The local cone of attainability

This section is devoted to a closer study of the sets

X

v

W e consider

a reference solution u(-),x(0,

uniformly with respect

to al l remaining variables occurring in the formula.

The definition of

jf

j will be ba sed on a m odification of w ha t was called

in [1] a control var iat ion conc entra ted at 2 = 2 . W e consider families

of

control.functions u(t;

r ,

X)

which depend on two rea l parameters r ,

X

and which are defined for t eR, 0 -p + A(t)p.

This is nothing else than the operation which can be applied in order

to generate the controllable subspace out of the columns of the matrix

B(t).

If

p(t)

is of the form

p

(t,x(t)), where

p

is a sufficiently smooth func

tion of2, ,then

p

= dpldt+p

x

f

and (2.7) can be written in the form

p->-dpldt-[f,p],

where the expressions appearing in this formula have to be evaluated

at x = x(t), t =t. Hence th e mapping (2.7) is in fact no thing else th an

th e application of the operator

r,

as introduced in [1].

We conclude this section by stating the two fundamental theorems

about jf| which were announced in the introduction. The proof of the

second one follows immediately, in view of (2.5), from Theorem 9.1 in [1].

THEOEEM

2.2.

If X%

= R

n

then we have localcontrollability along the

reference

solution and for

2 = 2 .

THEOEEM

2.3.

Let the

reference

solution be optima l. Then there exists

an adjoint statevector y(-) whichsatisfies the transversality conditions at

the

endpoints

and

the inequalities y(t)

T

p < 0

for

all

p eX

t

and all te[t

Q9

tj.
http://can.be/http://can.be/


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Nonlinear System s: Local Controllability an d Optimal Solutions 13 79

3. Second order elements

Special attention is deserved by those elements in

X

t which lead to

second order necessary conditions. This notion was explained in [1] (Sec

tion 1, p. 5); the definition can easily be modified so as to make sense if

one works withX

t

instead ofII

t

. We consider a family of control func

tions as specified in Section 2 an d we assume in addition th a t

u(t; x,X)

can be w ritten as

u(t) +

X

r

v(t

9

x,

X) 3.1)

where

r

issome positive

integer,v

(2;r , X)is supposed to vanish for2


12/12


We then restate here two of the second order conditions which have

been proved in [1] (Theorem 20.2 and 21.2) for a general system of the

form (1.1). The first is commonly known as the

generalized Olebsch-Legendre

condition.

It is a non-trivial result, regardless of whether

u

is scalar or

not, and we will restrict ourselves to the case of a scalar control. The sec

ond one is the prototype of an equality type necessary condition, and

hence is of interest only if

u

is not scalar as we have remarked in the intro

duction. Therefore we will here assume that

u

= (u

1

, u*)

T

is 2-dimensional.

As before we denote by

(u(-),x(-))

the given reference solution and

assume that

u(

) satisfies the condition

u(t) e

int

TJ

for all 2. We associate

with this solution a sequence of vectors

B\,

v = 0 , 1 , . . . , i = 1 , . . . , m

(

= dimension of

u),

which are recursively defined as follows

/ ( ; * ) := /( 0 , i = 1 , 2.

Conclusion : \B\, B

2

V

]

(t,x

(2))GX

f

if r +p

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