nonlinear - local controllability
TRANSCRIPT
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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
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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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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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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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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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13 74 Section 14: H. W. Knobloch
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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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.
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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
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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