3
2.
If
we
wish
to
steer this system between
(x(O),
y(O),
z(O))
and
(x(l),
y(l),
z(l))
while minimizing
1]
=
10
1
u
2
+ v
2
dt
then
the
optimal
trajectories can be
computed
explicitly in
terms
of trigonometric functions.
3.
This
system is
part
of a hierarchy with
other
members of
the
hierarchy being obtained by considering systems with more
controls
and/or
a higher dimensional
state
space with higher
order nonlinear effects.
Because
the
example
to
be
discussed here requires it,
we
enlarge
on this last point.
There
is
a generalization of
the
:i:
= u;
iJ
=
v;
i =
xv
-
yu
system in which
the
number
of controls
is
n
rather
than
two.
It
takes
the
form
x=u
with x
and
u being m-vectors
and
Z being an m by m skew sym-
metric
matrix.
This problem was also discussed in reference
[12].
There
is
a second,
and
in all respects more
intricate
extension
that
involves a higher dimensional
state
space with controllability being
achieved by
virtue
of higher order Lie brackets. We display this
hierarchy for
the
case of two
inputs
and
indicate, in less detail,
the
m-input
situation.
The
number
of linearly independent binary forms (i.e. forms in-
volving two variables)
that
are homogeneous of degree p is, of course,
p +
1.
Thus
there
are 2(p +
1)
linearly independent expressions of
the
form
1]
=
¢(x,y)x+~(x,y)y
with
¢
and
~
being homogeneous of degree p. Since
there
are p + 2
linearly independent homogeneous forms of degree
p + 1 in x
and
y
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