Geometry of Optimal Control for Control-Affine Systems
Motivated by the ubiquity of control-affine systems in optimal control theory, we investigate the geometry of point-affine control systems with metric structures in dimensions two and three. We compute local isometric invariants for point-affine distributions of constant type with metric structures...
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irk-123456789-1492062019-02-20T01:26:24Z Geometry of Optimal Control for Control-Affine Systems Clelland, J.N. Moseley, C.G. Wilkens, G.R. Motivated by the ubiquity of control-affine systems in optimal control theory, we investigate the geometry of point-affine control systems with metric structures in dimensions two and three. We compute local isometric invariants for point-affine distributions of constant type with metric structures for systems with 2 states and 1 control and systems with 3 states and 1 control, and use Pontryagin's maximum principle to find geodesic trajectories for homogeneous examples. Even in these low dimensions, the behavior of these systems is surprisingly rich and varied. 2013 Article Geometry of Optimal Control for Control-Affine Systems / J.N. Clelland, C.G. Moseley, G.R. Wilkens // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 6 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 58A30; 53C17; 58A15; 53C10 DOI: http://dx.doi.org/10.3842/SIGMA.2013.034 http://dspace.nbuv.gov.ua/handle/123456789/149206 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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Motivated by the ubiquity of control-affine systems in optimal control theory, we investigate the geometry of point-affine control systems with metric structures in dimensions two and three. We compute local isometric invariants for point-affine distributions of constant type with metric structures for systems with 2 states and 1 control and systems with 3 states and 1 control, and use Pontryagin's maximum principle to find geodesic trajectories for homogeneous examples. Even in these low dimensions, the behavior of these systems is surprisingly rich and varied. |
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Article |
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Clelland, J.N. Moseley, C.G. Wilkens, G.R. |
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Clelland, J.N. Moseley, C.G. Wilkens, G.R. Geometry of Optimal Control for Control-Affine Systems Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Clelland, J.N. Moseley, C.G. Wilkens, G.R. |
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Clelland, J.N. |
| title |
Geometry of Optimal Control for Control-Affine Systems |
| title_short |
Geometry of Optimal Control for Control-Affine Systems |
| title_full |
Geometry of Optimal Control for Control-Affine Systems |
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Geometry of Optimal Control for Control-Affine Systems |
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Geometry of Optimal Control for Control-Affine Systems |
| title_sort |
geometry of optimal control for control-affine systems |
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Інститут математики НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/149206 |
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Geometry of Optimal Control for Control-Affine Systems / J.N. Clelland, C.G. Moseley, G.R. Wilkens // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 6 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT clellandjn geometryofoptimalcontrolforcontrolaffinesystems AT moseleycg geometryofoptimalcontrolforcontrolaffinesystems AT wilkensgr geometryofoptimalcontrolforcontrolaffinesystems |
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2025-07-12T21:38:44Z |
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2025-07-12T21:38:44Z |
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1837478791522287616 |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 034, 31 pages
Geometry of Optimal Control
for Control-Affine Systems
Jeanne N. CLELLAND †, Christopher G. MOSELEY ‡ and George R. WILKENS §
† Department of Mathematics, 395 UCB, University of Colorado, Boulder, CO 80309-0395, USA
E-mail: Jeanne.Clelland@colorado.edu
‡ Department of Mathematics and Statistics, Calvin College, Grand Rapids, MI 49546, USA
E-mail: cgm3@calvin.edu
§ Department of Mathematics, University of Hawaii at Manoa,
2565 McCarthy Mall, Honolulu, HI 96822-2273, USA
E-mail: grw@math.hawaii.edu
Received June 07, 2012, in final form April 03, 2013; Published online April 17, 2013
http://dx.doi.org/10.3842/SIGMA.2013.034
Abstract. Motivated by the ubiquity of control-affine systems in optimal control theory,
we investigate the geometry of point-affine control systems with metric structures in dimen-
sions two and three. We compute local isometric invariants for point-affine distributions of
constant type with metric structures for systems with 2 states and 1 control and systems
with 3 states and 1 control, and use Pontryagin’s maximum principle to find geodesic tra-
jectories for homogeneous examples. Even in these low dimensions, the behavior of these
systems is surprisingly rich and varied.
Key words: affine distributions; optimal control theory; Cartan’s method of equivalence
2010 Mathematics Subject Classification: 58A30; 53C17; 58A15; 53C10
1 Introduction
In [1], we investigated the local structure of point-affine distributions. A rank-s point-affine
distribution on an n-dimensional manifold M is a sub-bundle F of the tangent bundle TM such
that, for each x ∈ M , the fiber Fx = TxM ∩ F is an s-dimensional affine subspace of TxM
that contains a distinguished point. In local coordinates, the points of F are parametrized
by s + 1 pointwise independent smooth vector fields v0(x), v1(x), . . . , vs(x) for which Fx =
v0(x) + span (v1(x), . . . , vs(x)) and v0(x) is the distinguished point in Fx.
Our interest in point-affine distributions is motivated by a family of ordinary differential
equations that occurs in control theory: the control-affine systems. A control system is a system
of underdetermined ODEs
ẋ = f(x, u),
where x ∈M and u takes values in an s-dimensional manifold U. The system is control-affine if
the right-hand side is affine linear in the control variables u, i.e., if the system locally has the
form
ẋ(t) = v0(x) +
s∑
i=1
vi(x)ui(t), (1.1)
where the controls u1, . . . , us appear linearly in the right hand side and v0, . . . , vs are s + 1
independent vector fields (see, e.g., [3]). Replacing v0, which is called the drift vector field, with
mailto:Jeanne.Clelland@colorado.edu
mailto:cgm3@calvin.edu
mailto:grw@math.hawaii.edu
http://dx.doi.org/10.3842/SIGMA.2013.034
2 J.N. Clelland, C.G. Moseley and G.R. Wilkens
a linear combination of v1, . . . , vs added to v0 would yield an equivalent system of differential
equations. In many instances, however, there is a distinguished null value for the controls (for
example, consider turning off all motors on a boat drifting downstream), and this null value
determines a distinguished drift vector field. In these instances, we always choose v0 to be the
distinguished drift vector field. Consequently, the null value for the controls will be
u1 = · · · = us = 0.
While the control-affine systems (1.1) may appear to be rather special, these systems are
ubiquitous. In fact, any control system whatsoever becomes control-affine after a single pro-
longation, so these systems actually encompass all control systems, at the cost of increasing the
number of state variables.
In [1] we studied local diffeomorphism invariants for these point-affine structures. A local
equivalence for two point-affine structures is a local diffeomorphism of M whose derivative maps
one distinguished drift vector field to the other, and maps one affine sub-bundle to the other
(see [1] for precise definitions). With this notion of local equivalence, we were able to determine
local normal forms for strictly affine, rank-1 point-affine structures of constant type when the
manifold M had dimension 2 or 3. In some cases the normal forms are parametrized by arbitrary
functions.
The current paper seeks to refine the previous results by adding a metric structure to the
point-affine structure. We do so by introducing a positive definite quadratic cost functional
Q : F → R. In local coordinates, where
w = v0(x) +
s∑
i=1
vi(x)ui ∈ Fx,
we will define
Qx(w) =
∑
gij(x)uiuj ,
where the matrix (gij(x)) is positive definite and the components are smooth functions of x.
This is a natural extension of the well-studied notion of a sub-Riemannian metric on a linear
distribution, which represents a quadratic cost functional for a driftless system (see, e.g., [4, 5, 6]).
With the added metric structure, we refine our notion of local point-affine equivalence to that
of a local point-affine isometry. A local point-affine isometry is a local point-affine equivalence
that additionally preserves the quadratic cost functional.
Let γ(t) = x(t) be a trajectory for (1.1). The added metric structure allows us to assign the
following energy cost functional to γ(t):
E(γ) =
1
2
∫
γ
Qx(t)
(
ẋ(t)
)
dt. (1.2)
Naturally associated to (1.2) is the optimal control problem of finding trajectories of (1.1) that
minimize (1.2). We will use Pontryagin’s maximum principle to find an ODE system on T ∗M
with the property that any minimal cost trajectory for (1.1) must be the projection of some
solution for the ODE system on T ∗M .
In this paper we shall only consider homogeneous examples, i.e., examples that admit a sym-
metry group which acts transitively on M . We shall use the normal forms from [1] as starting
points, adding a homogeneous metric structure to the point-affine structure in each case. Even in
these low-dimensional cases, the analysis can be quite involved; we will see that these structures
exhibit surprisingly rich and varied behavior.
Geometry of Optimal Control for Control-Affine Systems 3
2 Normal forms for homogeneous cases
We begin by identifying the homogeneous examples of the point-affine systems described in [1]
where possible, and then we describe the homogeneous metric structures on these systems. In
some cases, the metric structure must be added before the homogeneous examples can be iden-
tified. Recall that the assumption of homogeneity is equivalent to the condition that all structure
functions T ijk appearing in the structure equations for a canonical coframing are constants (see [2]
for details).
2.1 Two states and one control
In [1], we found two local normal forms under point-affine equivalence.
Case 1.1. F = ∂
∂x1
+ span
(
∂
∂x2
)
. The framing
v1 =
∂
∂x1
, v2 =
1
λ
∂
∂x2
(well-defined up to scaling in v2) has dual coframing
η1 = dx1, η2 = λdx2, (2.1)
with structure equations
dη1 = 0, dη2 ≡ 0 mod η2.
Because the method of equivalence does not lead to a completely determined canonical cofra-
ming, it is not clear from these structure equations whether this example is homogeneous as
a point-affine distribution.
Fortunately, this ambiguity is resolved when we add a metric function to the point-affine
structure. This amounts to a choice of function G(x) > 0 for which the quadratic cost functional
is given by
Q
(
∂
∂x1
+ u
∂
∂x2
)
=
1
2
G(x)u2. (2.2)
For the point-affine structure, the frame vector v2 is only well-defined up to a scale factor;
however, when we impose a metric structure (2.2), we can choose v2 canonically (up to sign) by
requiring that it be a unit vector for the metric. This choice leads to a canonical framing
v1 =
∂
∂x1
, v2 =
1√
G(x)
∂
∂x2
,
with corresponding canonical coframing
η1 = dx1, η2 =
√
G(x) dx2.
The structure equations for this refined coframing are
dη1 = 0, dη2 =
Gx1
2G
η1 ∧ η2,
and so the structure is homogeneous if and only if
Gx1
2G is equal to a constant c1. This condition
implies that
G
(
x1, x2
)
= G0
(
x2
)
e2c1x
1
for some function G0
(
x2
)
.
4 J.N. Clelland, C.G. Moseley and G.R. Wilkens
The local coordinates in the coframing (2.1) are only determined up to transformations of
the form
x1 = x̃1 + a, x2 = φ
(
x̃2
)
, (2.3)
and under this transformation we have
G̃0
(
x̃2
)
= e2c1a
((
φ′
(
x̃2
))2
G0
(
φ
(
x̃2
))
.
Therefore, we can apply a transformation of the form (2.3) to arrange that G̃0
(
x̃2
)
= 1, and
hence G̃ = e2c1x̃
1
. Moreover, coordinates for which G has this form are uniquely determined up
to a transformation of the form
x1 = x̃1 + a, x2 = e−c1ax̃2 + b.
To summarize: the homogeneous metrics in this case are given by quadratic functionals of
the form
Q
(
∂
∂x1
+ u
∂
∂x2
)
=
1
2
e2c1x
1
u2
for some constant c1, with corresponding canonical coframings
η1 = dx1, η2 = ec1x
1
dx2.
Case 1.2. F = x2
(
∂
∂x1
+ J ∂
∂x2
)
+ span
(
∂
∂x2
)
. We found a canonical framing
v1 = x2
(
∂
∂x1
+ J
∂
∂x2
)
, v2 = x2
∂
∂x2
, (2.4)
with dual coframing
η1 =
1
x2
dx1, η2 =
1
x2
(
dx2 − Jdx1
)
, (2.5)
and structure equations
dη1 = η1 ∧ η2, dη2 = T 2
12η
1 ∧ η2,
where
T 2
12 = x2
∂J
∂x2
− J. (2.6)
The structure is homogeneous if and only if T 1
12 is equal to a constant −j0. According to
equation (2.6), this is the case if and only if
J = x2J1
(
x1
)
+ j0 (2.7)
for some function J1
(
x1
)
.
The local coordinates in the coframing (2.5) are only determined up to transformations of
the form
x1 = φ
(
x̃1
)
, x2 = x̃2φ′
(
x̃1
)
, (2.8)
Geometry of Optimal Control for Control-Affine Systems 5
and under this transformation we have
J̃
(
x̃1, x̃2
)
= J
(
φ
(
x̃1
)
, x̃2φ′
(
x̃1
))
− x̃2
φ′′
(
x̃1
)
φ′
(
x̃1
) .
In the homogeneous case (2.7), this implies that
J̃1
(
x̃1
)
= φ′
(
x̃1
)
J1
(
φ
(
x̃1
))
−
φ′′
(
x̃1
)
φ′
(
x̃1
) .
Therefore, we can apply a transformation of the form (2.8) to arrange that J̃1
(
x̃1
)
= 0, and
hence J̃ = j0. Moreover, coordinates for which J is constant are uniquely determined up to an
affine transformation
x1 = ax̃1 + b, x2 = ax̃2.
Now suppose that a metric on the point-affine structure is given by
Q (v1 + uv2) = Q
(
x2
(
∂
∂x1
+ j0
∂
∂x2
)
+ u
(
x2
∂
∂x2
))
=
1
2
G(x)u2. (2.9)
This case differs from the previous case in that the control vector field v2 is already canonically
defined by the point-affine structure prior to the introduction of a metric. Therefore, in order
that the metric (2.9) be homogeneous, the unit control vector field
1√
G(x)
v2
must be a constant scalar multiple of v2. Thus we must have G(x) = g0 for some positive
constant g0, and the homogeneous metrics in this case are given by quadratic functionals of the
form
Q(v1 + uv2) =
1
2
g0u
2
for some positive constant g0, where v1, v2 are the canonical frame vectors (2.4).
2.2 Three states and one control
In [1], we found three nontrivial local normal forms under point-affine equivalence.
Remark 2.1. This classification assumes that the point-affine distribution is either bracket-
generating or almost bracket-generating; otherwise the 3-manifold M can locally be foliated by
a 1-parameter family of 2-dimensional submanifolds such that every trajectory of F is contained
in a single leaf of the foliation.
Case 2.1. F =
(
∂
∂x1
+ x3 ∂
∂x2
+ J ∂
∂x3
)
+ span
(
∂
∂x3
)
. The framing
v1 =
∂
∂x1
+ x3
∂
∂x2
+ J
∂
∂x3
, v2 =
∂
∂x3
, v3 = −[v1, v2] =
∂
∂x2
+ Jx3
∂
∂x3
(well-defined up to dilation in the (v2, v3)-plane) has dual coframing
η1 = dx1, η2 = dx3 − J dx1 − Jx3
(
dx2 − x3 dx1
)
, η3 = dx2 − x3 dx1,
6 J.N. Clelland, C.G. Moseley and G.R. Wilkens
with structure equations
dη1 = 0,
dη2 ≡ T 2
13η
1 ∧ η3 mod η2,
dη3 ≡ η1 ∧ η2 mod η3.
As in Case 1.1, the method of equivalence does not lead to a completely determined coframing,
so it is not clear from these structure equations whether this example is homogeneous as a point-
affine distribution.
So, suppose that a metric on the point-affine structure is given by
Q
((
∂
∂x1
+ x3
∂
∂x2
+ J
∂
∂x3
)
+ u
∂
∂x3
)
=
1
2
G(x)u2. (2.10)
The addition of the metric (2.10) allows us to choose a canonical framing (up to sign) by requi-
ring v2 to be a unit vector for the metric, i.e.,
v2 =
1√
G(x)
∂
∂x3
,
and setting
v3 = −[v1, v2].
The canonical coframing associated to this framing is given by
η1 = dx1, η2 ≡
√
G(x)
(
dx3 − J dx1
)
mod η3, η3 =
√
G(x)
(
dx2 − x3 dx1
)
. (2.11)
In order to identify the homogeneous examples, we consider the structure equations for the
coframing (2.11), taking into account the fact that local coordinates for which the coframing
takes the form (2.11) are determined only up to transformations of the form
x1 = x̃1 + a, x2 = φ
(
x̃1, x̃2
)
, x3 = φx̃1
(
x̃1, x̃2
)
+ x̃3φx̃2
(
x̃1, x̃2
)
, (2.12)
with φx̃2 6= 0. Under such a transformation we have√
G̃
(
x̃1, x̃2, x̃3
)
=
√
G
(
x1, x2, x3
)
φx̃2 , (2.13)
J̃
(
x̃1, x̃2, x̃3
)
=
1
φx̃2
(
J
(
x1, x2, x3
)
− φx̃2x̃2
(
x̃3
)2 − 2φx̃1x̃2 x̃
3 − φx̃1x̃1
)
, (2.14)
with x1, x2, x3 as in (2.12).
First consider the structure equation for dη3. A computation shows that
dη3 ≡ Gx3
2G3/2
η2 ∧ η3 mod η1.
Therefore, homogeneity implies that
Gx3
2G3/2 must be equal to a constant −c1. The remaining
analysis varies considerably depending on whether c1 is zero or nonzero.
Case 2.1.1. First suppose that c1 = 0. Then Gx3 = 0, and so
G
(
x1, x2, x3
)
= G0
(
x1, x2
)
for some function G0
(
x1, x2
)
. According to (2.13), by a local change of coordinates of the
form (2.12) with φ a solution of the PDE
φx̃2
(
x̃1, x̃2
)
=
1
G0
(
x̃1, φ
(
x̃1, x̃2
)) ,
Geometry of Optimal Control for Control-Affine Systems 7
we can arrange that G̃0
(
x̃1, x̃2
)
= 1. This condition is preserved by transformations of the
form (2.12) with
φ
(
x̃1, x̃2
)
= x̃2 + φ0
(
x̃1
)
. (2.15)
With the assumption that G
(
x1, x2, x3
)
= 1, the equation for dη3 reduces to
dη3 = η1 ∧ η2 + Jx3η
1 ∧ η3.
Therefore, Jx3 must be equal to a constant c3, and so
J
(
x1, x2, x3
)
= c3x
3 + J0
(
x1, x2
)
for some function J0
(
x1, x2
)
. Now the equation for dη2 becomes
dη2 = (J0)x2 η
1 ∧ η3.
Therefore, (J0)x2 must be equal to a constant c2, and so
J0
(
x1, x2
)
= c2x
2 + J1
(
x1
)
for some function J1
(
x1
)
. With φ as in (2.15) and
J
(
x1, x2, x3
)
= c2x
2 + c3x
3 + J1
(
x1
)
,
equation (2.14) reduces to
J̃1
(
x̃1
)
= J1
(
x̃1 + a
)
−
(
φ′′0
(
x̃1
)
− c3φ′0
(
x̃1
)
− c2φ0
(
x̃1
))
.
Therefore, we can choose local coordinates to arrange that J̃1
(
x̃1
)
= 0.
To summarize, we have constructed local coordinates for which
G
(
x1, x2, x3
)
= 1, J
(
x1, x2, x3
)
= c2x
2 + c3x
3.
These coordinates are determined up to transformations of the form
x1 = x̃1 + a, x2 = x̃2 + φ0
(
x̃1
)
, x3 = x̃3 + φ′0
(
x̃1
)
,
where φ0
(
x̃1
)
is a solution of the ODE
φ′′0
(
x̃1
)
− c3φ′0
(
x̃1
)
− c2φ0
(
x̃1
)
= 0.
Case 2.1.2. Now suppose that c1 6= 0. Then
G
(
x1, x2, x3
)
=
1(
c1x3 +G0
(
x1, x2
))2
for some function G0
(
x1, x2
)
. According to (2.13), by a local change of coordinates of the
form (2.12) with φ a solution of the PDE
φx1
(
x̃1, x̃2
)
=
1
c1
G0
(
x̃1, φ
(
x̃1, x̃2
))
,
we can arrange that G̃0
(
x̃1, x̃2
)
= 0. This condition is preserved by transformations of the
form (2.12) with
φ
(
x̃1, x̃2
)
= φ0
(
x̃2
)
. (2.16)
8 J.N. Clelland, C.G. Moseley and G.R. Wilkens
With the assumption that G
(
x1, x2, x3
)
= 1
(c1x3)2
, the equation for dη3 reduces to
dη3 = η1 ∧ η2 − (2J − x3Jx3)
x3
η1 ∧ η3 − c1η2 ∧ η3.
Therefore,
(2J−x3Jx3 )
x3
must be equal to a constant c3, and so
J
(
x1, x2, x3
)
= c3x
3 + J0
(
x1, x2
)(
x3
)2
for some function J0
(
x1, x2
)
. Now the equation for dη2 becomes
dη2 = −x3(J0)x1 η1 ∧ η3.
The quantity −x3(J0)x1 can only be constant if (J0)x1 = 0; therefore, we must have
J0
(
x1, x2
)
= J1
(
x2
)
for some function J1
(
x2
)
. With φ as in (2.16) and
J
(
x1, x2, x3
)
= c3x
3 + J1
(
x2
)(
x3
)2
,
equation (2.14) reduces to
J̃1
(
x̃2
)
= J1
(
φ0
(
x̃2
))
φ′0
(
x̃2
)
−
φ′′0
(
x̃2
)
φ′0
(
x̃2
) .
Therefore, we can choose local coordinates to arrange that J̃1
(
x̃2
)
= 0.
To summarize, we have constructed local coordinates for which
G
(
x1, x2, x3
)
=
1(
c1x3
)2 , J
(
x1, x2, x3
)
= c3x
3.
These coordinates are determined up to transformations of the form
x1 = x̃1 + a, x2 = bx̃2 + c, x3 = bx̃3 + c.
Case 2.2. F =
(
x2 ∂
∂x1
+ x3 ∂
∂x2
+ J
(
x2 ∂
∂x3
))
+ span
(
∂
∂x3
)
. We found a canonical framing
v1 = x2
∂
∂x1
+ x3
∂
∂x2
+ J
(
x2
∂
∂x3
)
,
v2 = x2
∂
∂x3
,
v3 = −[v1, v2] = x2
∂
∂x2
+
((
x2
)2
Jx3 − x3
) ∂
∂x3
, (2.17)
with dual coframing
η1 =
1
x2
dx1,
η2 =
1
x2
dx3 − 1
x2
J dx1 −
(
Jx3 −
x3(
x2
)2
)(
dx2 − x3
x2
dx1
)
,
η3 =
1
x2
dx2 − x3(
x2
)2dx1, (2.18)
Geometry of Optimal Control for Control-Affine Systems 9
and structure equations
dη1 = η1 ∧ η3,
dη2 = T 2
13η
1 ∧ η3 + T 2
23η
2 ∧ η3,
dη3 = η1 ∧ η2 + T 3
13η
1 ∧ η3. (2.19)
The local coordinates in the coframing (2.18) are only determined up to transformations of
the form
x1 = φ
(
x̃1
)
, x2 = φ′
(
x̃1
)
x̃2, x3 = φ′
(
x̃1
)
x̃3 + φ′′
(
x̃1
)(
x̃2
)2
, (2.20)
with φ′
(
x̃1
)
6= 0. Under such a transformation we have
J̃
(
x̃1, x̃2, x̃3
)
= J
(
x1, x2, x3
)
− 1
φ′
(
x̃1
) (φ′′′(x̃1)(x̃2)2 + 3φ′′
(
x̃1
)
x̃3
)
, (2.21)
with x1, x2, x3 as in (2.20).
First consider the structure equation for η3. Substituting the expressions (2.18) into the
structure equation (2.19) for dη3 shows that
T 2
12 = x2Jx3 − 3
x3
x2
.
Homogeneity implies that T 2
12 must be equal to a constant a, from which it follows that
J
(
x1, x2, x3
)
=
3
2
(
x3
x2
)2
+ a
x3
x2
+ J0
(
x1, x2
)
for some function J0
(
x1, x2
)
. Now the equation for dη2 yields
T 2
13 = x2(J0)x2 − 2J0 − a
x3
x2
,
and homogeneity implies that T 2
13 must be constant. The quantity
(
x2(J0)x2 − 2J0 − ax
3
x2
)
can
only be constant if a = 0; therefore, we must have a = 0 and
x2(J0)x2 − 2J0 = −2c1
for some constant c1. Therefore,
J0
(
x1, x2
)
= c1 + J1
(
x1
)(
x2
)2
for some function J1
(
x1
)
, and
J
(
x1, x2, x3
)
=
3
2
(
x3
x2
)2
+ c1 + J1
(
x1
)(
x2
)2
.
With φ as in (2.20) and J as above, equation (2.21) reduces to
J̃1
(
x̃1
)
= φ′
(
x̃1
)2
J1
(
φ
(
x̃1
))
−
φ′′′
(
x̃1
)
φ′
(
x̃1
) +
3
2
φ′′
(
x̃1
)(
φ′
(
x̃1
))2 .
10 J.N. Clelland, C.G. Moseley and G.R. Wilkens
Therefore, we can choose local coordinates to arrange that J̃1
(
x̃1
)
= 0. This condition is pre-
served by transformations of the form (2.20) with
φ′′′
(
x̃1
)
φ′
(
x̃1
) − 3
2
φ′′
(
x̃1
)(
φ′
(
x̃1
))2 = 0.
This implies that φ is a linear fractional transformation, i.e.,
φ
(
x̃1
)
=
ax̃1 + b
cx̃1 + d
.
Now suppose that a metric on the point-affine structure is given by
Q (v1 + uv2) =
1
2
G(x)u2. (2.22)
As in Case 1.2, the control vector field v2 is already canonically defined by the point-affine
structure prior to the introduction of a metric. Therefore, in order that the metric (2.22) be
homogeneous, the unit control vector field
1√
G(x)
v2
must be a constant scalar multiple of v2. Thus we must have G(x) = g0 for some positive cons-
tant g0, and the homogeneous metrics in this case are given by quadratic functionals of the form
Q(v1 + uv2) =
1
2
g0u
2
for some positive constant g0, where v1, v2, v3 are the canonical frame vectors (2.17).
To summarize, we have constructed local coordinates for which
G
(
x1, x2, x3
)
= g0, J
(
x1, x2, x3
)
=
3
2
(
x3
x2
)2
+ c1.
These coordinates are determined up to transformations of the form
x1 =
ax̃1 + b
cx̃1 + d
, x2 =
ad− bc
(cx̃1 + d)2
x̃2, x3 =
ad− bc
(cx̃1 + d)2
x̃3 − 2c(ad− bc)
(cx̃1 + d)3
x̃2.
Case 2.3.
F =
(
∂
∂x1
+ J
(
x3
∂
∂x1
+
∂
∂x2
+H
∂
∂x3
))
+ span
(
x3
∂
∂x1
+
∂
∂x2
+H
∂
∂x3
)
,
where ∂H
∂x1
6= 0. We found a canonical framing
v1 =
∂
∂x1
+ J
(
x3
∂
∂x1
+
∂
∂x2
+H
∂
∂x3
)
,
v2 =
ε√
εHx1
(
x3
∂
∂x1
+
∂
∂x2
+H
∂
∂x3
)
,
v3 = −[v1, v2],
Geometry of Optimal Control for Control-Affine Systems 11
where ε = ±1 = sgn(Hx1), with dual coframing
η1 = dx1 − x3 dx2,
η2 ≡ ε
√
εHx1
(
dx2 − J
(
dx1 − x3 dx2
))
mod η3,
η3 =
1√
εHx1
(
H dx2 − dx3
)
, (2.23)
and structure equations
dη1 = T 1
13η
1 ∧ η3 + T 1
23η
2 ∧ η3,
dη2 = T 2
13η
1 ∧ η3 + T 2
23η
2 ∧ η3,
dη3 = η1 ∧ η2 + T 3
13η
1 ∧ η3 + T 3
23η
2 ∧ η3. (2.24)
The identification of homogeneous examples is considerably more complicated than in the pre-
vious cases. We refer the reader to Appendix A for the details. We find that the homogeneous
examples in this case are all locally equivalent to one of the following:
• J
(
x1, x2, x3
)
= c1, H
(
x1, x2, x3
)
= ε
(
x1 + c2x
3
)
for some constants c1, c2;
• J
(
x1, x2, x3
)
= c1 cos
(
c3x
1
)/√
εc3
(
c3
(
x3
)2
+ c4
)
,
H
(
x1, x2, x3
)
=
(
c3
(
x3
)2
+ c4
)
tan
(
c3x
1
)
+ F20
(
x2
)√
c3
(
x3
)2
+ c4
for some constants c1, c3, c4 with c3 6= 0, and some arbitrary function F20
(
x2
)
;
• J
(
x1, x2, x3
)
= c1 cosh
(
c3x
1
)/√
εc3(c3
(
x3
)2 − c4),
H
(
x1, x2, x3
)
=
(
−c3
(
x3
)2
+ c4
)
tanh
(
c3x
1
)
+ F20
(
x2
)√
c3
(
x3
)2 − c4
for some constants c1, c3, c4 with c3 6= 0, and some arbitrary function F20
(
x2
)
.
Now suppose that a metric on the point-affine structure is given by
Q (v1 + uv2) =
1
2
G(x)u2.
As in the previous case, since the control vector field v2 is already canonically defined by the
point-affine structure prior to the introduction of a metric, we must have G(x) = g0 for some
positive constant g0.
The results of this section are encapsulated in the following two theorems:
Theorem 2.2. Let F be a rank 1 strictly affine point-affine distribution of constant type on
a 2-dimensional manifold M , equipped with a positive definite quadratic cost functional Q. If
the structure (F, Q) is homogeneous, then (F, Q) is locally point-affine equivalent to
F = v1 + span (v2), Q(v1 + uv2) =
1
2
G(x)u2,
where the triple (v1, v2, G(x)) is one of the following:
(1.1) v1 =
∂
∂x1
, v2 =
∂
∂x2
, G(x) = e2c1x
1
;
(1.2) v1 = x2
(
∂
∂x1
+ j0
∂
∂x2
)
, v2 = x2
∂
∂x2
, G(x) = g0.
12 J.N. Clelland, C.G. Moseley and G.R. Wilkens
Theorem 2.3. Let F be a rank 1, strictly affine, bracket-generating or almost bracket-generating
point-affine distribution of constant type on a 3-dimensional manifold M , equipped with a positive
definite quadratic cost functional Q. If the structure (F, Q) is homogeneous, then (F, Q) is locally
point-affine equivalent to
F = v1 + span (v2), Q(v1 + uv2) =
1
2
G(x)u2,
where the triple (v1, v2, G(x)) is one of the following:
(2.1.1) v1 =
∂
∂x1
+ x3
∂
∂x2
+
(
c2x
2 + c3x
3
) ∂
∂x3
, v2 =
∂
∂x3
, G(x) = 1;
(2.1.2) v1 =
∂
∂x1
+ x3
∂
∂x2
+ c3x
3 ∂
∂x3
, v2 =
∂
∂x3
, G(x) =
1(
c1x3
)2 ;
(2.2) v1 = x2
∂
∂x1
+ x3
∂
∂x2
+
(
3
2
(
x3
x2
)2
+ c1
)(
x2
∂
∂x3
)
,
v2 = x2
∂
∂x3
, G(x) = g0;
(2.3.1) v1 =
∂
∂x1
+ c1
(
x3
∂
∂x1
+
∂
∂x2
+ ε
(
x1 + c2x
3
) ∂
∂x3
)
,
v2 = ε
(
x3
∂
∂x1
+
∂
∂x2
+ ε
(
x1 + c2x
3
) ∂
∂x3
)
, G(x) = g0;
(2.3.2) v1 =
∂
∂x1
+
c1 cos(c3x
1)√
εc3(c3(x3)2 + c4)
(
x3
∂
∂x1
+
∂
∂x2
+H
∂
∂x3
)
,
v2 = ε
(
x3
∂
∂x1
+
∂
∂x2
+H
∂
∂x3
)
, G(x) = g0,
where H =
((
c3
(
x3
)2
+ c4
)
tan
(
c3x
1
)
+ F20
(
x2
)√
c3
(
x3
)2
+ c4
)
;
(2.3.3) v1 =
∂
∂x1
+
c1 cosh(c3x
1)√
εc3(c3(x3)2 − c4)
(
x3
∂
∂x1
+
∂
∂x2
+H
∂
∂x3
)
,
v2 = ε
(
x3
∂
∂x1
+
∂
∂x2
+H
∂
∂x3
)
, G(x) = g0,
where H =
((
− c3
(
x3
)2
+ c4
)
tanh
(
c3x
1
)
+ F20
(
x2
)√
c3
(
x3
)2 − c4) .
3 Optimal control problem for homogeneous metrics
3.1 Two states and one control
In this section we use Pontryagin’s maximum principle to compute optimal trajectories for each
of the homogeneous metrics of Theorem 2.2.
Case 1.1. This point-affine distribution corresponds to the control system
ẋ1 = 1, ẋ2 = u, (3.1)
with cost functional
Q(ẋ) =
1
2
e2c1x
1
u2.
Geometry of Optimal Control for Control-Affine Systems 13
Consider the problem of computing optimal trajectories for (3.1). The Hamiltonian for the
energy functional (1.2) is
H = p1ẋ
1 + p2ẋ
2 −Q(ẋ) = p1 + p2u−
1
2
e2c1x
1
u2.
By Pontryagin’s maximum principle, a necessary condition for optimal trajectories is that the
control function u(t) is chosen so as to maximize H. Since u is unrestricted and 1
2e
2c1x1 > 0,
maxuH occurs when
0 =
∂H
∂u
= p2 − e2c1x
1
u,
that is, when
u = p2e
−2c1x1 .
So along an optimal trajectory, we have
H = p1 + (p2)
2e−2c1x
1 − 1
2
(p2)
2e−2c1x
1
= p1 +
1
2
(p2)
2e−2c1x
1
.
Moreover, H is constant along trajectories, and so we have
p1 +
1
2
(p2)
2e−2c1x
1
= k.
Hamilton’s equations
ẋ =
∂H
∂p
, ṗ = −∂H
∂x
take the form
ẋ1 = 1, ṗ1 = c1(p2)
2e−2c1x
1
,
ẋ2 = p2e
−2c1x1 , ṗ2 = 0. (3.2)
The equation for ṗ2 in (3.2) implies that p2 is constant; say, p2 = c2. Then optimal trajectories
are solutions of the system
ẋ1 = 1, ẋ2 = c2e
−2c1x1 .
This system can be integrated explicitly:
• If c1 = 0, then the solutions are
x1 = t, x2 = c2t+ c3.
These solutions correspond to the family of curves
x2 = c2x
1 + c3
in the
(
x1, x2
)
-plane. Thus, the set of critical curves consists of all non-vertical straight
lines in the
(
x1, x2
)
plane, oriented in the direction of increasing x1.
14 J.N. Clelland, C.G. Moseley and G.R. Wilkens
• If c1 6= 0, then the solutions are
x1 = t, x2 = − 1
2c1
c2e
−2c1t.
These solutions correspond to the family of curves
x2 = − 1
2c1
c2e
−2c1x1
in the
(
x1, x2
)
-plane. Thus, the set of critical curves consists of a family of exponential
curves in the
(
x1, x2
)
plane, oriented in the direction of increasing x1.
Case 1.2. This point-affine distribution corresponds to the control system
ẋ1 = x2, ẋ2 = x2j0 + x2u,
with cost functional
Q(ẋ) =
1
2
g0u
2.
Pontryagin’s maximum principle leads to the Hamiltonian
H = p1x
2 + p2x
2j0 +
1
2g0
(
p2x
2
)2
along an optimal trajectory, and Hamilton’s equations take the form
ẋ1 = x2, ṗ1 = 0,
ẋ2 = x2j0 +
p2
(
x2
)2
g0
, ṗ2 = −p1 − p2j0 −
(p2)
2x2
g0
. (3.3)
It is straightforward to show that the three functions
I1 = H = p1x
2 + p2x
2j0 +
1
2g0
(
p2x
2
)2
, I2 = p1, I3 = p1x
1 + p2x
2
are first integrals for this system. This observation alone would in principle allow us to construct
unparametrized solution curves for the system. But in fact, we can solve this system fully, as
follows.
The equation for ṗ1 in (3.3) implies that p1 is constant; say, p1 = c1. Now it is straightforward
to show that
d
dt
(
p2x
2
)
+ c1x
2 = 0. (3.4)
If c1 = 0, then (3.4) implies that p2x
2 is equal to a constant k2, and so
ẋ2 = x2
(
j0 +
k2
g0
)
= c2x
2.
There are two subcases, depending on the value of c2.
• If c2 = 0, then x2 = c3, and since ẋ1 = x2, we have x1 = c3t + c4. These solutions
correspond to the family of curves x2 = c3 in the
(
x1, x2
)
-plane. These curves are all
horizontal lines, oriented in the direction of increasing x1 when x2 > 0 and decreasing x1
when x2 < 0.
Geometry of Optimal Control for Control-Affine Systems 15
• If c2 6= 0, then x2 = c3e
c2t, and since ẋ1 = x2, we have x1 = c3
c2
ec2t + c4. These solutions
correspond to the family of curves x2 = c2
(
x1 − c4
)
in the
(
x1, x2
)
-plane. These curves
are all non-vertical, non-horizontal lines, oriented in the direction of increasing x1 when
x2 > 0 and decreasing x1 when x2 < 0.
On the other hand, if c1 6= 0, then it is straightforward to show that
d2
dt2
(
p2x
2
)
=
d
dt
(
p2x
2
)(
j0 +
p2x
2
g0
)
.
Integrating this equation once gives
d
dt
(
p2x
2
)
= j0
(
p2x
2
)
+
(
p2x
2
)2
2g0
+ c2. (3.5)
There are three subcases, depending on the value of k = g0(j
2
0g0 − 2c2).
• If k = 0, then the solution to (3.5) is
p2x
2 = −g0(2 + j0(t+ c3))
t+ c3
,
and from equation (3.4),
x2 = − 1
c1
d
dt
(
p2x
2
)
= − 2g0
c1(t+ c3)2
.
Then since ẋ1 = x2 = − 1
c1
d
dt
(
p2x
2
)
, we have
x1 = − 1
c1
(
p2x
2
)
+ c4 =
g0(2 + j0(t+ c3))
c1(t+ c3)
+ c4.
These solutions correspond to the family of curves
x2 = − 1
2c1g0
(
c1x
1 − (j0g0 + c1c4)
)2
in the
(
x1, x2
)
-plane. These curves are all parabolas with vertex lying on the x1-axis.
Since we must have x2 6= 0, the set of critical curves consists of all branches of parabolas
with vertex on the x2-axis, oriented in the direction of increasing x1 when x2 > 0 and
decreasing x1 when x2 < 0.
• If k > 0, then the solution to (3.5) is
p2x
2 = −
√
k tanh
(√
k
2g0
(t+ c3)
)
− j0g0,
and from equation (3.4),
x2 = − 1
c1
d
dt
(
p2x
2
)
=
k
2c1g0
sech2
(√
k
2g0
(t+ c3)
)
.
Then since ẋ1 = x2 = − 1
c1
d
dt
(
p2x
2
)
, we have
x1 = − 1
c1
(
p2x
2
)
+ c4 =
1
c1
(
√
k tanh
(√
k
2g0
(t+ c3)
)
+ j0g0
)
+ c4.
16 J.N. Clelland, C.G. Moseley and G.R. Wilkens
These solutions correspond to the family of curves
x2 = − 1
2c1g0
[(
c1x
1 − (j0g0 + c1c4)
)2 − k]
in the
(
x1, x2
)
-plane. These curves are all parabolas opening towards the x1-axis. Thus the
set of critical curves consists of parabolic arcs opening towards the x1-axis, approaching
the axis as t → ±∞, and oriented in the direction of increasing x1 when x2 > 0 and
decreasing x1 when x2 < 0.
• If k < 0, then the solution to (3.5) is
p2x
2 =
√
−k tan
(√
−k
2g0
(t+ c3)
)
− j0g0,
and from equation (3.4),
x2 = − 1
c1
d
dt
(
p2x
2
)
=
k
2c1g0
sec2
(√
−k
2g0
(t+ c3)
)
.
Then since ẋ1 = x2 = − 1
c1
d
dt
(
p2x
2
)
, we have
x1 = − 1
c1
(
p2x
2
)
+ c4 = − 1
c1
(√
−k tan
(√
−k
2g0
(t+ c3)
)
− j0g0
)
+ c4.
These solutions correspond to the family of curves
x2 = − 1
2c1g0
[(
c1x
1 − (j0g0 + c1c4)
)2 − k]
in the
(
x1, x2
)
-plane. These curves are all parabolas opening away from the x1-axis. Thus
the set of critical curves consists of parabolic arcs opening away from the x1-axis, becoming
unbounded in finite time, and oriented in the direction of increasing x1 when x2 > 0 and
decreasing x1 when x2 < 0.
3.2 Three states and one control
In this section we use Pontryagin’s maximum principle to compute optimal trajectories for each
of the homogeneous metrics of Theorem 2.3.
Case 2.1.1. This point-affine distribution corresponds to the control system
ẋ1 = 1, ẋ2 = x3, ẋ3 = c2x
2 + c3x
3 + u,
with cost functional
Q(ẋ) =
1
2
u2.
The Hamiltonian for the energy functional (1.2) is
H = p1ẋ
1 + p2ẋ
2 + p3ẋ
3 −Q(ẋ) = p1 + p2x
3 + p3
(
c2x
2 + c3x
3 + u
)
− 1
2
u2.
Pontryagin’s maximum principle leads to the Hamiltonian
H = p1 + p2x
3 + p3
(
c2x
2 + c3x
3
)
+
1
2
(p3)
2
Geometry of Optimal Control for Control-Affine Systems 17
Figure 1.
along an optimal trajectory, and Hamilton’s equations take the form
ẋ1 = 1, ṗ1 = 0,
ẋ2 = x3, ṗ2 = −c2p3,
ẋ3 = c2x
2 + c3x
3 + p3, ṗ3 = −p2 − c3p3. (3.6)
The equations for ṗ2 and ṗ3 in (3.6) can be written as
p̈2 + c3ṗ2 − c2p2 = 0,
and the function p3 = − 1
c2
ṗ2 satisfies this same ODE. Then the equations for ẋ2 and ẋ3 can be
written as
ẍ2 − c3ẋ2 − c2x2 = p3(t),
where p3(t) is an arbitrary solution of the ODE
p̈3 + c3ṗ3 − c2p3 = 0.
Therefore, x2(t) is an arbitrary solution of the 4th-order ODE(
d2
dt2
+ c3
d
dt
− c2
)(
d2
dt2
− c3
d
dt
− c2
)
x2(t) = 0,
and for any such x2(t), we have
x1(t) = t+ t0, x3(t) = ẋ2(t).
A sample optimal trajectory is shown in Fig. 1.
Case 2.1.2. This point-affine distribution corresponds to the control system
ẋ1 = 1, ẋ2 = x3, ẋ3 = c3x
3 + u,
with cost functional
Q(ẋ) =
1
2
(
c1x3
)2u2.
18 J.N. Clelland, C.G. Moseley and G.R. Wilkens
Figure 2. Figure 3.
Pontryagin’s maximum principle leads to the Hamiltonian
H = p1 + p2x
3 + c3p3x
3 +
1
2
(
c1x
3p3
)2
along an optimal trajectory, and Hamilton’s equations take the form
ẋ1 = 1, ṗ1 = 0,
ẋ2 = x3, ṗ2 = 0,
ẋ3 = c3x
3 +
(
c1x
3
)2
p3, ṗ3 = −p2 − c3p3 − (c1p3)
2x3. (3.7)
The equation for ṗ2 in (3.7) implies that p2(t) is equal to a constant c2. Then (3.7) implies that
˙(
p3x3
)
= −c2x3, ẋ3 = c3x
3 + c1x
3
(
p3x
3
)
. (3.8)
These equations can be solved as follows:
• If c2 = 0, then the function p3x
3 is constant, and so the equation for ẋ3 becomes
ẋ3 = c̃x3
for some constant c̃. If c̃ = 0, then the solution trajectories are given by
x1(t) = t+ t0, x2(t) = at+ b, x3(t) = a
for some constants a, b. Sample optimal trajectories are shown in Fig. 2.
If c̃ 6= 0, then the solution trajectories are given by
x1(t) = t+ t0, x2(t) =
a
c̃
ec̃t + b, x3(t) = aec̃t
for some constants a, b. Sample optimal trajectories are shown in Fig. 3.
• If c2 6= 0, then (3.8) can be written as the 2nd-order ODE for the function z(t) = p3(t)x
3(t):
z̈ =
(
c3 + c21z
)
ż.
Geometry of Optimal Control for Control-Affine Systems 19
Figure 4.
Integrating once yields
ż =
1
2
(c1z)
2 + c3z + c4
for some constant c4. Depending on the values of the constants, the solution z(t) has one
of the following forms:
(1) z(t) = a tan(bt+ c) + d, c23 − 2c1c4 < 0;
(2) z(t) = a tanh(bt+ c) + d, c23 − 2c1c4 > 0;
(3) z(t) =
1
at+ b
+ c, c23 − 2c1c4 = 0.
Then we have
x3 = − 1
c2
ż = ẋ2,
and so the corresponding solution trajectories are given (with slightly modified constants)
by:
(1)
x1(t) = t+ t0,
x2(t) = a tan(bt+ c) + d,
x3(t) = ab sec2(bt+ c);
(2)
x1(t) = t+ t0,
x2(t) = a tanh(bt+ c) + d,
x3(t) = ab sech2(bt+ c);
(3)
x1(t) = t+ t0,
x2(t) =
1
at+ b
+ c,
x3(t) = − a
(at+ b)2
.
Sample optimal trajectories for the first two cases are shown in Fig. 4.
Case 2.2. This point-affine distribution corresponds to the control system
ẋ1 = x2, ẋ2 = x3, ẋ3 = x2
(
3
2
(
x3
x2
)2
+ c1 + u
)
,
20 J.N. Clelland, C.G. Moseley and G.R. Wilkens
with cost functional
Q(ẋ) =
1
2
g0u
2.
Pontryagin’s maximum principle leads to the Hamiltonian
H = p1x
2 + p2x
3 + p3x
2
(
3
2
(
x3
x2
)2
+ c1
)
+
1
2g
(
p3x
2
)2
along an optimal trajectory, and Hamilton’s equations take the form
ẋ1 = x2, ṗ1 = 0,
ẋ2 = x3, ṗ2 = −p1 +
3
2
p3
(
x3
)2(
x2
)2 − c1p3 − 1
g
(p3)
2x2,
ẋ3 =
3
2
(
x3
)2
x2
+ c1x
2 +
1
g
p3
(
x2
)2
, ṗ3 = −p2 − 3
p3x
3
x2
. (3.9)
The system (3.9) has three independent first integrals in addition to the Hamiltonian H
(which is automatically a first integral): it is straightforward to show, using (3.9), that the
functions
I1 = p1, I2 = p1x
1 + p2x
2 + p3x
3, I3 = p1
(
x1
)2
+ 2p2x
1x2 + 2p3x
1x3 + 2p3
(
x2
)2
are first integrals for this system. We can use these conserved quantities to reduce the sys-
tem (3.9), as follows: on any solution curve of (3.9), we have
I1 = k1, I2 = k2, I3 = k3
for some constants k1, k2, k3. These equations can be solved for p1, p2, p3 to obtain
p1 = k1,
p2 = k1
(
−x
1
x2
− (x1)2x3
2(x2)3
)
+ k2
(
1
x2
+
x1x3
(x2)3
)
+ k3
(
− x3
2(x2)3
)
,
p3 = k1
(
(x1)2
2(x2)2
)
+ k2
(
− x1
(x2)2
)
+ k3
(
1
2(x2)2
)
.
These equations can be substituted into (3.9) to obtain a closed, first-order ODE system for
the functions x1, x2, x3, depending on the parameters k1, k2, k3; moreover, making the same
substitution in the Hamiltonian H yields a conserved quantity for this system. (The precise
expressions for the system and the conserved quantity are complicated and unenlightening,
so we will not write them out explicitly here.) The resulting ODE system cannot be solved
analytically, but numerical integration yields sample trajectories as shown in Fig. 5.
Case 2.3.1. This point-affine distribution corresponds to the control system
ẋ1 = 1 + x3(c1 + u), ẋ2 = c1 + u, ẋ3 = ε
(
x1 + c2x
3
)
(c1 + u),
with cost functional
Q(ẋ) =
1
2
g0u
2.
Geometry of Optimal Control for Control-Affine Systems 21
Figure 5.
Pontryagin’s maximum principle leads to the Hamiltonian
H = p1 + p3x
1 +
c1√
c2
(
p1x
3 + p2 + c3p3x
3
)
+
(
p1x
3 + p2 + c3p3x
3
) (
c2
(
p1x
3 + p2 + p3x
1 + c3p3x
3
))
along an optimal trajectory, and Hamilton’s equations take the form
ẋ1 = 1 + x3
(
c1√
c2
+ c2
(
p1x
3 + p2 + p3x
1 + c3p3x
3
))
,
ẋ2 =
c1√
c2
+ c2
(
p1x
3 + p2 + p3x
1 + c3p3x
3
)
,
ẋ3 =
(
x1 + c3x
3
)( c1√
c2
+ c2
(
p1x
3 + p2 + p3x
1 + c3p3x
3
))
,
ṗ1 = −p3
(
c1√
c2
+ c2
(
p1x
3 + p2 + p3x
1 + c3p3x
3
))
,
ṗ2 = 0,
ṗ3 = −
(
(p1 + c3p3)
(
c1√
c2
+ c2
(
p1x
3 + p2 + p3x
1 + c3p3x
3
))
+
(
(p1 + c3p3)x
3 + p2
)
(c2p1 + c2c3p3)
)
. (3.10)
The system (3.10) has three independent first integrals in addition to the Hamiltonian H: it
is straightforward to show, using (3.10), that the functions
I1 = (p1 + r1p3)e
r1x2 , I2 = (p1 + r2p3)e
r2x2 , I3 = p2,
where r1, r2 are as in (A.19), are first integrals for this system. A similar process to that
described in the previous case leads to a closed, first-order system of ODEs for the func-
tions x1, x2, x3; numerical integration of this system yields sample trajectories (with c3 > 0
and c3 < 0) as shown in Fig. 6.
Case 2.3.2. Due to the complexity of the computations, we will restrict our attention to the
simplest case, where ε = 1, c1 = c4 = 0, and F20
(
x2
)
= 0. (In the interest of brevity, we will
omit Case 2.3.3, which is similar to this case.) With these assumptions, we have
v1 =
∂
∂x1
, v2 = x3
∂
∂x1
+
∂
∂x2
+ c3
(
x3
)2
tan
(
c3x
1
) ∂
∂x3
.
22 J.N. Clelland, C.G. Moseley and G.R. Wilkens
Figure 6.
This point-affine distribution corresponds to the control system
ẋ1 = 1 + x3u, ẋ2 = u, ẋ3 = c3
(
x3
)2
tan
(
c3x
1
)
u,
with cost functional
Q(ẋ) =
1
2
g0u
2.
Pontryagin’s maximum principle leads to the Hamiltonian
H =
1
2g0
(
c23
((
x3
)2
p3
)2
tan2
(
c3x
1
)
+ 2c3p3
(
x3
)2(
x3p1 + p2
)
tan
(
c3x
1
)
+
(
2p1
(
x3p2 + g0
)
+
(
x3p1
)2
+ p22
))
along an optimal trajectory, and Hamilton’s equations take the form
ẋ1 =
1
g0
(
c3
(
x3
)3
p3 tan
(
c3x
1
)
+
(
x3
)2
p1 + x3p2 + g0
)
,
ẋ2 =
1
g0
(
c3
(
x3
)2
p3 tan
(
c3x
1
))
+ x3p1 + p2,
ẋ3 =
1
g0
c3
(
x3
)2
tan
(
c3x
1
) (
c3
(
x3
)2
p3 tan
(
c3x
1
)
+ x3p1 + p2
)
,
ṗ1 = − 1
g0 cos3
(
c3x1
)(c3x3)2p3 (c3(x3)2p3 sin
(
c3x
1
)
+
(
x3p1 + p2
)
cos
(
c3x
1
))
,
ṗ2 = 0,
ṗ3 = − 1
g0
(
2(c3p3)
2
(
x3
)3
tan2
(
c3x
1
)
+ c3x
3p3
(
3x3p1 + 2p2
)
tan
(
c3x
1
)
+ p1
(
x3p1 + p2
))
.
This ODE system cannot be solved analytically, but numerical integration yields sample trajec-
tories as shown in Fig. 7.
4 Conclusion
What is perhaps most interesting about these results is how the behavior of control-affine
systems in low dimensions varies from that of control-linear (i.e., driftless) systems. As we
observed in [1], functional invariants appear in much lower dimension for affine distributions
Geometry of Optimal Control for Control-Affine Systems 23
Figure 7.
(beginning with n = 2, s = 1) than for linear distributions, where there are no functional
invariants in dimensions below n = 5, s = 2.
With the addition of a quadratic cost functional, we see a similar phenomenon: for linear
distributions with a quadratic cost functional, there are no functional invariants for any n when
s = 1, since local coordinates can always be chosen so that a unit vector field for the cost
functional is represented by the vector field ∂
∂x1
. But for affine distributions with s = 1, there
are numerous functional invariants, and even the homogeneous examples exhibit a wide variety
of behaviors for the optimal trajectories.
A Normal forms for Case 2.3
In this appendix, we carry out the analysis to identify examples of normal forms for homogeneous
point-affine structures in Case 2.3.
First consider local coordinate transformations which preserve the expressions (2.23). Let(
x1, x2, x3
)
and (x̃1, x̃2, x̃3) be two local coordinate systems with respect to which the coframing
(η1, η2, η3) takes the form (2.23). Then we must have
η1 = dx1 − x3 dx2 = dx̃1 − x̃3 dx̃2. (A.1)
Taking the exterior derivative of (A.1) yields
dη1 = dx2 ∧ dx3 = dx̃2 ∧ dx̃3. (A.2)
In particular,
span
(
dx2, dx3
)
= span
(
dx̃2, dx̃3
)
.
Therefore we must have
x2 = φ̄
(
x̃2, x̃3
)
, x3 = ψ̄
(
x̃2, x̃3
)
(A.3)
for some functions φ̄
(
x̃2, x̃3
)
, ψ̄
(
x̃2, x̃3
)
. Equation (A.2) then implies that the functions φ̄, ψ̄
satisfy the PDE
φ̄x̃2ψ̄x̃3 − φ̄x̃3ψ̄x̃2 = 1. (A.4)
24 J.N. Clelland, C.G. Moseley and G.R. Wilkens
Unfortunately, equation (A.4) cannot be solved explicitly in terms of arbitrary functions
of x̃2, x̃3. However, it can be solved implicitly with a slightly different setup. Instead of (A.3),
suppose that we define our coordinate transformation by
x̃2 = φ
(
x2, x̃3
)
, x3 = ψ
(
x2, x̃3
)
.
Then equation (A.2) is equivalent to the condition
φx2 = ψx̃3 .
(In addition, both terms in this equation must be nonzero.) This is equivalent to the condition
that there exists a function Φ
(
x2, x̃3
)
such that
φ
(
x2, x̃3
)
= Φx̃3 , ψ
(
x2, x̃3
)
= Φx2 .
Then equation (A.1) implies that
x1 = x̃1 + Φ
(
x2, x̃3
)
− x̃3Φx̃3
(
x2, x̃3
)
.
The local coordinate transformations which preserve the expression for η1 in (2.23) are defined
implicitly by
x1 = x̃1 + Φ
(
x2, x̃3
)
− x̃3Φx̃3
(
x2, x̃3
)
, x̃2 = Φx̃3
(
x2, x̃3
)
, x3 = Φx2
(
x2, x̃3
)
, (A.5)
where Φ
(
x2, x̃3
)
is an arbitrary smooth function of two variables with Φx2x̃3 6= 0.
Next we will compute how the function H
(
x1, x2, x3
)
transforms under a coordinate trans-
formation of the form (A.5). (When we consider the implications of homogeneity, it will turn
out that J can be expressed in terms of H and its derivatives; thus there is no need to explicitly
compute the effects of the transformation (A.5) on J .) Consider the expression for η3 in (2.23).
We must have
η3 =
1√
εHx1(x)
(
H(x) dx2 − dx3
)
=
1√
εH̃x̃1(x̃)
(
H̃(x̃) dx̃2 − dx̃3
)
. (A.6)
From (A.5), we have
dx̃2 = Φx2x̃3 dx
2 + Φx̃3x̃3 dx̃
3, dx3 = Φx2x2 dx
2 + Φx2x̃3 dx̃
3.
Substituting these expressions into (A.6) yields
1√
εHx1(x)
(
(H(x)− Φx2x2) dx2 − Φx2x̃3 dx̃
3
)
=
1√
εH̃x̃1(x̃)
(
H̃(x̃)Φx2x̃3 dx
2 +
(
H̃(x̃)Φx̃3x̃3 − 1
)
dx̃3
)
. (A.7)
Equating the ratios of the coefficients of dx2 and dx̃3 on both sides of (A.7) yields
(H(x)− Φx2x2)
−Φx2x̃3
=
H̃(x̃)Φx2x̃3
(H̃(x̃)Φx̃3x̃3 − 1)
,
which implies that
H
(
x1, x2, x3
)
=
(
(Φx2x̃3)2 − Φx2x2Φx̃3x̃3
)
H̃
(
x̃1, x̃2, x̃3
)
+ Φx2x2
1− Φx̃3x̃3H̃(x̃1, x̃2, x̃3)
. (A.8)
Geometry of Optimal Control for Control-Affine Systems 25
Now suppose that the structure is homogeneous. Unlike the previous cases, the assumption
of homogeneity will imply some relations among the constants appearing in the structure equa-
tions (2.24). In the homogeneous case, the functions T ijk are all constant, and differentiating
equations (2.24) implies that
0 = d
(
dη1
)
=
(
T 1
23T
3
13 − T 1
13T
3
23
)
η1 ∧ η2 ∧ η3,
0 = d
(
dη2
)
=
(
T 2
23T
3
13 − T 2
13T
3
23
)
η1 ∧ η2 ∧ η3,
0 = d
(
dη3
)
= −
(
T 1
13 + T 2
23
)
η1 ∧ η2 ∧ η3.
The first two equations imply that the vectors[
T 1
13 T
1
23
]
,
[
T 2
13 T
2
23
]
,
[
T 3
13 T
3
23
]
are all scalar multiples of each other unless T 3
13 = T 3
23 = 0, while the third equation implies that
T 2
23 = −T 1
13.
In most of the computations that follow, these relations will be self-evident; however, at one
point they will have implications for the function H.
The structure equation for dη1 is
dη1 = −J
√
εHx1η
1 ∧ η3 − εη2 ∧ η3.
Therefore, we must have
J =
c1√
εHx1
for some constant c1, so that the equation for dη1 becomes
dη1 = −c1η1 ∧ η3 − εη2 ∧ η3.
Now the equation for dη3 reduces to
dη3 = η1 ∧ η2
− 1
2Hx1
√
εHx1
(
Hx1x2 + x3Hx1x1 +HHx1x3 − 2Hx1Hx3
) (
c1η
1 ∧ η3 + εη2 ∧ η3
)
.
Therefore,
Hx1x2 + x3Hx1x1 +HHx1x3 − 2Hx1Hx3
Hx1
√
εHx1
= −2c2 (A.9)
for some constant c2. Substituting the derivative of (A.9) with respect to x1 into the equation
for dη2 yields
dη2 =
(
3
4
(
Hx1x1
Hx1
)2
− 1
2
Hx1x1x1
Hx1
+
c21
ε
)
η1 ∧ η3 + c1 η
2 ∧ η3.
Observe that:
• The coefficient of η2 ∧ η3 in dη2 is equal to minus the coefficient of η1 ∧ η3 in dη1, as we
previously observed that it must be.
26 J.N. Clelland, C.G. Moseley and G.R. Wilkens
• If c2 6= 0, then the ratio of the η1 ∧ η3 and η2 ∧ η3 coefficients in dη2 must be equal to c1
ε
(which is the ratio of these coefficients in dη1), and hence the η1 ∧ η3 coefficient in dη2
must be equal to
c21
ε .
Therefore, if c2 6= 0, then H satisfies the PDE
3
4
(
Hx1x1
Hx1
)2
− 1
2
Hx1x1x1
Hx1
= 0. (A.10)
The solutions of (A.10) are precisely the linear fractional transformations in the x1 variable,
and so we must have
H
(
x1, x2, x3
)
=
F1
(
x2, x3
)
x1 + F0
(
x2, x3
)
G1
(
x2, x3
)
x1 +G0
(
x2, x3
) (A.11)
for some functions F0
(
x2, x3
)
, F1
(
x2, x3
)
, G0
(
x2, x3
)
, G1
(
x2, x3
)
.
By contrast, if c2 = 0, then the vectors [T 1
13 T
1
23], [T 2
13 T
2
23] are no longer required to be
linearly independent, and so the Schwarzian derivative of H with respect to x1 appearing in
equation (A.10) is only required to be constant, but not necessarily equal to zero. There are
two possibilities, depending on the sign:
• If 3
4
(
Hx1x1
Hx1
)2
− 1
2
Hx1x1x1
Hx1
= −c23 for c3 > 0, then
H
(
x1, x2, x3
)
= F1
(
x2, x3
)
tan
(
c3x
1 + F0
(
x2, x3
))
+ F2
(
x2, x3
)
(A.12)
for some functions F0
(
x2, x3
)
, F1
(
x2, x3
)
, F2
(
x2, x3
)
with F1
(
x2, x3
)
6= 0.
• If 3
4
(
Hx1x1
Hx1
)2
− 1
2
Hx1x1x1
Hx1
= c23 for c3 > 0, then
H
(
x1, x2, x3
)
= F1
(
x2, x3
)
tanh
(
c3x
1 + F0
(
x2, x3
))
+ F2
(
x2, x3
)
(A.13)
for some functions F0
(
x2, x3
)
, F1
(
x2, x3
)
, F2
(
x2, x3
)
with F1
(
x2, x3
)
6= 0.
We consider each of these cases separately.
A.1 c2 6= 0
In this case, H
(
x1, x2, x3
)
is given by (A.11). Now we compute how the function (A.11) trans-
forms under a local coordinate transformation of the form (A.5).
Lemma A.1. There exists a local coordinate transformation of the form (A.5) such that the
function H̃(x̃1, x̃2, x̃3) is linear in x̃1, i.e.,
H̃
(
x̃1, x̃2, x̃3
)
= F̃1
(
x̃2, x̃3
)
x̃1 + F̃0
(
x̃2, x̃3
)
, (A.14)
with F̃1 6= 0.
Proof. Equation (A.8) can be written as
H̃(x̃1, x̃2, x̃3) =
H
(
x1, x2, x3
)
− Φx2x2
Φx̃3x̃3H
(
x1, x2, x3
)
+
(
(Φx2x̃3)2 − Φx2x2Φx̃3x̃3
) . (A.15)
Geometry of Optimal Control for Control-Affine Systems 27
Substituting (A.11) into this equation yields
F̃1x̃
1 + F̃0
G̃1x̃1 + G̃0
=
(
F1x1+F0
G1x1+G0
)
− Φx2x2
Φx̃3x̃3
(
F1x1+F0
G1x1+G0
)
+ ((Φx2x̃3)2 − Φx2x2Φx̃3x̃3)
=
(
F1x
1 + F0
)
− Φx2x2
(
G1x
1 +G0
)
Φx̃3x̃3 (F1x1 + F0) + ((Φx2x̃3)2 − Φx2x2Φx̃3x̃3) (G1x1 +G0)
=
[F1 − Φx2x2G1]x
1 + [F0 − Φx2x2G0]
[Φx̃3x̃3F1+((Φx2x̃3)2−Φx2x2Φx̃3x̃3)G1]x1 + [Φx̃3x̃3F0+((Φx2x̃3)2−Φx2x2Φx̃3x̃3)G0]
.
The coefficients of x̃1 on the left-hand side of this equation are the same as the coefficients of x1
on the right-hand side, so the condition that G̃1 = 0 is equivalent to
0 = Φx̃3x̃3
(
x2, x̃3
)
F1
(
x2, x3
)
+
((
Φx2x̃3
(
x2, x̃3
))2− Φx2x2
(
x2, x̃3
)
Φx̃3x̃3
(
x2, x̃3
))
G1
(
x2, x3
)
= Φx̃3x̃3
(
x2, x̃3
)
F1
(
x2,Φx2
(
x2, x̃3
))
+
((
Φx2x̃3
(
x2, x̃3
))2 − Φx2x2
(
x2, x̃3
)
Φx̃3x̃3
(
x2, x̃3
))
G1
(
x2,Φx2
(
x2, x̃3
))
.
Any solution Φ
(
x2, x̃3
)
of this equation will induce a local coordinate transformation for which
H̃(x̃1, x̃2, x̃3) has the form (A.14), as desired. Note that F̃1 = H̃x1 6= 0, and hence F̃1 must have
the same sign as ε. �
Local coordinates for which H has the form (A.14) are determined up to transformations of
the form (A.5) with Φx̃3x̃3 = 0, i.e.,
Φ
(
x2, x̃3
)
= Φ1
(
x2
)
x̃3 + Φ0
(
x2
)
.
With Φ as above, the local coordinate transformation (A.5) reduces to
x1 = x̃1 + Φ0
(
x2
)
, x̃2 = Φ1
(
x2
)
, x3 = Φ′0
(
x2
)
+ Φ′1
(
x2
)
x̃3. (A.16)
With the assumption that H has the form
H
(
x1, x2, x3
)
= F1
(
x2, x3
)
x1 + F0
(
x2, x3
)
,
differentiating equation (A.9) with respect to x1 yields
(F1)x3√
εF1
= 0.
Therefore,
F1
(
x2, x3
)
= F1
(
x2
)
.
Now equation (A.8) reduces to
F1
(
x2
)
x1 + F0
(
x2, x3
)
=
(
Φ′1
(
x2
))2 (
F̃1
(
x̃2
)
x̃1 + F̃0
(
x̃2, x̃3
))
+ Φ′′0
(
x2
)
+ Φ′′1
(
x2
)
x̃3,
which, taking (A.16) into account, becomes
F1
(
x2
)
x̃1 +
(
F1
(
x2
)
Φ0
(
x2
)
+ F0
(
x2,Φ′0
(
x2
)
+ Φ′1
(
x2
)
x̃3
))
=
(
Φ′1
(
x2
))2
F̃1
(
Φ1
(
x2
))
x̃1 +
(
Φ′1
(
x2
))2
F̃0
(
Φ1
(
x2
)
, x̃3
)
+ Φ′′0
(
x2
)
+ Φ′′1
(
x2
)
x̃3. (A.17)
28 J.N. Clelland, C.G. Moseley and G.R. Wilkens
Equating the coefficients of x̃1 on both sides yields
F1
(
x2
)
=
(
Φ′1
(
x2
))2
F̃1
(
Φ1
(
x2
))
.
Thus any solution Φ1
(
x2
)
of the equation
Φ′1
(
x2
)
=
√
εF1
(
x2
)
will induce a local coordinate transformation for which
F̃1
(
x̃2
)
= ε = ±1.
Local coordinates for which F1
(
x2
)
= ε are determined up to transformations of the form (A.16)
with
Φ′1
(
x2
)
= ±1;
for simplicity, we will assume that Φ′1
(
x2
)
= 1. Then
Φ
(
x2, x̃3
)
= x2x̃3 + ax̃3 + Φ0
(
x2
)
for some constant a. With Φ as above, the local coordinate transformation (A.5) reduces to
x1 = x̃1 + Φ0
(
x2
)
, x̃2 = x2 + a, x3 = x̃3 + Φ′0
(
x2
)
. (A.18)
Now equation (A.9) takes the form
(F0)x3 = εc3.
Therefore,
F0
(
x2, x3
)
= εc3x
3 + F2
(
x2
)
.
Now equation (A.17) reduces to
εΦ0
(
x2
)
+ εc3Φ
′
0
(
x2
)
+ F2
(
x2
)
= F̃2(x
2 + a) + Φ′′0
(
x2
)
.
Thus any solution Φ0
(
x2
)
of the equation
εΦ0
(
x2
)
+ εc3Φ
′
0
(
x2
)
− Φ′′0
(
x2
)
= −F2
(
x2
)
will induce a local coordinate transformation for which
F̃2
(
x̃2
)
= 0.
Local coordinates for which F2
(
x2
)
= 0 are determined up to transformations of the form (A.18)
with
εΦ0
(
x2
)
+ εc3Φ
′
0
(
x2
)
− Φ′′0
(
x2
)
= 0,
i.e.,
Φ0
(
x2
)
= b1e
r1x2 + b2e
r2x2 ,
where b1, b2 are constants and
r1 =
εc3 +
√
c23 + 4ε
2
, r2 =
εc3 −
√
c23 + 4ε
2
. (A.19)
Note that if ε = 1, then r1, r2 are real and distinct; if ε = −1, then r1, r2 may be real and
distinct, real and equal, or a complex conjugate pair.
To summarize, we have constructed local coordinates for which
J
(
x1, x2, x3
)
= c1, H
(
x1, x2, x3
)
= ε
(
x1 + c3x
3
)
.
These coordinates are determined up to transformations of the form
x1 = x̃1 + b1e
r1x2 + b2e
r2x2 , x̃2 = x2 + a, x3 = x̃3 + b1r1e
r1x2 + b2r2e
r2x2 .
Geometry of Optimal Control for Control-Affine Systems 29
A.2 c2 = 0
We will only give the details of the analysis for the case (A.12); the case (A.13) is similar. First
we compute how the function (A.12) transforms under a local coordinate transformation of the
form (A.5).
Lemma A.2. There exists a local coordinate transformation of the form (A.5) such that
F̃0
(
x̃2, x̃3
)
= 0, i.e.,
H̃
(
x̃1, x̃2, x̃3
)
= F̃1
(
x̃2, x̃3
)
tan(c3x̃
1) + F̃2
(
x̃2, x̃3
)
, (A.20)
with F̃1 6= 0.
Proof. Substituting (A.12) into the expression (A.15) for H̃(x̃1, x̃2, x̃3) yields
F̃1 tan
(
c3x̃
1 + F̃0
)
+ F̃2
=
(
F1 tan(c3x
1 + F0) + F2
)
− Φx2x2
Φx̃3x̃3 (F1 tan(c3x1 + F0) + F2) + ((Φx2x̃3)2 − Φx2x2Φx̃3x̃3)
=
F1 sin(c3x
1 + F0) + (F2 − Φx2x2) cos(c3x
1 + F0)
Φx̃3x̃3F1 sin(c3x1+F0)+(Φx̃3x̃3F2+(Φx2x̃3)2−Φx2x2Φx̃3x̃3) cos(c3x1+F0)
. (A.21)
Now, define functions R
(
x2, x̃3
)
, Θ
(
x2, x̃3
)
by the conditions that
Φx̃3x̃3F1 = R sin Θ, Φx̃3x̃3F2 + (Φx2x̃3)2 − Φx2x2Φx̃3x̃3 = R cos Θ;
in particular, we have
Θ = tan−1
(
Φx̃3x̃3F1
Φx̃3x̃3F2 + (Φx2x̃3)2 − Φx2x2Φx̃3x̃3
)
.
Then the denominator of the right-hand side of (A.21) can be written as
R cos
(
c3x
1 + F0 −Θ
)
= R cos
(
c3
(
x̃1 + Φ− x̃3Φx̃3
)
+ F0 −Θ
)
.
Therefore, H̃(x̃1, x̃2, x̃3) is a linear function of the quantity
tan
(
c3
(
x̃1 + Φ− x̃3Φx̃3
)
+ F0 −Θ
)
,
which implies that
F̃0 = c3
(
Φ− x̃3Φx̃3
)
+ F0 −Θ.
Keeping in mind that Θ is a second-order differential operator in Φ, the condition F̃0 = 0
is a second-order PDE for the function Φ
(
x2, x̃3
)
. Any solution Φ
(
x2, x̃3
)
of this equation
will induce a local coordinate transformation for which H̃(x̃1, x̃2, x̃3) has the form (A.20), as
desired. �
Local coordinates for which H has the form (A.20) are determined up to transformations of
the form (A.5) with Φ
(
x2, x̃3
)
a solution of the PDE
c3
(
Φ− x̃3Φx̃3
)
−Θ = 0. (A.22)
Unfortunately we cannot explicitly write down the general solution to this PDE; however, a sub-
set of the solutions is given by the family
Φ
(
x2, x̃3
)
= x̃3Φ0
(
x2
)
,
where Φ0 is an arbitrary function of x2 with Φ′0
(
x2
)
6= 0.
30 J.N. Clelland, C.G. Moseley and G.R. Wilkens
With the assumption that H has the form (A.20), equation (A.9) becomes (recalling that
c2 = 0)
F1
(
(F1)x3 − 2c3x
3
)
tan
(
c3x
1
)
+ 2F1(F2)x3 − F2(F1)x3 − (F1)x2 = 0.
Therefore, since F1 6= 0,
(F1)x3 − 2c3x
3 = 0, 2F1(F2)x3 − F2(F1)x3 − (F1)x2 = 0. (A.23)
The first equation implies that
F1
(
x2, x3
)
= c3
(
x3
)2
+ F10
(
x2
)
for some function F10
(
x2
)
.
Computations similar to those in the previous case show that, under a local coordinate
transformation (A.5) with Φ = x̃3Φ0
(
x2
)
, we have
F̃10
(
x̃2
)
=
F10
(
x2
)
Φ′0
(
x2
)2 .
Thus any solution Φ0
(
x2
)
of the equation
Φ′0
(
x2
)
=
√
1
c3
εF10
(
x2
)
will induce a local coordinate transformation for which either F̃10
(
x̃2
)
= 0 or F̃10
(
x̃2
)
= ±c3.
Denote this constant by c4, so that we now have
F̃1
(
x̃2, x̃3
)
= c3
(
x̃3
)2
+ c4.
Finally, the second equation in (A.23) becomes
2
(
c3
(
x3
)2
+ c4
)
(F2)x3 − 2c3x
3F2 = 0,
which implies that
F2
(
x2, x3
)
= F20
(
x2
)√
c3(x3)2 + c4
for some function F20
(
x2
)
. We conjecture that the remaining solutions of (A.22) can be used to
normalize the function F20
(
x2
)
, but unfortunately we have been unable to complete this step in
the analysis.
To summarize, we have constructed local coordinates for which
J
(
x1, x2, x3
)
=
c1 cos(c3x
1)√
c3(c2(x3)2 + c4)
,
H
(
x1, x2, x3
)
=
(
c3
(
x3
)2
+ c4
)
tan
(
c3x
1
)
+ F20
(
x2
)√
c3(x3)2 + c4.
Acknowledgements
This research was supported in part by NSF grants DMS-0908456 and DMS-1206272. We
would like to thank the referees for many helpful suggestions, which significantly improved the
organization and exposition of this paper.
Geometry of Optimal Control for Control-Affine Systems 31
References
[1] Clelland J.N., Moseley C.G., Wilkens G.R., Geometry of control-affine systems, SIGMA 5 (2009), 095,
28 pages, arXiv:0903.4932.
[2] Gardner R.B., The method of equivalence and its applications, CBMS-NSF Regional Conference Series in
Applied Mathematics, Vol. 58, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA,
1989.
[3] Jurdjevic V., Geometric control theory, Cambridge Studies in Advanced Mathematics, Vol. 52, Cambridge
University Press, Cambridge, 1997.
[4] Liu W., Sussman H.J., Shortest paths for sub-Riemannian metrics on rank-two distributions, Mem. Amer.
Math. Soc. 118 (1995), no. 564, x+104 pages.
[5] Montgomery R., A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys
and Monographs, Vol. 91, Amer. Math. Soc., Providence, RI, 2002.
[6] Moseley C.G., The geometry of sub-Riemannian Engel manifolds, Ph.D. thesis, University of North Carolina
at Chapel Hill, 2001.
http://dx.doi.org/10.3842/SIGMA.2009.095
http://arxiv.org/abs/0903.4932
http://dx.doi.org/10.1137/1.9781611970135
http://dx.doi.org/10.1137/1.9781611970135
1 Introduction
2 Normal forms for homogeneous cases
2.1 Two states and one control
2.2 Three states and one control
3 Optimal control problem for homogeneous metrics
3.1 Two states and one control
3.2 Three states and one control
4 Conclusion
A Normal forms for Case 2.3
A.1 c2=0
A.2 c2=0
References
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