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Using infinitesimal symmetries for determining the first Maxwell time of geometric control problem on SH(2)
Lie algebra is non-nilpotent and solvable. Rather where X 0 , X 1 , . . . , X m are smooth vector fields on
than relying on nilpotent approximations, which M and u i (t) are control inputs. Further details on
have been used in previous works by Hrdina and control affine systems can be found in Jurdjevic. 15
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Zalabov´, we directly compute the infinitesimal Let F be a family of smooth vector fields on M,
symmetries. This approach enhances the accu- then the Lie algebra generated by F is the small-
racy of calculating the first Maxwell time and est Lie algebra that contains F. It is obtained
avoids potential errors associated with nilpotent by considering all linear combinations of elements
approximations. of F, taking all Lie brackets of these, consider-
8
Additionally, in Agrachev and Barilari, the ing all linear combinations of these, and continu-
authors classify all sub-Riemannian structures on ing so on. It will be denoted by Lie(F) and its
three-dimensional Lie groups using basic differen- evaluation at any point q ∈ M will be denoted
tial invariants. Our study, therefore, highlights by Lie q (F). The following result gives a neces-
the significance of the sub-Riemannian problem sary and sufficient condition for a driftless control
on the SH(2) group, as it plays a crucial role in the affine system to be controllable.
broader context of three-dimensional Lie groups. Proposition 1. (Jurdjevic 15) The control
We focus on the motion of a unicycle, a classical affine system dx = P m u i X i (x) with u =
problem in geometric mechanics. These concepts dt m i=1
(u 1 , . . . , u m ) ∈ R is controllable if and only
can also be adjusted for use in other nonholo- if Lie q F = T q M, for all q ∈ M. 1
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nomic mechanical systems; see Jean, Hill and
Nurowski, 10 Hermans, 11 Bloch. 12 2.2. Sub-Riemannian problem and
Moreover, deep learning techniques (see Mao optimal solutions
14
et al., 13 Peng et al. ) have shown their potential
Consider a sub-Riemannian manifold (M, ∆, g),
in optimizing complex dynamical systems. Such
where M is an n-dimensional smooth manifold,
approaches could offer new perspectives for the
∆ is a smooth distribution of rank k ≤ n, and
analysis of sub-Riemannian structures and opti- 1
g is a Riemannian metric on ∆; see Sachkov for
mal control problems.
more details. The sub-Riemannian length of an
Sub-Riemannian geodesics are generally not admissible curve γ(t), defined on [0, t 1 ], is given
optimal at all times. Each geodesic has a spe-
by:
cific point where it ceases to be optimal. At this Z t 1 q
stage, the role of the infinitesimal symmetries of ℓ(γ) = g γ(t) (˙γ(t), ˙γ(t)) dt.
0
our problem on the group SH(2) becomes cru-
Given two points q 0 and q 1 of M, the sub-
cial. Using the Lie algebra of symmetries, we
Riemannian distance between q 0 and q 1 is stated
identify transformations that map a given geo-
by
desic to another geodesic. We show that there
d(q 0 , q 1 ) = inf{ℓ(γ) | γ admissible,
is a subgroup of symmetries isomorphic to the
group SO(1, 2). Using this action, we determine γ(0) = q 0 , γ(t 1 ) = q 1 }.
the set of Maxwell points and, subsequently, the
first Maxwell time corresponding to our infinites- A sub-Riemannian problem is a control problem
imal symmetries. All of this is detailed in Section on a sub-Riemannian manifold (M, ∆, g) where
4 of our main results. one seek for admissible curves γ that satisfy the
property ℓ(γ) = d(q(0), q(t 1 )). Suppose there ex-
2. Preliminaries on geometric control ists a family of smooth vector fields X 1 , . . . , X k
that forms an orthonormal frame on (∆, g), i.e.
theory
∀q ∈ M, ∆ q = span{X 1 (q), . . . , X k (q)} and
In this section, we give some preliminaries on geo- g q (X i (q), X j (q)) = δ ij . Thus, sub-Riemannian
metric control theory. For more details on the minimizers (or optimal solutions) are the solu-
subject, see Agrachev et al. 4 tions of the following optimal control problem on
M:
2.1. Controllability of control affine k
X k
systems ˙ x = u i X i (x), u = (u 1 , ..., u k ) ∈ R , (2)
i=1
A control affine system on a manifold M is any x(0) = q 0 , x(t 1 ) = q 1 , (3)
differential system of the form
k
Z
m t 1 X
dx X ℓ = ( u ) dt −→ min . (4)
2 1/2
= X 0 (x) + u i (t)X i (x), (1) i
dt 0 i=1
i=1
1 Here T qM denotes the tangent space to M at the point q
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