Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Time Optimal Synthesis for Left--Invariant Control Systems on SO(3)   Consider the control system given by $\dot x=x(f+ug)$, where $x\in SO(3)$,
$|u|\leq 1$ and $f,g\in so(3)$ define two perpendicular left-invariant vector
fields normalized so that $\|f\|=\cos(\al)$ and $\|g\|=\sin(\al)$, $\al\in
]0,\pi/4[$. In this paper, we provide an upper bound and a lower bound for
$N(\alpha)$, the maximum number of switchings for time-optimal trajectories.
More precisely, we show that $N_S(\al)\leq N(\al)\leq N_S(\al)+4$, where
$N_S(\al)$ is a suitable integer function of $\al$ which for $\al\to 0$ is of
order $\pi/(4\alpha).$ The result is obtained by studying the time optimal
synthesis of a projected control problem on $R P^2$, where the projection is
defined by an appropriate Hopf fibration. Finally, we study the projected
control problem on the unit sphere $S^2$. It exhibits interesting features
which will be partly rigorously derived and partially described by numerical
simulations.

1
The categories of this paper are: cs
29
The categories of this paper are: funct-an
11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
9
The categories of this paper are: nlin
33
The categories of this paper are: plasm-ph
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
24
The categories of this paper are: dg-ga
14
The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
6
The categories of this paper are: astro-ph
12
The categories of this paper are: nucl-th