

Международная молодежная конференция «Геометрия и управление»
15 апреля 2014 г. 17:00, Стендовые доклады, г. Москва, МИАН






Maxwell Strata and Conjugate Points in SubRiemannian Problem on
Group SH(2)
Yasir Awais Butt^{a}, Yuri L. Sachkov^{b}, Aamer Iqbal Bhatti^{c} ^{a} Muhammad Ali Jinnah University, Islamabad, Pakistan
^{b} Program Systems Institute of Russian Academy of Sciences,
Pareslavl Zalessky, Russia
^{c} Muhammad Ali Jinnah University, Islamabad, Pakistan

Количество просмотров: 
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Аннотация:
SubRiemannian geometry has experienced resurgence of interest and
extensive research for past several decades. It has emerged as an extremely
rich framework with a unique character seeking applications in various
fields of pure and applied mathematics such as classical and quantum
mechanics, control theory, geometric analysis, stochastic calculus and
evolution equations. The renewed interest is also attributed to the fact
that subRiemannian geometry has given entirely new and richer perspective
to some older problems such as image inpainting, neurophysiology of vision
and quantum control [1]. Consequently, research in subRiemannian
problems via geometric control methods on various Lie groups such as the
Heisenberg group, $\mathrm{S^{3}}$, SL(2), SU(2), SE(2), Engel group etc.
has been paticularly popular for two decades now. From control theory
perspective, subRiemannian geometry models optimal control problems for
nonholonomic systems such as motion planning and control of robots, falling
cats, parking of cars, rolling of bodies on plane without sliding,
satellites, vision, quantum phases and even finance. Magnificence of
subRiemannian geometry as an optimal control framework drew our attention
to the subRiemannian problem on the group of motions of pseudo Euclidean
plane. The pseudo Euclidean plane $F_{1}^{1+1}$ is $(1+1)$dimensional
space defined over field of real numbers $\mathbb{R}$ and endowed with a
nondegenerate indefinite quadratic form $q$:
$$
q(x)=x_{1}^{2}x_{2}^{2}.
$$
The motions of pseudo Euclidean plane are distance and orientation
preserving maps of the points in the plane. The motions describe the
hyperbolic rototranslations of the pseudo Euclidean plane and form
a 3dimensional Lie group known as special hyperbolic group $\mathrm{SH(2)}$
[2]. The driftless control system on $\mathrm{SH(2)}$
is described as follows:
$$
\tag{1}
\dot{q} = u_{1}f_{1}(q)+u_{2}f_{2}(q),\qquad q\in
M=\mathrm{SH(2)},\qquad(u_{1},u_{2})\in\mathbb{R}^{2}.
$$
Here, (1) is the control system with bounded inputs $u_{i}$ and
control distribution $\Delta=\mathrm{span}\{f_{1},f_{2}\}$. The vector
fields $f_{i}$ satisfy the Lie bracket relations:
$$
[f_{2},f_{1}]=f_{0},\qquad[f_{1},f_{0}]=0,\qquad[f_{2},f_{0}]=f_{1}.
$$
The subRiemannian problem on control system (1) is defined as:
\begin{gather}
\tag{2}
q(0) =Id,\qquad q(t_{1})=q_{1},
\tag{3}
l =\int_{0}^{t_{1}}\sqrt{u_{1}^{2}+u_{2}^{2}} dt\to\min.
\end{gather}
In (2), $q(0)$ and $q(t_{1})$ represent the initial and the final
states whereas $l$ (3) is the subRiemannian distance (length
functional) to be minimized. In coordinates $q=(x,y,z),$ the control system
(1) is given as:
$$
\tag{4}
( \begin{array}{c}
\dot{x}
\dot{y}
\dot{z}
\end{array} )=( \begin{array}{c}
\cosh z
\sinh z
0
\end{array} )u_{1}+( \begin{array}{c}
0
0
1
\end{array} )u_{2}.
$$
We applied the Pontryagin Maximum Principle (PMP) on
(1)–(3) to calculate the extremal controls
$\tilde{u}(t)$ and the extremal trajectories. Since the problem is 3D
contact, there are no nontrivial abnormal trajectories. A change of
coordinates in the vertical subsystem of the normal Hamiltonian system
transforms it into a mathematical pendulum. The phase cylinder $C$ of the
pendulum is decomposed into five connected subsets $C_{i}\quad
i=1,\ldots,5$ depending upon the energy of the pendulum. Suitable elliptic
coordinates i.e. reparametrized energy $k$ and reparametrized time
$\varphi$ are introduced on each $C_{i}$ and such that the flow of the
vertical subsystem is rectified. Computation of the Hamiltonian flow then
follows from integration of vertical and horizontal subsystem and the
resulting extremal trajectories are parametrized by Jacobi elliptic
functions. Further analysis/simulations reveal the qualitative nature of
extremal trajectories.
Parametrization of extremal trajectories is followed by second order
optimality analysis based on description of Maxwell strata and conjugate
loci. Since the vertical subsystem is a mathematical pendulum, it admits
reflection symmetries in the phase portrait which are used to obtain
complete description of Maxwell strata. The fixed points of the extremals
$\lambda$ in the preimage and the multiple points in the image of
exponential mapping are used to obtain complete description of the Maxwell
strata and compute the first Maxwell time $t_{1}^{MAX}$ for $\lambda\in
C_{i}$, $i=1,\ldots,5$. On the basis of Maxwell strata and Maxwell time, we
obtain a global upper bound on cut time in the subRiemannian problem on
$\mathrm{SH(2)}$ which happens to be the first Maxwell time $t_{1}^{MAX}$.
We then turn to the problem of characterizing the conjugate points.
Computation and simplification of Jacobian for $\lambda\in C_{1}\cup C_{2}$
reveals a rather unexpected symmetry with respect to bounds of conjugate
times in these cases which hasn't been observed in corresponding analysis
in subRiemannian problem on SE(2) [3], Engel group [5] and
Euler Elasticae problem [4]. It turns out that the first conjugate
time $t_{1}^{C_{1}}$ for $\lambda\in C_{1}$ is bounded as $4K(k)\leq
t_{1}^{C_{1}}\leq2p_{1}^{1}(k)$ where $p_{1}^{1}(k)$ is the first root of a
function
$f_{1}(p)=[\mathrm{cn}p \mathrm{E}(p)\mathrm{sn}p \mathrm{dn}p]$.
The function $f_{1}(p)$ and its roots shall be described in more detail in
our upcoming journal paper on Maxwell Strata on SH(2). Similarly, for
$\lambda\in C_{2}$ first conjugate time is bounded as
$t_{1}^{C_{2}}=kt_{1}^{C_{1}}$. Thus globally the first conjugate time is
greater or equal to the first Maxwell time. We conjecture that the cut time
is equal to the first Maxwell time. This conjecture will be studied in a
forthcoming work.
Материалы:
abstract.pdf (83.7 Kb)
Язык доклада: английский
Список литературы

Enrico Le Donne. Lecture notes on subRiemannian geometry. Preprint, 2010.

N. Ja. Vilenkin. Special Functions and Theory of Group Representations (Translations of Mathematical Monographs). American Mathematical Society, revised edition, 1968.

Yuri L. Sachkov. Conjugate and cut time in the subRiemannian problem on the group of motions of a plane, ESAIM: COCV, Volume 16, Issue 04, October 2010, pp 1018–1039

Yuri L. Sachkov. Conjugate points in the Euler elastic problem. Journal of Dynamical and Control Systems, 14:409–439, July 2008.

A. A. Ardentov and Yu. L. Sachkov. Conjugate points in nilpotent subRiemannian problem on the Engel group, Journal of Mathematical Sciences December 2013, Volume 195, Issue 3, pp 369–390

