Maxwell Strata and Conjugate Points in Sub-Riemannian Problem on Group SH(2)

Sub-Riemannian 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 sub-Riemannian 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 sub-Riemannian 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, sub-Riemannian 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 sub-Riemannian geometry as an optimal control framework drew our attention to the sub-Riemannian 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 non-degenerate 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 roto-translations of the pseudo Euclidean plane and form a 3-dimensional Lie group known as special hyperbolic group $\mathrm{SH(2)}$ [2]. The driftless control system on $\mathrm{SH(2)}$ is described as follows:
$$ \tag{1} \dot{q} = u_{1}f_{1}(q)+u_{2}f_{2}(q),\qquad q\in M=\mathrm{SH(2)},\qquad\left(u_{1},u_{2}\right)\in\mathbb{R}^{2}. $$
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 sub-Riemannian 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 sub-Riemannian distance (length functional) to be minimized. In coordinates $q=(x,y,z),$ the control system (1) is given as:
$$ \tag{4} \left(
\begin{array}{c} \dot{x}\\ \dot{y}\\ \dot{z} \end{array}
\right)=\left(
\begin{array}{c} \cosh z\\ \sinh z\\ 0 \end{array}
\right)u_{1}+\left(
\begin{array}{c} 0\\ 0\\ 1 \end{array}
\right)u_{2}. $$
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 sub-Riemannian 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 sub-Riemannian 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 $f_{1}(p)=\left[\mathrm{cn}p\,\mathrm{E}(p)-\mathrm{sn}p\,\mathrm{dn}p\right]$. 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.