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This paper solves a path planning problem for a group of gliders. The gliders are tasked with visiting a set of interest points. The gliders have limited range but are able to increase their range by visiting special points called thermals. The problem addressed in this paper is of path planning for the gliders such that, the total number of interest points visited by the gliders is maximized. This is referred to as the multi-agent problem. The problem is solved by first decomposing it into several single-agent problems. In a single-agent problem a set of interest points are allocated to a single glider. This problem is solved by planning a path which maximizes the number of visited interest points from the allocated set. This is achieved through a uniform cost graph search, as shown in our earlier work. The multi-agent problem now consists of determining the best allocation (of interest points) for each glider. Two ways are presented of solving this problem, a brute force search approach as shown in earlier work and a Branch\&Bound type graph search. The Branch&Bound approach is the main contribution of the paper. This approach is proven to be optimal and shown to be faster than the brute force search using simulations.

Global optimality analysis in sub-Riemannian problem on the Lie group SH(2) is considered. We cutout open dense domains in the preimage and in the image of the exponential mapping based on the description of Maxwell strata. We then prove that the exponential mapping restricted to these domains is a diffeomorphism. Based on the proof of diffeomorphism, the cut time, i.e., time of loss of global optimality is computed on SH(2). We also consider the global structure of the exponential mapping and obtain an explicit description of cut locus and optimal synthesis.

We study local and global optimality of geodesics in the left invariant sub-Riemannian problem on the Lie group $\mathrm{SH}(2)$. We obtain the complete description of the Maxwell points corresponding to the discrete symmetries of the vertical subsystem of the Hamiltonian system. An effective upper bound on the cut time is obtained in terms of the first Maxwell times. We study the local optimality of extremal trajectories and prove the lower and upper bounds on the first conjugate times. We also obtain the generic time interval for the $n$-th conjugate time which is important in the study of sub-Riemannian wavefront. Based on our results of $n$-th conjugate time and $n$-th Maxwell time, we prove a generalization of Rolle's theorem that between any two consecutive Maxwell points, there is exactly one conjugate point along any geodesic., Comment: 18 Pages, 13 Figures

We consider the sub-Riemannian length minimization problem on the group of motions of pseudo Euclidean plane that form the special hyperbolic group SH(2). The system comprises of left invariant vector fields with 2-dimensional linear control input and energy cost functional. We apply the Pontryagin Maximum Principle to obtain the extremal control input and the sub-Riemannian geodesics. A change of coordinates transforms the vertical subsystem of the normal Hamiltonian system into the mathematical pendulum. In suitable elliptic coordinates the vertical and the horizontal subsystems are integrated such that the resulting extremal trajectories are parametrized by the Jacobi elliptic functions. Qualitative analysis reveals that the projections of normal extremal trajectories on the xy-plane have cusps and inflection points. The vertical subsystem being a generalized pendulum admits reflection symmetries that are used to obtain a characterization of the Maxwell strata., Comment: 19 pages, 5 figures, Accepted for Publication in Journal of Dynamical and Control System