Abstract. We address the problem of optimal coordinated motions of multiple agents moving in the same planar region. The agents ’ motions must satisfy a separation constraint throughout the encounter to be conflict-free. The objective is to determine the conflict-free maneuvers (motions) with the least combined energy, while taking into account the fact that agents may have different priorities. A formal classification of conflict-free maneuvers into homotopy types is introduced by using their braid representation. Various local and global optimality conditions are derived through variational analysis in the presence of the separation constraint. In the case of two agents, these optimality conditions allow us to construct the optimal maneuvers geometrically. For the general multi-agent case, a convex optimization algorithm is proposed to compute within each homotopy type a solution to the optimization problem restricted to the class of multi-legged maneuvers. Since the number of types grows explosively with the number of agents, a stochastic algorithm is suggested as the “type chooser”, thus leading to a randomized optimization algorithm.

Abstract. In this paper, an optimal coordinated motion planning problem for multiple agents subject to constraints on the admissible formation patterns is formulated. Solutions to the problem are reinterpreted as distance minimizing geodesics on a certain manifold with boundary. A geodesic on this manifold may fail to be a solution for different reasons. In particular, if a geodesic possesses conjugate points, then it will no longer be distance minimizing beyond its first conjugate point. We study a particular instance of the formation constrained optimal coordinated motion problem, where a number of initially aligned agents tries to switch positions by rotating around their common centroid. The complete set of conjugate points of a geodesic naturally associated to this problem is characterized analytically. This allows us to prove that the geodesic will not correspond to an optimal coordinated motion when the angle of rotation exceeds a threshold that decreases to zero as the number of agents increases. Moreover, infinitesimal perturbations that improve the performance of the geodesic after it passes the conjugate points are also determined, which interestingly are characterized by a certain family of orthogonal polynomials. Key words. Conjugate point, multi-agent coordination, geodesics, orthogonal polynomials. AMS subject classifications. 53C22, 58E25, 05E35.

Abstract. The optimal control problem for a class of hybrid systems (switched Lagrangian systems) is studied. Some necessary conditions of the optimal solutions of such a system are derived based on the assumption that there is a group of symmetries acting uniformly on the domains of different discrete modes, such that the Lagrangian functions, the guards, and the reset maps are all invariant under the action. Lagrangian reduction approach is adopted to establish the conservation law of certain quantities for the optimal solutions. Some examples are presented. In particular, the problems of optimal collision avoidance (OCA) and optimal formation switching (OFS) of multiple agents moving on a Riemannian manifold are studied in some details. 1

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It is a classical result that solutions to the isoperimetric problem, i.e., finding the planar curves with a fixed length that enclose the largest area, are circles. As a generalization, we study an asymptotic version of the dual isoholonomic problem in a Euclidean space with a co-dimension one distribution. We propose the concepts of asymptotic rank and efficiency, and compute these quantities as well as the efficiency-achieving curves in several special cases. In particular, an example of a snake moving on ice is worked out in detail to illustrate the results.