Inverse problem in the self-organization of a multi-agent
system in pattern formation
D. H. Nguyen and P. K. C. Wang
Department of Electrical Engineering
University of California, Los
Angeles
Los Angeles, CA 90024
The self-organization of a multi-agent system produces many interesting and complex effects. In this study, the pattern formation resulting from the self-organization of multiple agents is investigated. A mathematical model of multiple interacting agents is developed to examine the convergence and stability of the patterns formed. The main focus of this study is the inverse problem where given the final stable formation, find a self-organizing rule which when applied to the multi-agent system results in such a formation. For simplicity, we assume that the dynamics of each agent is governed by a rule-based ordinary differential equation where inertia is neglected.
We first study the forward problem by examining several rules which when applied to the multi-agent system yield the formation of circles and polygons. The convergence to such stationary formations is initially studied by way of computer simulation after which the dynamic behavior is formulated as a fixed-point problem to demonstrate the results observed. Since the rules governing the dynamics of these agents are generally discontinuous, proof of convergence and stability is performed in the appropriate metric spaces to lessen the constraint on the discontinuity problem. In addition, self-organizing rules leading to the convergence of equalized and nonequalized grouping are examined by simulation and verified mathematically.
Next, we examine the inverse problem for some simple geometrical patterns such as the "V-formation", polygons, etc... In our mathematical formulation for an n-agent system, given a stable fixed-point in a 2n-dimensional metric space, find a rule-based mapping with such a fixed-point. In general, the ordinary differential equation will have discontinuous right-hand sides which we will analyze by embedding the map into a set-valued map and examine the existence and stability of the fixed-point to the differential inclusion in the Hausdorff metric.
Finally, an application of interest to study is the hexagonal shapes of snow crystals in which it is desired to know the self-organizing rule governing the interacting molecules.