Stability Conditions for Coupled Autonomous Vehicles Formations

In this paper, we give necessary conditions for stability of coupled autonomous vehicles in R. We focus on linear arrays with decentralized vehicles, where each vehicle interacts with only a few of its neighbors. We obtain explicit expressions for necessary conditions for stability in the cases that a system consists of a periodic arrangement of two or three different types of vehicles, i.e. configurations as follows: ...2-1-2-1 or ...3-2-1-3-2-1. Previous literature indicated that the (necessary) condition for stability in the case of a single vehicle type (...1-1-1) held that the first moment of certain coefficients of the interactions between vehicles has to be zero. Here, we show that that does not generalize. Instead, the (necessary) condition in the cases considered is that the first moment plus a nonlinear correction term must be zero

Spectra of Certain Large Tridiagonal Matrices

We characterize the eigenvalues and eigenvectors of a class of complex valued tridiagonal n by n matrices subject to arbitrary boundary conditions, i.e. with arbitrary elements on the first and last rows of the matrix. For large n, we show there are up to 4 eigenvalues, the so-called special eigenvalues, whose behavior depends sensitively on the boundary conditions. The other eigenvalues, the so-called regular eigenvalues vary very little as function of the boundary conditions. For large n, we determine the regular eigenvalues up to , O(n−2), and the special eigenvalues up to O ( κ n ) , for some κ ∈ ( 0 , 1 ) . . The components of the eigenvectors are determined up to O ( n − 1 ) . The matrices we study have important applications throughout the sciences. Among the most common ones are arrays of linear dynamical systems with nearest neighbor coupling, and discretizations of second order linear partial differential equations. In both cases, we give examples where specific choices of boundary conditions substantially influence leading eigenvalues, and therefore the global dynamics of the system