J. J. P. Veerman

J. J. P. Veerman

The study of the movement of flocks, whether biological or technological is motivated by the desire to understand the capability of coherent motion of a large number of agents that only receive very limited information. In a biological flock a large group of animals seek their course while moving in a more or less fixed formation. It seems reasonable that the immediate course is determined by leaders at the boundary of the flock. The others follow: what is their algorithm? The most popular technological application consists of cars on a one-lane road. The light turns green and the lead car accelerates. What is the efficient algorithm for the others to closely follow without accidents? In this position paper we present some general questions from a more fundamental point of view. We believe that the time is right to solve many of these questions: they are within our reach.

J. J. P. Veerman, D. K. Hammond, Pablo E. Baldivieso

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. %By boundary conditions, we mean the first and last row of the matrix. For large $n$, we show there are up to $4$ eigenvalues, the so-called \emph{special eigenvalues}, whose behavior depends sensitively on the boundary conditions. The other eigenvalues, the so-called \emph{regular eigenvalues} vary very little as function of the boundary conditions. For large $n$, we determine the regular eigenvalues up to ${\cal O}(n^{-2})$, and the special eigenvalues up to ${\cal O}(κ^n)$, for some $κ\in (0,1)$. The components of the eigenvectors are determined up to ${\cal O}(n^{-1})$. 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.

Ivo Herman, Dan Martinec, J. J. P. Veerman

We consider an asymmetric control of platoons of identical vehicles with nearest-neighbor interaction. Recent results show that if the vehicle uses different asymmetries for position and velocity errors, the platoon has a short transient and low overshoots. In this paper we investigate the properties of vehicles with friction. To achieve consensus, an integral part is added to the controller, making the vehicle a third-order system. We show that the parameters can be chosen so that the platoon behaves as a wave equation with different wave velocities. Simulations suggest that our system has a better performance than other nearest-neighbor scenarios. Moreover, an optimization-based procedure is used to find the controller properties.