An Analytical Framework to Describe the Interactions Between Individuals and a Continuum

We consider a discrete set of individual agents interacting with a continuum. Examples might be a predator facing a huge group of preys, or a few shepherd dogs driving a herd of sheeps. Analytically, these situations can be described through a system of ordinary differential equations coupled with a scalar conservation law in several space dimensions. This paper provides a complete well posedness theory for the resulting Cauchy problem. A few applications are considered in detail and numerical integrations are provided

Kinetic theory of dusty plasmas and effective grain interactions

The basic points of the consistent kinetic theory of dusty plasmas is discussed. The equations for microscopic phase densities of plasma particles and grains are formulated. Using such equations it is possible to derive the kinetic equations, taking into account both elastic and inelastic particle collisions. Obtained equations are used for kinetic description of effective grain-grain potentials.Обговорюються основні положення послідовної кінетичної теорії запорошеної плазми. Наведено рівняння для мікроскопічних фазових густин для плазмових частинок і порошинок. Показано, що, використовуючи такі рівняння, можна сформулювати кінетичні рівняння для запорошеної плазми з урахуванням пружних і непружних зіткнень частинок та поглинання електронів та іонів порошинками. Розглянуто кінетичний опис ефективних потенціалів взаємодії порошинок у плазмі.Обсуждаются основные положения последовательной кинетической теории пылевой плазмы. Приведены уравнения для микроскопических фазовых плотностей для плазменных частиц и пылинок. Показано, что, используя такие уравнения, можно сформулировать кинетические уравнения для пылевой плазмы, учитывающие упругие и неупругие столкновения частиц и поглощение электронов и ионов пылинками. Выполнены кинетические расчеты эффективных потенциалов взаимодействия пылинок в плазме

The effective bandwidth problem revisited

The paper studies a single-server queueing system with autonomous service and $\ell$ priority classes. Arrival and departure processes are governed by marked point processes. There are $\ell$ buffers corresponding to priority classes, and upon arrival a unit of the $k$th priority class occupies a place in the $k$th buffer. Let $N^{(k)}$, $k=1,2,...,\ell$ denote the quota for the total $k$th buffer content. The values $N^{(k)}$ are assumed to be large, and queueing systems both with finite and infinite buffers are studied. In the case of a system with finite buffers, the values $N^{(k)}$ characterize buffer capacities. The paper discusses a circle of problems related to optimization of performance measures associated with overflowing the quota of buffer contents in particular buffers models. Our approach to this problem is new, and the presentation of our results is simple and clear for real applications.Comment: 29 pages, 11pt, Final version, that will be published as is in Stochastic Model

Simple model of bouncing ball dynamics. Displacement of the limiter assumed as a cubic function of time

Nonlinear dynamics of a bouncing ball moving vertically in a gravitational field and colliding with a moving limiter is considered and the Poincare map, describing evolution from an impact to the next impact, is described. Displacement of the limiter is assumed as periodic, cubic function of time. Due to simplicity of this function analytical computations are possible. Several dynamical modes, such as fixed points, 2 - cycles and chaotic bands are studied analytically and numerically. It is shown that chaotic bands are created from fixed points after first period doubling in a corner-type bifurcation. Equation for the time of the next impact is solved exactly for the case of two subsequent impacts occurring in the same period of limiter's motion making analysis of chattering possible.Comment: 8 pages, 1 figure, presented at the DSTA 2011 conference, Lodz, Polan

Simple model of bouncing ball dynamics: displacement of the table assumed as quadratic function of time

Nonlinear dynamics of a bouncing ball moving in gravitational field and colliding with a moving limiter is considered. Displacement of the limiter is a quadratic function of time. Several dynamical modes, such as fixed points, 2 - cycles and chaotic bands are studied analytically and numerically. It is shown that chaotic bands appear due to homoclinic structures created from unstable 2 - cycles in a corner-type bifurcation.Comment: 11 pages, 6 figure

Avoidance Control on Time Scales

We consider dynamic systems on time scales under the control of two agents. One of the agents desires to keep the state of the system out of a given set regardless of the other agent's actions. Leitmann's avoidance conditions are proved to be valid for dynamic systems evolving on an arbitrary time scale.Comment: Revised edition in JOTA format. To appear in J. Optim. Theory Appl. 145 (2010), no. 3. In Pres

Dynamics of Simple Balancing Models with State Dependent Switching Control

Time-delayed control in a balancing problem may be a nonsmooth function for a variety of reasons. In this paper we study a simple model of the control of an inverted pendulum by either a connected movable cart or an applied torque for which the control is turned off when the pendulum is located within certain regions of phase space. Without applying a small angle approximation for deviations about the vertical position, we see structurally stable periodic orbits which may be attracting or repelling. Due to the nonsmooth nature of the control, these periodic orbits are born in various discontinuity-induced bifurcations. Also we show that a coincidence of switching events can produce complicated periodic and aperiodic solutions.Comment: 36 pages, 12 figure