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On the controllability of a robot arm

Considered is the rotation of a robot arm or rod in a horizontal plane about an axis through the arm's fixed end and driven by a motor whose torque is controlled. The model was derived and investigated computationally by Sakawa and co-authors in [7] for the case that the arm is described as a homogeneous Euler beam. The resulting equation of motion is a partial differential equation of the type of a wave equation which is linear with respect to the state, if the control is fixed, and non-linear with respect to the control. Considered is the problem of steering the beam, within a given time interval, from the position of rest for the angle zero into the position of rest under a certain given angle. At first we show that, for every L²-control, there is exactly one (weak) solution of the initial boundary value problem which describes the vibrating system without the end condition. Then we show that the problem of controllability is equivalent to a non-linear moment problem. This, however, is not exactly solvable. Therefore, an iteration method is developed which leads to an approximate solution of sufficient accuracy in two steps. This method is numerically implemented and demonstrated by an example. © 1998 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Mathematische Behandlung ernes Modells der Haemodialyse

In this paper a simple mathematical model for the process of hemodialysis is presented. This model is based on a system of two linear differential equations of first order with partly discontinuous coefficients that describe the time-development of the concentrations of a certain toxin (like urea) in the intra- and extracellular part of the human body. The main result is the existence of periodic positive solutions of this system under the natural assumption that the generation of the toxin and its removal by hemodialysis are periodic processes. These periodic positive solutions are also computed numerically for a realistic choice of the coefficients of the modelling differential equations.     

On time-minimal distributed control of vibrations

In [1] Fattorini investigates distributed control of vibrations governed by an abstract wave equation and influenced by controls which are Hilbert-space-valuedL -functions of time and uniformly bounded with respect to the maximum norm. He shows that null-controllability is possible with such controls for sufficiently large times and that there is a time-minimal bounded control which is unique on the minimum time interval and satisfies a strong bang-bang principle. We show that null-controllability is possible for every positive time by anL -control function whose essential supremum norm is as small as possible, which is unique on the given time interval, and satisfies a strong bang-bang principle. We further show that the timeminimal bounded control is the minimum norm control on the minimum time interval and is characterized by this property.     

Stability in predator – prey models and discretization of a modified Volterra – Lotka model

We consider n  2 populations of animals that are living in mutual predator – prey relations or are pairwise neutral to each other. We assume the temporal development of the population densities to be described by a system of differential equations which has an equilibrium state solution. We derive sufficient conditions for this equilibrium state to be stable by Lyapunov's method. The results supplement those published elsewhere.

Further we consider a modification of the Volterra – Lotka model which admits an asymptotically stable steady state solution. This model is discretized in two ways and we investigate how small the time step size has to be chosen in order to guarantee that the steady state solution is an attractive fixed point of the discretized model. This investigation is connected with the determination of the model parameters from given data.     

Stability and Controllability in Non-autonomous Time-discrete Dynamical Systems

In this paper stability and attractivity in non-autonomous time-discrete dynamical systems is investigated with the aid of Lyapunov functions. The results are applied to the problem of stabilization of controlled systems by feedback controls. In the final section of the paper we give sufficient conditions for norm-bounded null-controllability of linear systems.     

A convolution formula for time‐invariant linear communication systems

For every linear and time‐invariant time‐discrete (communication) system T : l^∞ → l^∞, formally, the following convolution formula can be derived: \input amssym.def $$(Tf)(n)=\sum_{k \in {\Bbb Z}} h(n‐k) f(k),\quad n \in {\Bbb Z}, f \in l^{\infty},$$ where h = Tδ is the delta impulse response. This paper is concerned with the question under which assumptions linear and time‐invariant time‐discrete systems T : l^∞ → l^∞ can be characterized by this formula. For this purpose we derive a convolution formula in a more general situation which also leads to a well‐known convolution formula in the time‐continuous case. Copyright © 1999 John Wiley & Sons, Ltd.     

A General Predator-Prey Model

We consider n = 2 populations of animals or plants that are living in mutual predator-prey relations or are pairwise neutral to each other. We assume the temporal development of the population densities to be described by a system of differential equations which has an equilibrium state solution. We at first give sufficient conditions for this equilibrium state to be asymptotically stable by linearizing the system around it. Then we derive sufficient conditions for asymptotic stability by Lyapunov’s method. Finally we investigate a discretization of the Volterra-Lotka model.