On boundary controllability of one-dimensional vibrating systems

In this paper boundary controllability of one-dimensional vibrating system such as the vibrating string or the vibrating beam is studied. In particular we are concerned with the question whether it is possible to transfer a given initial state of vibration into rest within a given time such that the system stays in rest when the control is turned off. This problem is rephrased as a typical trigonometric moment problem which is solved within the framework of an abstract moment problem in a Hilbert space. The results of null-controllability which are obtained are substantially based on classical results of Ingham and Redheffer concerning trigonometric inequalities and incompleteness of certain sequences of trigonometric functions, respectively. The representation of the general statements follows closely the lines of a paper of Russell. Besides a special case is treated where explicit representations of boundary controls can be given that transfer the system to a permanent rest position. This special case includes amplitude boundary control of the vibrating string and the freely supported beam.     

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.     

An optimal control problem in cancer chemotherapy

We consider a mathematical model for the control of the growth of tumor cells which is formulated as a problem of optimal control theory. It is concerned with chemotherapeutic treatment of cancer and aims at the minimization of the size of the tumor at the end of a certain time interval of treatment with a limited amount of drugs. The treatment is controlled by the dosis of drugs that is administered per time unit for which also a limit is prescribed. It is shown that optimal controls are of bang-bang type and can be chosen at the upper limit, if the total amount of drugs is large enough.     

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.     

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.     

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.     

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.