Analysis, Controllability and Optimization of Time-Discrete by Prof. Dr. Werner Krabs, Dr. Stefan Wolfgang Pickl (auth.),

By Prof. Dr. Werner Krabs, Dr. Stefan Wolfgang Pickl (auth.), M. Beckmann, H. P. Künzi, Prof. Dr. G. Fandel, Prof. Dr. W. Trockel, C. D. Aliprantis, A. Basile, A. Drexl, G. Feichtinger, W. Güth, K. Inderfurth, P. Korhonen, W. Kürsten, U. Schittko, R. Selten,

J. P. los angeles Salle has built in [20] a balance idea for structures of distinction equations (see additionally [8]) which we introduce within the first bankruptcy in the framework of metric areas. the soundness conception for such structures is additionally present in [13] in a marginally changed shape. we commence with self reliant platforms within the first component to bankruptcy 1. After theoretical arrangements we learn the localization of restrict units as a result of Lyapunov services. utilising those Lyapunov services we will be able to increase a balance thought for self sustaining platforms. If we linearize a non-linear procedure at a hard and fast element we will enhance a balance conception for mounted issues which uses the Frechet spinoff on the mounted element. the following subsection bargains with common linear structures for which we intro­ duce a brand new notion of balance and asymptotic balance that we undertake from [18]. purposes to numerous fields illustrate those effects. we begin with the classical predator-prey-model as being constructed and investigated by way of Volterra that's in accordance with a 2 x 2-system of first order differential equations for the densities of the prey and predator inhabitants, respectively. This version has additionally been investigated in [13] with recognize to balance of its equilibrium through a Lyapunov functionality. right here we examine the discrete model of the model.

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For every (x , y) T E ]R2 the J acobi matrix of J ( ) j x, Y = (1+ f is given by a - by bx ) dy 1 - c + dx which implies be) . 5 the only fixed point (x *, y*)T of f in ]R~ is un st abl e. Next we assume that the prey population grows logistically in the abs ence of pred ators. x nYn with a constant param et er e > O. 2( x n , Yn) = (1 - c)Yn + dXnYn , n E No . f) 1 d . The eigenval ues of Jf(x * , y*) are given by Al ,2 = ec 1 - 2d ± V(2dec ) 2 - c (a - dec ). I We have to distinguish three cases: Then - 1 < Al ,-') if and only if =1- ec d 2) (~d) 2 ec -2d < 1 ' < 4.

Al Xo + L A pA p_ l . . Aj+lb P j . 35) , if and only if the matrix 1- ApA p- l . AI , 1= k x k-unit matrix, is non-singul ar. The first vector Xo is then given by Xo = (I - ApA p_ l .. Ad- l p L A pA p_ l . . A j+lA j . 35) can only exist, if the matrix I - ApA p- l . . e. det(I - A pA p- l . . A d = 0 . This again is equivalent to the fact that A = 1 is an eigenv alue of ApA p_ l .. AI . 4 Application to a Model for the Process of Hemo-Dialysis In order to describe the tempor al development of the concent rat ion of som e po ison like ur ea in the bod y of a person suffering from a renal disease and having t o be attached to an artificial kidney a mathem atical model has been proposed in [1 2] which ca n be described as follows.

N -1)) instead of (u(O) , . . , u(N - 1)) . Now let us consider a special case which is motivated by a situation which occurs in the mod elling of conflicts . We begin with an uncontrolled system of the form x 1(t x 2(t + 1) = gl( :r 1(t ), x 2(t )) , + 1) = g2(X1(t) , x 2(t )) , t E No , where gi : jRnl X jRn 2 --+ jRn ; , i = 1,2, are given continuous mappings and X i : No --+ jRn i , i = 1,2, ar e considered as state functions. 11) where x6 E jRnl and x6E jRn 2 are given vectors with 54 2 Controlled Systems x; : : for some 8 n z which is also given.

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