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 constructed in [20] a balance concept for structures of distinction equations (see additionally [8]) which we introduce within the first bankruptcy in the framework of metric areas. the steadiness conception for such platforms can be present in [13] in a marginally changed shape. we commence with self sustaining platforms within the first part of bankruptcy 1. After theoretical arrangements we learn the localization of restrict units through Lyapunov services. using those Lyapunov features we will improve a balance conception for self sustaining platforms. If we linearize a non-linear approach at a hard and fast element we can boost a balance conception for fastened issues which uses the Frechet by-product on the mounted aspect. the following subsection bargains with basic linear structures for which we intro­ duce a brand new proposal of balance and asymptotic balance that we undertake from [18]. purposes to varied fields illustrate those effects. we commence with the classical predator-prey-model as being built and investigated by means of Volterra that is 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 admire to balance of its equilibrium through a Lyapunov functionality. the following we contemplate the discrete model of the model.

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The point (x *, y*)T E lR 2 is a fixed poin t of I with x* and only if b x* + e y* = a , e x * + I y* Let us assu me t hat b] - ee Then x * = a I -de bI - e e and -I- 0 and y* -I- 0, if =d. > O. 20). 1 The Autonomous Case 29 For every (x , y)T E IR 2 the Jacobi matrix of f is given by Jj( x , y) = ( - h bx +1+h a - h b x - h ey - h ex ) - he y - h f y + 1+ h d- h e x - h f y which implies J j (x * , y *) = (1 - h b x* - h c x* ) _ h e y* 1 - h f y* . l1 ,2. l1 ,2) < O. This implies IA1,21< 1 (Exercise), if h > 0 is sufficiently small.

R (ej = j-t h unit vector) are linearly ind ependent. 26) is stable. 1 Definitions and Elementary Properties Let X be a metric space with metric d : X x X - t JR+ and let (fn)nEN be a sequence of cont inuous mappings f n : X - t X , n E N. Then the pair (X, (fn)n EN) is called a non-autonomous time-di scret e dynamical system . The dynamics in this system is defined by the sequence F = (Fn)n ENo of mappings Fn : X - t X given by Fn(x) = fn 0 fn-l and 0 ... 0 ft( x) for all x E X and n E N Fa = x for all x E X .

U(N - l)f (C N (xo, u(O), . . , u(N - 1)) - x) where ,\ > 0 and 1m is the m x m-unit matrix. (O) , . . (O) , . (N - 1)) <

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