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 idea for such structures is additionally present in [13] in a marginally converted shape. we begin with self sufficient platforms within the first portion of bankruptcy 1. After theoretical arrangements we learn the localization of restrict units by means of Lyapunov features. using those Lyapunov capabilities we will strengthen a balance concept for self reliant platforms. If we linearize a non-linear method at a hard and fast aspect we will boost a balance thought for fastened issues which uses the Frechet by-product on the fastened element. the subsequent subsection bargains with normal linear platforms for which we intro­ duce a brand new inspiration of balance and asymptotic balance that we undertake from [18]. functions to numerous fields illustrate those effects. we begin with the classical predator-prey-model as being built and investigated by means of Volterra that's according to 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 appreciate to balance of its equilibrium through a Lyapunov functionality. the following we think about the discrete model of the model.

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9. Assumption: All An, n E N, are invertibl e. A ssertion: Th e sequence (An 0 A n- I 0 · · · 0 AI(GE))nENo is stable, if and only if for every N E No there exists a constant CN > 0 such that IIAn 0 A n- I 0 . . Th e sequence (An 0 A n- I only if for every N E No °A N +1 11:::: CN 0 · · · 0 + 1. 32) AI(GE))nENo is asymptotically stable, if and lim II A n 0 A n - n -+ oo for all n 2 N I 0 . 33) = O. Proof If the sequ ence (An 0 A n- I 0 .. 0 A1(GE)) nENo is st abl e, we choose an arbitrary e > 0 and conclude that, for every N E No, there exist s 0 = o(e , N) > 0 such that for every seq uence (x n = Fn (xo) )nENo, with Xo E E , it follows that IIAn 0 A n- I 0 · · · °AN+I(XN) II < e for all n 2 N + 1.

According to the a bove conside rations it suffices t o consider homogeneous syst ems with f n(x ) = An (x ) , X E E , n E N, 40 1 Uncontrolled Systems such that Fn(x) = An A n- I 0 0 .. °Al (x) , x E E , n E N. Then all sequ ences (x n = Fn (xo) )nENo, Xo E E, ar e stable or asymptotically stable, if and only if the sequence (x n = Fn (G E))nENo is stable or asymptotically stable. 9. Assumption: All An, n E N, are invertibl e. A ssertion: Th e sequence (An 0 A n- I 0 · · · 0 AI(GE))nENo is stable, if and only if for every N E No there exists a constant CN > 0 such that IIAn 0 A n- I 0 .

1 The Autonomous Case 21 Exercise 1. 7. 3 t hat {8d is stable with respect to f. b) Show with the aid of Th eorem 1. S that {8d is asy mptotically st able with resp ect to i. , {8d is stable wit h res pect to f and whi ch is equivalent t o A n ----; a= k x k - zero matrix. Converse ly, let {8d be globally asy mptotically stable with resp ect t o f . Further let C be a symmet ric and positi ve definite real k x k - m atrix such t hat 00 t he ser ies L: (A T) kCA k converges . 6 Applications a) Predator - Prey - Models The classical predator - pr'ey - mo del as being develop ed and invest igated by Volterra is based on a 2 x 2 - system of first order different ial equat ions for t he densities of the prey and pr edat or populati on , res pect ively, This model has also been invest igat ed in [13] with respect t o stability of its equilibrium via a Lyapunov functio n.

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