# 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 platforms of distinction equations (see additionally [8]) which we introduce within the first bankruptcy in the framework of metric areas. the steadiness conception for such structures is also present in [13] in a marginally changed shape. we begin with self sufficient platforms within the first part of bankruptcy 1. After theoretical arrangements we study the localization of restrict units due to Lyapunov capabilities. utilizing those Lyapunov services we will be able to increase a balance thought for self reliant platforms. If we linearize a non-linear method at a set element we can strengthen a balance thought for mounted issues which uses the Frechet by-product on the fastened element. the subsequent subsection bargains with basic linear structures for which we intro duce a brand new notion 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 constructed and investigated by means of Volterra that is 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 admire to balance of its equilibrium through a Lyapunov functionality. the following we examine the discrete model of the model.

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**Additional info for Analysis, Controllability and Optimization of Time-Discrete Systems and Dynamical Games**

**Example text**

We conside r this model as a cont rolled system of t he form Xl (t xz (t + 1) = + 1) = Xl (t ) + aXl(t) - eXl( t )Z - bXl(t) XZ(t) - Xl (t )Ul(t ), x z(t ) - cxz (t ) + dXl (t )x z(t) - xz (t )uz( t ) , t E No. ) d 1 is non-singul ar , if and only if Fu rther we obtain whi ch implies Hence x is an int erior point of 5(x) , if (*) is sat isfied. This example is a spec ial case of the following sit ua t ion: Let m g(x ,u) = f( x) + F (x)u = f(·T) + L li(X)Ui , i =l X E lR. n , U1, " " Um E lR. , wh ere f , Ii E C (lR.

U (N -1)) = 2 C"(k)(XO ,U(O) , . , u(N - 1))T x (C N (xo , u(O) , . . , u( N - 1)) - x ) = 8 m (OC) for all k = 0, . , N - l. For the det ermination of (u(O) , ... , u(N -1)) E n N with (OC) one can apply Marquardt's algorithm: Let (u(O) , . . , u (N - 1)) E n N be chosen. If (OC) is satisfied, then (u(O) , . , u(N - 1)) is t aken as a solut ion of the optimization problem . Otherwise, for every k E {O , .. (k) E jRm is det ermined as solution of the linear system N (2 CU(k) (xo , u(O), .

Tr . Then it follows t hat which implies Al i' A2 IA11= IA21= 1. C21 . 4 -< -h 2 which is impossible. c21:s; h2 If leI ' C2 1> h2 4 ' then IAl,2 1= 1 . ' then A2 < Al < 1 , but A2 < -1 . For t he following we assume t hat Then it follows (as we have seen above) that wh ich implies that the corres po nd ing eigenvectors are linearly ind ep endent. From the rem ark following Theorem 1. 7 we therefor e infer that the mapping 9 : lR 2 ~ lR 2 defined by g( x ,y) = Jj(x * , y*) has (~) as stable fixed point.