# Asymptotic Behavior of Dynamical and Control Systems under - download pdf or read online

By Lars Grüne

ISBN-10: 3540433910

ISBN-13: 9783540433910

This publication offers an method of the research of perturbation and discretization results at the long-time habit of dynamical and regulate platforms. It analyzes the effect of time and house discretizations on asymptotically good attracting units, attractors, asumptotically controllable units and their respective domain names of points of interest and handy units. Combining strong balance innovations from nonlinear keep an eye on idea, options from optimum keep an eye on and differential video games and techniques from nonsmooth research, either qualitative and quantitative effects are received and new algorithms are constructed, analyzed and illustrated by way of examples.

**Read Online or Download Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization PDF**

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**Additional info for Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization**

**Sample text**

Then—since C is closed— we obtain that x C =: ε > 0. Hence for all k suﬃciently large we obtain xk C ≥ ε/2, thus dist(Ck , C) ≥ ε/2 contradicting the assumption. (iv) follows from (i) and (iii). 17). , we are going to investigate strong dynamical properties. 17). Once these concepts are deﬁned we will proceed to give several characterizations of these robustness properties by means of geometric criteria and Lyapunov functions. Along the way we will clarify the relation between the diﬀerent types of robustness and also to other concepts in the literature.

We now state the relation between the existence of an α-contracting family and the ISDS property. 3 Consider a strongly attracting set A. Then A is ISDS with rate µ, overshoot gain σ and robustness gain γ if and only if there exists an α-contracting family of neighborhoods Bα with dH (Bα , A) ≤ γ(α), and B(σ −1 (γ(α)), A) ∩ B ⊆ Bα ϑ(α, t) = γ −1 (µ(γ(α), τ )). Proof: Assume the existence of the α-contracting family, let x ∈ B, u ∈ U, w ∈ W and t ∈ T+ . e. α ≤ γ −1 (σ( x A )). 2 we obtain Φ(t, x, u, w) ∈ Bα , with α = max{ϑ(α, t), ν¯(w, t)}.

The following example shows that this may indeed happen. 5) with right hand side given by max −x, − x1 + w, x > 0 x=0 f (x) = w, min −x, − x1 + w, x < 0 Clearly, A = {0} is a strongly attracting set for the corresponding unperturbed system where each set B = (−c, c), c > 0 is an attracted neighborhood. Fixing some c > 0 one sees that A = {0} is γ-robust with attracted neighborhood B for each γ of class K∞ with γ(r) ≤ min{r, 1/r} for r ∈ [0, c] and W = [−d, d] where d = min{c, 1/c}. Thus the bigger dist(B, A) = c becomes the smaller W must be.

### Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization by Lars Grüne

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