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Download PDF by Lars Grüne: Asymptotic Behavior of Dynamical and Control Systems under

By Lars Grüne

ISBN-10: 3540433910

ISBN-13: 9783540433910

This publication presents an method of the examine of perturbation and discretization results at the long-time habit of dynamical and keep an eye on platforms. It analyzes the impression of time and area discretizations on asymptotically good attracting units, attractors, asumptotically controllable units and their respective domain names of sights and accessible units. Combining strong balance strategies from nonlinear keep watch over concept, thoughts from optimum keep watch over and differential video games and strategies from nonsmooth research, either qualitative and quantitative effects are received and new algorithms are built, analyzed and illustrated via examples.

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Additional resources for Asymptotic Behavior of Dynamical and Control Systems under Pertubation and Discretization

Example text

From the construction of the Bα we then obtain Φα(t) (t, Bα ) ⊆ Φαi+1 (t, Bα ) ⊆ Φαi+1 (t + t(α), Bi−1 ) ∪ Bi = Bϑ(αi ,t+t(α)) = Bϑ(α,t) for α(t) = ϑ(α, t), α ∈ [αi+2 , αi+1 ] and t ∈ [t(α), ∆ti−1 ]. This shows property (iv’). 5, the robustness gain γ in the formulation of γ-robustness can chosen “almost” optimal with the only restriction being the possible discontinuity of the optimal gain. In the ISDS formulation the situation is different. Here we have a tradeoff between µ and γ, which also appears when a continuous optimal gain γ for the γ-robustness property exists.

The next lemma shows some relations between distances and the Lim sup for sets. 6 Consider a sequence of sets Ck ⊂ Rn and a set C ⊂ Rn . Then the following implications hold. (i) If C is closed and limk→∞ dist(C, Ck ) = 0, then C ⊆ Lim supk→∞ Ck . (ii) If C is open and limk→∞ dist(Ckc , C c ) = 0, then C ⊆ int Lim supk→∞ Ck . (iii) If C is closed and limk→∞ dist(Ck , C) = 0, then C ⊇ Lim supk→∞ Ck . (iv) If C is closed and limk→∞ dH (Ck , C) = 0, then C = Lim supk→∞ Ck . Proof: (i) Let x ∈ C. Then the dist assumption yields the existence of xk ∈ Ck such that xk → x.

42 3 Strongly Attracting Sets In case (ii), for any x ∈ B with x A ≤ σ ˜ −1 (˜ γ (α)) for some α ∈ [αi+2 , αi+1 ] ˜ ( x A ) ≤ γ˜ (αi+1 ), hence we have β( x A , 0) = σ β( x A , 0) ≤ γ(αi ) = δi which by the definition of the Bi implies x ∈ Bi , hence x ∈ Bαi+2 ⊆ Bα which yields the desired inclusion B(˜ σ −1 (˜ γ (α)), A) ⊆ Bα . It remains to show the α-contraction. This follows by setting ϑ(αi , t) = αi − t(αi − αi+1 ) ∆ti−1 for t ∈ [0, ∆ti−1 ] and extending this map for all t ∈ R via ϑ(αi , t) = ϑ(αk , t − Ti,k ) for all t ∈ [Ti,k , Ti,k+1 ] with Ti,k given inductively by Ti,i = 0, Ti,k+1 = Ti,k + ∆tk−1 for k ≥ i and Ti,k−1 = Ti,k − ∆tk−2 for k ≤ i.

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


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