New PDF release: Asymptotic Behavior of Dynamical and Control Systems under

By Lars Grüne

ISBN-10: 3540433910

ISBN-13: 9783540433910

This ebook presents an method of the research of perturbation and discretization results at the long-time habit of dynamical and keep watch over structures. It analyzes the effect of time and house discretizations on asymptotically sturdy attracting units, attractors, asumptotically controllable units and their respective domain names of points of interest and on hand units. Combining strong balance thoughts from nonlinear regulate idea, thoughts from optimum keep watch over and differential video games and strategies from nonsmooth research, either qualitative and quantitative effects are acquired and new algorithms are constructed, analyzed and illustrated by way of examples.

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Extra info for Asymptotic Behavior of Dynamical and Control Systems under Pertubation and Discretization

Example text

We obtain the desired distance dH (Bα , A) ≤ γ˜ (α) since for α ∈ [αi+2 , αi+1 ] we have dist(Bα , A) ≤ dist(Bi−1 , A) ≤ δi−1 = γ˜ (γ(δi+2 )) = γ˜ (αi+2 ) ≤ γ˜ (α). 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.

V) Let x ∈ C. Since C is open there exists ε > 0 such that x C c = ε. , x ∈ Ckc implies dist(Ckc , C c ) ≥ ε. Hence x ∈ Ck for all k ∈ N with dist(Ckc , C c ) < ε. If C is bounded then for each ε > 0 we find a finite number of points yl ∈ C, l = 1, . . , m such that dist(C, {y1 , . . , yl }) ≤ ε. Since by the first assertion we obtain {y1 , . . , yl } ⊂ Ck for all k sufficiently large the desired convergence follows. If C c is bounded then also B(ε, C c ) is bounded for each ε > 0. Fix ε > 0 and define Cε := C ∩ B(ε, C c ).

3 √ we can assume that the gains are C ∞ by enlarging them by the factor 1 + ε. 7 applied with ε1 such that (1 + ε1 ) ≤ 1 + ε, one easily verifies the assertion. Conversely, if we have Vε as in the assumption, then we set V = γ(Vε ). 1 we obtain ISDS with robustness gain (1 + ε)γ, and hence (1 + ε)γ-robustness of A for each ε > 0. 3 this implies γ-robustness of A. 5) and an open set O and a function V : cl O → R+ 0 which satisfies inf x∈∂O V (x) =: α0 > 0 and which is a viscosity supersolution of the equation inf {−DV (x)f (x, u, w) − g(V (x))} ≥ 0 u∈U, w∈W : w 0 for a > 0.

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

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