Download e-book for iPad: Asymptotic Behavior of Dynamical and Control Systems under by Lars Grüne

By Lars Grüne

This booklet offers an method of the examine of perturbation and discretization results at the long-time habit of dynamical and keep an eye on structures. It analyzes the effect of time and house discretizations on asymptotically strong attracting units, attractors, asumptotically controllable units and their respective domain names of points of interest and on hand units. Combining powerful balance thoughts from nonlinear keep an eye on thought, strategies from optimum keep watch over and differential video games and strategies from nonsmooth research, either qualitative and quantitative effects are got and new algorithms are built, analyzed and illustrated by means of examples.

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

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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.

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 ). From the assumption we know that dist(Ckc , C c ) < ε for k sufficiently large, hence Ckc ⊂ B(ε, C c ) and consequently Ck ⊇ B(ε, C c )c = C \ Cε implying dist(C, Ck ) = max{dist(C \ Cε , Ck ), dist(Cε , Ck )} = dist(Cε , Ck ).

Hence setting t1 = t−h and t2 = h for the first assertion and t1 = t, t2 = h for the second implies the assertion by the continuity of µ in t. We will occasionally assume that the comparison functions σ, γ and µ in the ISDS definition are smooth on R+ or R+ × R, respectively, with nonvanishing derivatives. 3 implies that if we have ISDS with non–smooth functions this assumption can be satisfied by slightly enlarging these functions. 3 Geometric Characterizations In this section we will derive a geometric condition for the ISDS property of some attracting set A.

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