Asymptotic Behavior of Dynamical and Control Systems under by Lars Grüne

By Lars Grüne

This ebook offers an method of the learn 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 good attracting units, attractors, asumptotically controllable units and their respective domain names of sights and available units. Combining powerful balance techniques from nonlinear regulate concept, innovations from optimum keep watch over and differential video games and techniques from nonsmooth research, either qualitative and quantitative effects are bought and new algorithms are constructed, analyzed and illustrated through examples.

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

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C Springer-Verlag Berlin Heidelberg 2002 28 3 Strongly Attracting Sets Let us now define attracting sets. In many references in the literature these sets are supposed to be compact. Here we allow a bit more generality since we assume that A is closed and either A or Ac is bounded. The second case will turn out to be useful later when considering domains of attraction and reachable sets in Chapter 7. , in [70]. In general, for arbitrary closed sets things can go wrong if we do not have these properties.

Then we define the sets Bα := x ∈ Rn Φ(t, x, u, w) A ≤ max{µ(γ(α), t), ν(w, t)} for all u ∈ U, w ∈ W, t ∈ T+ 0 for all α ≥ 0. Obviously Bα ⊆ Bα for α ≤ α. The assertions on the distance are immediate and imply, in particular, that the sets shrink down to A. The fact that we can choose ϑ(α, t) = γ −1 (µ(γ(α), τ )) follows directly from the construction. It will turn out to be useful to have a characterization via a strictly αcontracting family of neighborhoods. We can obtain this at least with respect to O = R(B) by slightly relaxing the bounds in the previous proposition.

It would seem more natural to consider the equation multiplied by −1. However, this is needed in order to be consistent with the usual definition of viscosity supersolutions, cf. Appendix A. 5. 1) if and only if it is a viscosity supersolution of inf {−DV (x)f (x, u, w) − g(V (x))} ≥ 0. u∈U, w∈W : w <γ −1 (V (x)) Proof: Let V satisfy the inequality and fix x ∈ R(B), u0 ∈ U and w0 ∈ W with w0 < V (x). Consider the constant perturbation functions u(t) ≡ u0 and w(t) ≡ w0 . Then by continuity there exists t > 0 such that µ(V (x), τ ) ≥ γ( w0 ) = ν(τ, w) for all τ ∈ [0, t].

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