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

By Lars Grüne

This booklet offers an method of the research of perturbation and discretization results at the long-time habit of dynamical and keep watch over platforms. It analyzes the influence of time and area discretizations on asymptotically sturdy attracting units, attractors, asumptotically controllable units and their respective domain names of points of interest and available units. Combining strong balance ideas from nonlinear keep watch over idea, ideas from optimum keep watch over and differential video games and strategies from nonsmooth research, either qualitative and quantitative effects are bought and new algorithms are constructed, analyzed and illustrated via examples.

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

Example text

Thus setting Vε (x) = γ(α(x)) for x ∈ R(B)\A and Vε (x) = 0 on A we obtain a continuous function. 2. 2). (ii)⇒(iii): Consider the function V : R(B) → R+ 0 deﬁned pointwise by V (x) := lim supε 0 Vε (x). Obviously the ﬁrst two inequalities are satisﬁed. For the third, pick x ∈ R(B), u ∈ U, w ∈ W and t ∈ T+ , and consider the 46 3 Strongly Attracting Sets point y = Φ(t, x, u, w). Then for each δ > 0 there exists ε ∈ (0, δ) such that V (y) ≤ Vε (y) + δ and Vε (x) ≤ V (x) + δ. This yields V (y) ≤ max{µ(Vε (x), (1 + ε)t), µ(w, t)} + δ ≤ max{µ(V (x) + δ, (1 + δ)t), µ(w, t)} + δ which implies the assertion as δ → 0.

By Gronwall’s inequality for continuous time systems or by induction for discrete time systems (observe that all trajectories under consideration stay inside some compact subset on which we have a uniform Lipschitz constant for our system) we furthermore obtain Φ(tn , y, un , wn ) A ≥ Φ(tn , y , un , wn ) A −C y−y for some C > 0. 2). (ii) Let α > 0, x ∈ Bα , consider α = ϑ(α, t) = γ −1 (µ(γ(α), (1 − ε)t)) and let x ∈ Bα and w ∈ W with w(τ ) ≤ ϑ(α, τ ) for almost all τ ∈ [0, t]. Then the deﬁnition of Bα implies Φ(t + s, x, u, w) ≤ ρε (µ(γ(α), t + s)) max{µ(γ(α), (1 − ε)(t + s)), ν(w, t + s)} ≤ ρε (µ(γ(α ), s)) max{µ(γ(α ), (1 − ε)s), µ(ν(w, t), s), ν(w(t + ·), s)} ≤ ρε (µ(γ(α ), s)) max{µ(γ(α ), (1 − ε)s), ν(w(t + ·), s)} for all u ∈ U since µ(ν(w, t), s) ≤ µ(γ(α ), s)) by the choice of w.

Note that forward completeness is trivially satisﬁed for our discrete time systems, but, since we did not make any assumptions on the u– and w–dependence of Φh , it does not necessarily imply boundedness of Φh,α (t, B). The reachable set R(B) is deﬁned analogously to the continuous time case, now for all t ≥ 0 with t ∈ hZ. 12), of course, only needs to be satisﬁed with respect to I = hN0 . 22 2 Setup and Preliminaries In what follows we will frequently make statements simultaneously for discrete and continuous time systems.