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

By Lars Grüne

This booklet presents 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 influence of time and area discretizations on asymptotically good attracting units, attractors, asumptotically controllable units and their respective domain names of sights and on hand units. Combining powerful balance ideas from nonlinear keep watch over thought, options from optimum keep an eye on and differential video games and strategies from nonsmooth research, either qualitative and quantitative effects are acquired and new algorithms are constructed, analyzed and illustrated via examples.

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

Example text

Then by continuity there exists t > 0 such that µ(V (x), τ ) ≥ γ( w0 ) = ν(τ, w) for all τ ∈ [0, t]. Now let ξ ∈ D− V (x). Then for small τ > 0 we obtain ξf (x, u0 , w0 ) = ξ(ϕ(τ, x, u, w) − x) o(τ ) + τ τ and hence V (ϕ(τ, x, u, w)) − V (x) τ µ(V (x), τ ) − V (x) ≤ lim sup = −g(V (x)). τ τ →0 ξf (x, u0 , w0 ) ≤ lim sup τ →0 This shows the claim. Let conversely V be a viscosity supersolution of the given inequality and fix some t > 0. 4 applied with b = V (x), a = µ(V (x), t) and W = Wγ −1 (µ(V (x),t)) we obtain V (ϕ(t, x, u, w)) ≤ µ(V (x), t) for all u ∈ U and all w ∈ W with γ( w(τ ) ) ≤ µ(V (x), t) for almost all τ ∈ [0, t].

1 Now we define our contracting family of neighborhoods as follows: For each α ∈ [αi+2 , αi+1 ] we set Bα := Φαi+1 αi+1 − α ∆ti , Bi−1 αi+1 − αi+2 ∪ Bi if Φ is a continuous time system, and Bα := Φαi+1 αi+1 − α ∆ti αi+1 − αi+2 , Bi−1 ∪ Bi h if Φ is a discrete time system with time step h, where [r]h denotes the largest value s ∈ hZ with s ≤ r. This construction implies Bαi = Bi−2 and Bα ⊆ Bα for all 0 < α ≤ α . 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).

Now for all α ≤ α0 := ρ−1 (e−LT (r0 ) min{r0 , β(r0 , 0)}/4) consider the sets Dα := B(r(α), A), where r(α) is chosen such that eLT (r(α)) ρ(α) ≤ r(α)/4. Observe that both α0 and r(·) only depend on β, ρ, r0 and L, and r(α) → 0 as α → 0. We set W = B(α0 , 0). Then by Gronwall’s Lemma we obtain for t ≤ T ( x A ) Φ(t, x, u, w) A ≤ β( x A , t) + eLt ρ(α) for all u ∈ U and all w ∈ Wα , which implies that for each point x ∈ Dα and all u ∈ U we obtain Φ(T (r(α)), x, u, w) ∈ Dα and Φ(t, x, u, w) A ≤ β(r(α), 0) + r(α)/4 for all t ∈ [0, T (r(α))] ∩ T.

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