Download PDF by J. Stoer, C. Witzgall: Convexity & Optimization in Finite Dimensions One

By J. Stoer, C. Witzgall

ISBN-10: 0387048359

ISBN-13: 9780387048352

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Example text

Proof. 1 implies that (θ, ϕ) is a compact RDS. 1. 5. Below we also need the following simple assertion concerning attractors of equivalent RDS (cf. Keller/Schmalfuss [63] and Imkeller/Schmalfuss [59]). 3. Let (θ, ϕ1 ) and (θ, ϕ2 ) be two RDS over the same MDS θ with phase spaces X1 and X2 resp. 4) and there exists a compact random attractor A1 for the RDS (θ, ϕ1 ) in the universe D1 . Then the RDS (θ, ϕ2 ) possesses a random attractor A2 in the universe D2 = {T (ω, D(ω))} : {D(ω)} ∈ D1 . e T (ω, A1 (ω)) = A2 (ω) for all ω ∈ Ω.

1 implies that (θ, ϕ) is a compact RDS. 1. 5. Below we also need the following simple assertion concerning attractors of equivalent RDS (cf. Keller/Schmalfuss [63] and Imkeller/Schmalfuss [59]). 3. Let (θ, ϕ1 ) and (θ, ϕ2 ) be two RDS over the same MDS θ with phase spaces X1 and X2 resp. 4) and there exists a compact random attractor A1 for the RDS (θ, ϕ1 ) in the universe D1 . Then the RDS (θ, ϕ2 ) possesses a random attractor A2 in the universe D2 = {T (ω, D(ω))} : {D(ω)} ∈ D1 . e T (ω, A1 (ω)) = A2 (ω) for all ω ∈ Ω.

Indeed, the deterministic dynamical system in R generated by the equation x˙ = x − x3 has one-point attractor A = {1} in the universe of all compact subsets of R+ \ {0} (see the formula for solutions given in the Introduction). The same formula implies that the interval [−1, 1] is the attractor in the universe of all 42 1. General Facts about Random Dynamical Systems bounded subsets of R and the set {−1, 0, 1} is the attractor in the universe of all one-point subsets of R. ). (iii) Sometimes it is convenient to consider random attractors which do not belong to the corresponding universe (see Crauel [33, 34], Crauel/Debussche/Flandoli [35], Crauel/Flandoli [36]).

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Convexity & Optimization in Finite Dimensions One by J. Stoer, C. Witzgall


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