A Brief Review of the Development of Qualitative Control by Marchenko V. M.

By Marchenko V. M.

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Differential Equations & Control Theory

According to papers on the Intl Workshop on Differential Equations and optimum keep an eye on held lately at Ohio collage, Athens.

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7) (3) there exists a non empty compact K ⊆ X such that lim sup ρ(xt, K) = 0. t→+∞ x∈B Proof. Let us show that from 1. follows 2. Let {xk } ⊆ B and tk → +∞, then according to 1. the sequence {xk tk } can be considered convergent. Assume x = lim xk tk , then x ∈ ω(B) and consequently ω(B) = ∅. Let us show ω(B) is k→+∞ compact. Let εk ↓ 0 and {yk } ⊆ ω(B), then there exist xk ∈ B and tk ≥ k such that ρ(xk tk , yk ) < εk . According to the condition (1), the sequence {xk tk } is relatively compact and since εk ↓ 0, {yk } also is relatively compact.

The theorem is proved. 17 For the compact dissipative dynamical system (X, T, π) to be local dissipative, it is necessary and sufficient that its Levinson center J would be uniformly attracting set. Proof. Let (X, T, π) be local dissipative, J be its center of Levinson and p ∈ J. 29) 26 Global Attractors of Non-autonomous Dissipative Dynamical Systems holds. By compactness of J from its open covering {B(p, δp )| p ∈ J}, it is possible to extract finite sub-covering {B(pi , δpi )| i ∈ 1, m}. 9 there exists γ > 0 such that B(J, γ) ⊂ U {B(pi , δpi )| i ∈ 1, m}.

26 The dynamical system (X, T, π) we will call: − locally completely continuous if for every point p ∈ X there exist δ = δ(p) > 0 and l = l(p) > 0 such that π l B(p, δ) is relatively compact; − weakly dissipative if there exist a nonempty compact K ⊆ X such that for every ε > 0 and x ∈ X there is τ = τ (ε, x) > 0 for which xτ ∈ B(K, ε). In this case we will call K weak attractor. Note that every dynamical system (X, T, π) defined on the locally compact metric space X is locally completely continuous.

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