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Particular balance stipulations for non-stop platforms offers with non-autonomous linear and nonlinear non-stop finite dimensional structures. specific stipulations for the asymptotic, absolute, input-to-state and orbital stabilities are mentioned. This monograph offers new instruments for experts up to the mark procedure conception and balance idea of normal differential equations, with a unique emphasis at the Aizerman challenge. a scientific exposition of the method of balance research in line with estimates for matrix-valued features is advised and numerous sessions of structures are investigated from a unified standpoint.
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Extra info for Explicit Stability Conditions for Continuous Systems: A Functional Analytic Approach
Example text
4) Moreover, U (t, s) − U1 (t, s) ≤ M e−ν(t−s) (eM t s A(τ )−A1 (τ ) dτ − 1). 2, only to check that the right part of that inequality is a solution of the equation t η(t) = M e−ν(t−s) + M s e−ν(t−τ ) A(τ ) − A1 (τ ) η(τ )dτ (s ≤ t). 2)). 1) be uniformly exponentially stable and let ∞ A(τ ) − A1 (τ ) dτ < ∞. 1) is also uniformly exponentially stable. 1) satisfies the inequality U (t, s) − I ≤ M e−ν(t−s) (e t s A(τ ) dτ − 1) (s ≤ t). 1), respectively. For some positive T ≤ ∞, let the condition J ≡ sup 0≤t≤T t 0 U (t, s) A(s) − A1 (s) ds < 1 be fulfilled.
1) satisfies the inequality U (t) ≤ (1 − WA )−1 sup e t 0 α(s)ds t≥0 n−1 k=0 1 [ k! t 0 N (V (t1 ))dt1 ]k . 1) where B(t) is a variable n × n-matrix. Introduce the notations J(t) = t 0 t q(J) := sup t≥0 and B(s)ds, 0 B(s)ds , m(t) = A(t)J(t) − J(t)(A(t) + B(t)) . 2) 2. Perturbations of Linear Systems 36 hold. 3) is true. 1) in the form x(t) = U (t, 0)x(0) + Evidently, t 0 U (t, s)B(s)x(s)ds. 4) t d [U (t, s)J(s)x(s)]ds = J(t)x(t), ds 0 since U (t, t) is the unit matrix. 3), we obtain J(t)x(t) = t 0 U (t, s)[−A(s)J(s)x(s)+ B(s)x(s) + J(s)x(s)]ds.
5) with a given piecewise continuous function f : R+ → Cn can be obtained in the form u(t) = U (t, s)u(s) + t s U (t, τ )f (τ )dτ (t ≥ s ≥ 0). 1), then for any complex number α, Uα (t, s) = eα(t−s) U (t, s) is the evolution operator of the equation u(t) ˙ = (A(t) + αI)u(t) (t ≥ 0). 7) The Cauchy operator can be represented in the form U (t) = I+ t 0 ∞ A(t1 )dt1 + k=2 t 0 0 tk ... dt1 . 1) is the uniform boundedness of its Cauchy operator: sup U (t) < ∞. 1). 1) satisfies the estimate u(t) ≤ M u(0) (t ≥ 0).
