Download e-book for iPad: Explicit Stability Conditions for Continuous Systems: A by Michael I. Gil

By Michael I. Gil

Particular balance stipulations for non-stop platforms bargains with non-autonomous linear and nonlinear non-stop finite dimensional structures. particular stipulations for the asymptotic, absolute, input-to-state and orbital stabilities are mentioned. This monograph offers new instruments for experts on top of things procedure concept and balance thought of standard differential equations, with a distinct emphasis at the Aizerman challenge. a scientific exposition of the method of balance research in accordance with estimates for matrix-valued capabilities is advised and diverse sessions of structures are investigated from a unified perspective.

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Extra info for Explicit Stability Conditions for Continuous Systems: A Functional Analytic Approach

Example text

Let C(t) be a τ -periodic matrix. Take A = τ −1 τ 0 C(t)dt, B(t) = C(t) − A. 4) AJ(t) − J(t)(A + B(t)) . 1). 6). 38 2. 6) wih real τ -periodic matrix C(t) = (cjk (t))2j,k=1 . That is n = 2. 5). That is, τ ajk = τ −1 0 cjk (t)dt, B(t) = C(t) − A. 1, g(A) ≤ |a12 − a21 |. 1, the condition q(J) + |a12 − a21 | m0 )<1 (1 + |α(A)| |α(A)| provides the stability of the considered periodic system. 7- =36'3 ;#6! * :8D3"%C 38 + H1% %ALH3:8 O=I> $ ( =I>O=I> =I R>$ =;>;> N1%C% =I> 3D  MC36% <3%%N3D%2:8H38L:LD LCN3HQ 8  8 2 7HC3O?

Obviously, h(t) 2 =− ∞ t d h(s) ds 2 ds = −2 ∞ t h(s) Taking into account that | d h(s) | ≤ dh(s)/ds , ds older’s inequality. ✷ we get the required result due to H´ d h(s) ds. ds 8- 26926#*(3 * #(2 =36'3 8-+ :*&96#*( ,26*23 %H L=I> %  D:6LH3:8 :* H1% %ALH3:8 L=I> $ ( =I>L=I> =I R> =;>;> N3H1  <3%%N3D% :8H38L:LD 8  827HC3O =I>? ;>? 1% %M:6LH3:8 :<%CH:C 1D H1% *:66:N380 ( > :C%:M%C! =  =I$ F> ( =I> =I$ F>> =I =;>K> =  =I$ F> (  =I$ F>=F> =F =;>I> 8"  =I$ F> =F$ J > (  =I$ J > =I$ F$ J R>> 1% 6HH%C (   =I$ F>?

1). 8) ρ := − sup α(A(t)) > 0. )3/2 χ ˜0 := sup p˜0 (t). 1 The Freezing Method 41 Simple calculations show that ∞ 0 where t˜ p0 (t)dt = ζ1 , n−1 ζ1 := k=0 (k + 1)v k √ . 9) hold. In addition, let q0 ζ1 < 1. 1) is exponentially stable. 1) satisfies the inequality U (t, s) ≤ χ ˜0 (t, s ≥ 0). 10) where n−1 P (z) = k=0 (k + 1)v k n−k−1 √ . z k! Recall that I is the unit matrix. 8) hold. In addition, let the matrix A(t) + ( + z(q0 , v))I be a Hurwitz one for an > 0 and all t ≥ 0. 1) is exponentially stable.

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