Explicit Stability Conditions for Continuous Systems: A by Michael I. Gil

By Michael I. Gil

Particular balance stipulations for non-stop structures 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 presents new instruments for experts up to speed process idea and balance concept 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 services is advised and diverse sessions of platforms are investigated from a unified point of view.

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Hence, x(t) ≤ x(0) U (t, 0) + q(J) x(t) + t 0 U (t, s) m(s) x(s) ds. This inequality implies y(t) ≤ U (t, 0) + q(J)y(t) + t 0 U (t, s) m(s)y(s)ds ≤ U (t, 0) + η(J) sup y(t), t≥0 where x(t) . 2) ensures the required result. 7 Integrally Small Perturbations of Autonomous Systems Assume that A(t) ≡ A is a constant matrix and consider the equation dx/dt = Ax + B(t)x. 1) Suppose that m0 := sup AJ(t) − J(t)(A + B(t)) < ∞ t≥0 and n−1 √ q(J) + m0 k=0 g k (A) < 1. 5. 2) hold. 1) is stable. 1) is equal to exp[A(t − s)].

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>? ;>! ;>? 38%  =I$ F> (  =I$ R> =R$ F>$ 8"  =R$ F> (   =F$ R>$ H1% C%6H3:8  =I$ F> (  =I>  =F> =I$ F R> 3D M63"? * =I>  3D  :8DH8H 7HC3O! ””˜ª¯¦j­h\ª¯½½s˜ ^Ê2”¡|†vc¡{=c¡ƒFvª c¡ƒ|†ªc|]cƒRc¡vªÓääx 28 2. Perturbations of Linear Systems U (t) = etA and U (t, s) = U (t − s) = e(t−s)A . 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).

Put W (t) = 1 π ∞ −∞ (−iIω − A∗ (t))−1 (iIω − A(t))−1 dω. 2) 52 3. Systems with Slowly Varying Coefficients Then W (t) is a solution of the equation. W (t)A(t) + A∗ (t)W (t) = −2I. ) is real, (W (t)A(t)h, h) = −(h, h) (h ∈ Rn ). 1) by W (t) and doing the scalar product, we get (W (t)x(t), ˙ x(t)) = (W (t)A(t)x(t), x(t)) = −(x(t), x(t)). 4) Since d ˙ (t)x(t), x(t)) + (W (t)x(t), x(t)) (W (t)x(t), x(t)) = (W (t)x(t), ˙ x(t)) + (W ˙ = dt ˙ (t)x(t), x(t)), 2(W (t)x(t), ˙ x(t)) + (W it can be written d ˙ (t)x(t), x(t)) + 2(W (t)x(t), (W (t)x(t), x(t)) = (W ˙ x(t)).

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