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8) implies |x(t/2)| ≤ σ(|x(0)|) for all t ≥ 0. Thus, |x(t)| ≤ max β σ (|x(0)|) , t 2 := max β¯ (|x(0)| , t) , γ ◦ ρ ,γ ◦ ρ x x [t/2,∞) [t/2,∞) . 1, there exists a βˆ ∈ KL such that ˆ |x(t)| ≤ β(|x(0)|, t). 12) The GAS property is thus proved. 7. 1 can be carried out by proving the global convergence property. 2. 12). 29) with ρ considered as the gain. 4) means that the composition of the gains of the two subsystems is less than the identity function. Such a condition is called the nonlinear small-gain condition.

Xi ]T and u ∈ R is the control input. It is assumed that, for each i = 1, . . , n, there exists a ψ∆i ∈ K∞ such that |∆i (¯ xi )| ≤ ψ∆i (|¯ xi |). 81) Through small-gain design, the [x1 , . . , xn ]T -system is recursively transformed into a new [e1 , . . 82)   ..  =  ..   .   . 83) where κ1 , . . , κn : R → R are appropriately chosen functions. 84) Interconnected Nonlinear Systems 33 with αV (s) = s2 /2 for s ∈ R+ . For convenience of notation, denote e¯i = [e1 , . . , ei ]T and en+1 = xn+1 − κn (en ) with xn+1 = u.

47) holds if V (x) ≤ max {α(|x|) : x ∈ Ω0 } , V (x) ≥ α ◦ χ( u ∞ ) := γ( u ∞ ). 56) given later for ISS-Lyapunov functions. 2]. , for any initial state x(0) = x0 and any measurable and locally essentially bound input u, the solution x(t) is defined for all t ≥ 0. Moreover, it has the uniformly bounded-input bounded-state (UBIBS) property. 50) for all t ≥ 0. 3 for class KL functions. 50) by defining σ1 (s) = β(s, 0) and σ2 (s) = γ(s) for s ∈ R+ . More importantly, ISS is equivalent to the conjunction of UBIBS and AG [250].

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