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**Example text**

9 are such that M < Mg or N < N g . Let R = cl Ω≤ Mg (Ug ) \ Ω≤ M (U ) . 8). 12 is to show the absence of ω-limit sets in R. This is achieved by exhibiting a contradiction between Fig. 4 Results 45 its existence and the assumption that, for every y ∈ R, div h(y) = 0 and h(y) = 0. 12 is provided in Sect. 3. 9, when Ml < Mg and Nl < N g . 12 are illustrated. 8ρx (x). 3 2 Let the functions R x → V (x) = |x| ∈ R≥0 , R z → W (z) = |z| ∈ R≥0 . The Dini derivative (cf. 39) of V along the vector field f yields the inequality D +f V (x, z) ≤ −ρx (V (x)) + W (z), which holds, for every (x, z) ∈ R × R.

Consequently, the set R = cl Ω≤ Mg (Ug ) \ Ω≤ M (U ) is nonempty. 16. Fourth Part. 8) that converge to the set Z ⊂ R (cf. 16) are shown to have also measure zero. For every t, let Y (t, Z) = {Y (t, z) : t ∈ dom(z), z ∈ Z}, where dom(z) is the maximum time interval where Y (t, z) exists. Since Z is positively invariant, for every t1 , t2 ∈ dom(z), t1 < t2 ≤ 0 ⇒ Y (t2 , Z) ⊂ Y (t1 , Z). This inclusion implies that4 Y := {Y (t, Z)} = t≤0 {Y (t, Z) : t ≤ l}. l∈Z<0 Hence, the set Y is a countable union of images of Z by the flow.

If x ∈ C2,l , then Eq. 11b) implies that ϕ2,l (x) = ϕg (x). 4, the set A is globally practically asymptotically stable for the closed-loop system h (ϕg ). Moreover, Eqs. 36) imply that the inclusion A ⊂ D2,l holds. ii. 11c), g2,l (x) = {(1, l)} and, after the jump, the local hybrid feedback law is selected. Since the value of x does not change during a jump, x ∈ D2,l after a jump. Moreover, Eqs. 36) imply that the inclusion D2,l ⊂ Ω