Remarks on the Lp-input converging-state property

Consider a system ẋ = f(x) + g(x)u (f and g locally Lipschitz) with the property that the associated autonomous system ż = f(z) has an asymptotically stable compactum C with region of attraction A. Assume that x is a solution of the former, defined on [0,∞), corresponding to an input function u. A simple proof is given of the following L -input converging-state property: if u ∈ L for some p ∈ [1,∞) and x has an ω-limit point in A, then x approaches C.