Remarks on the Lp-input converging-state property

Let X ⊂ R^N and consider a system x˙ = f(x,u), f : X × R^M → R^N, with the property that the associated autonomous system x˙ = f (x,0) 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. Assume further that, for each compact K ⊂ X, there exists k > 0 such that |f(z,v) - f(z,0)| ≤ k|v| for all (z,v) ∈ × R^M. A simple proof is given of the following L^p-input converging-state property: if u ∈ L^p for some p ∈ [1,∞) and x has an ω-limit point in A, then x approaches C.