Nonlinear Systems Stability Analysis

Corollary 2.4 [m6]:

Let the nonlinear system (2.199) be Δ-homogeneous of order k≥0, that is, f ∈ n_k, ρ(x) be a Δ-homogeneous norm, and S^n–1 be the corresponding Δ-homogeneous sphere using (2.278). If there exist V₁(x), V₂(x),…, V_m+1(x) functions of class C¹, such that the following conditions are satisfied:

(i) V_i ∈ H_p for i = 1,….,m+1 and for some p ∈ N.

(ii) V₁(u) > 0 and V_m+1(u) < 0, ∀u ∈ S^n–1.

(iii) For the time derivatives along the solutions of (2.199):

$[\begin{array}{l}{a}_{11}00\cdots 0\\ a_{21}{a}_{22}00⋮\\ ⋮a_{ij}\ddots 00\\ a_{m-1,1}\cdots a_{m-1,m-1}0\\ a_{m1}\cdots \end{array}$