Nonlinear Systems Stability Analysis

Appendix A2: Proof of Theorem 3.8

First, a new lemma, which is required for the proof of Theorem 3.8 is presented.

Lemma A2.1:

Under the assumptions of Theorem 3.8, a lower triangular matrix Q = [q_ij]_{m × m} with the same property as A, that is, (3.44) exists, such that:

(a) If ‖x(t)‖ ≥ b ≥ 0, ∀t ∈ [t₀, t₀ + T], for given t₀ and T < + ∞ then for i = 1,…,m:

${\displaystyle \sum _{j=1}^i{q}_{ij}{V}_i(tx)\le -{\text{ϕ}}_2(b)\frac{{(t-t_0)}^{m+1-i}}{(m+1-i)!}}+{\displaystyle \sum _{r=i}^m\frac{{(t-t_0)}^{r-i}}{(r-i)!}{\displaystyle \sum _{j=1}^r{q}_{rj}{V}_j(t_0{x}_0).}}$

(A2.1)

Morever,

$q_{11}{\text{ϕ}}_1(\parallel x\parallel )\le -{\text{ϕ}}_2(b)\frac{{(t-t_0)}^m}{m!}+{\displaystyle \sum _{r=1}^m\frac{{(t-t_0)}^{r-1}}{(r-1)!}}{\displaystyle \sum _{j=1}^r{q}_{rj}{V}_j(t_0{x}_0)\triangleq A_b(t).}$

(A2.2)

(b) A special use of (A2.2) for b = 0 yields:

$q_{11}{\text{ϕ}}_1(\parallel x\parallel )\le {\displaystyle \sum _{r=1}^m\frac{{(t-t_0)}^{r-1}}{(r-1)!}}{\displaystyle \sum _{j=1}^r{q}_{rj}{V}_j(t_0{x}_0)=A_0(t)\forall t\ge t_0.}$

(A2.3)

(c) No trajectory of ẋ= f(tx) escapes to infinity in finite time.

Proof:

The Q_{m× m}matrix is a lower ...