Since we consider cascades for which the origin of (2.37) is UGAS it follows from converse Lyapunov theorems (see e.g. [24, 20, 26]) that there exists a positive definite proper Lyapunov function V(t, x1) with a negative definite bound on the total time derivative. Thus consider the assumption below which we divide in two parts for ease of reference.
Assumption 1
a) The system (2.37) is UGAS.
b) There exists a known C1 Lyapunov function V(t, x1),α1,α2∈ K∞, a positivesemidefinitefunctionW(x1)a continuous non-decreasing function α4(ã), such that
α1(|x1|)≤V(t, x1)≤α2(|x1|) (2.38) V˙(2.37)(t, x1)≤ −W(x1) (2.39)
∂V
∂x1 ≤α4(|x1|). (2.40)
Remark 2.4.We point out that to verify Assumption 1a) it is enough to have a Lyapunov function with only negative semidefinite time derivative. Yet, we have the following.
Proposition 2.2.Assumption 1a implies the existence of a Lyapunov func- tionV(t, x1), functionsα¯1, α¯2∈ K∞ andα¯4∈ K such that,
¯
α1(|x1|)≤ V(t, x1)≤α¯2(|x1|) (2.41)
V˙(2.37)(t, x1)≤ −V(t, x1) (2.42)
∂V
∂x1 ≤α¯4(|x1|). (2.43) Sketch of proof. The inequalities in (2.41), as well as the existence of ¯α3∈ K such that,
V˙(2.37)(t, x1)≤ −α¯3(|x1|), (2.44) follow from [26, Theorem 2.9]. The property (2.43) follows along the lines of proofs of [19, Theorems 3.12, 3.14] and [24], using the assumption that f1(t, x1) is continuously differentiable and locally Lipschitz. Finally, (2.42) follows using (2.44) and [48, Proposition 13]. See also [64].
We stress the importance of formulating Assumption 1b) with the less restrictive conditions (2.38), (2.39) since, for some applications, UGAS for (2.37) may be established with a Lyapunov function V(t, x1) with a nega- tive semidefinite derivative. For autonomous systems, e.g., using invariance principles (such as Krasovskii-LaSalle’s) or, for non-autonomous systems, via Matrosov’s theorem [49]. See Section 2.4 and [25] for some examples.
We also remark that the same observations apply to [42, Theorem 2] where we overlooked this important issue, imposing the unnecessarily restrictive as- sumption of negative definiteness on ˙V(2.37)(t, x1). This is implicitly assumed in the proof of that Theorem. In Section 2.3.4 we present a theorem which includes the same result.
Further, we assume that
(Assumption 2) the subsystem Σ2 is UGAS.
Let us stress some direct consequences of Assumption 2 in order to introduce some notation. Firstly, it means that there exists β ∈ KL such that, for all (t◦, x◦) ∈R≥0 ×Rn,
|x2(t; t◦, x2◦)| ≤β(|x2◦|, t −t◦), ∀t ≥t◦ (2.45) and hence, for each r > 0
|x2(t; t◦, x2◦)| ≤c :=β(r,0) ∀ |x2◦|< r . (2.46) Secondly, note that due to [30, Lemma B.1] (2.36) implies that there ex- ist continuous functions θ1 : R≥0 → R≥0 and α5 : R≥0 → R≥0 such that
|g(t, x)| ≤θ1(|x2|)α5(|x1|). Hence, under Assumption 2, we have that for each r > 0 and for all t◦≥0
|g(t, x(t; t◦, x◦))| ≤cg(r)α5(|x1(t; t◦, x◦)|), ∀ |x2◦|< r , ∀t ≥t◦ (2.47) wherecg(ã) is the classK function defined bycg(ã) :=θ1(β(ã,0)).
Having laid the basic assumptions, we are ready to present some stability theorems. The following lemma extends the fact that GAS + GAS + BS ⇒ GAS, to the nonautonomous case. This is probably the most fundamental result of this chapter and therefore, we provide the proof of it.
Lemma 2.1 (UGAS + UGAS + UGB⇔ UGAS). The cascade (2.35) is UGAS if and only if the systems (2.35b) and (2.37) are UGAS and the solutions of (2.35) are uniformly globally bounded (UGB).
Proof .(Sufficency). By assumption (from UGB), for each r >0 there exists
¯
c(r)>0 such that, if|x◦|< rthen|x(t;t◦, x◦)| ≤¯c(r). Consider the function V(t, x1) as defined in Proposition 2.2. It’s time derivative along the trajectories of (2.35a) yields, using (2.43), (2.42), (2.47), and definingv(t) :=V(t, x1(t)),
˙
v(2.35a)(t)≤ −v(t) +c(r)|x2(t)| , (2.48) where c(r) := cg(r)¯α4(¯c(r))α5(¯c(r)). Therefore, using (2.45) and defining v◦:=v(t◦), we obtain that for allt◦≥0,|x◦|< randv◦<α¯2(r),
˙
v(2.35a)(t;t◦, v◦)≤ −v(t;t◦, v◦) + ˜β(r, t−t◦) (2.49) where ˜β(r, t−t◦) :=c(r)β(r, t−t◦).
Let τ◦ ≥t◦ and multiply bye(t−τ◦) on both sides of (2.49). Rearranging the terms we obtain
d
dt v(t)e(t−τ◦) ≤β(r, t˜ −t◦)e(t−τ◦), ∀t≥τ◦. (2.50) Then, integrating on both sides from τ◦ to t and multiplying bye−(t−τ◦) we obtain that
v(t)≤v(τ◦)e−(t−τ◦)+ t
τ◦
β˜(r, s−t◦)e−(t−s)ds , ∀t≥τ◦. (2.51) Next, letτ◦=t◦ (2.51) implies that
v(t)≤v(t◦) + ˜β(r,0) 1−e−(t−t◦) ≤α¯2(r) + ˜β(r,0), ∀t≥t◦. (2.52) Hence|x1(t)| ≤α¯−11 (¯α2(r) + ˜β(r,0)) =:γ(r). Uniform global stability follows observing that γ ∈ K∞ and that the subsystem Σ2 is UGS by assumption.
On the other hand, for each 0 < ε1 < r, let T1(ε1, r) ≥ 0 be such that β(r, T˜ 1) = ε1/2 (T1 = 0 if ˜β(r,0)≤ε1/2 ). Then, defining nowτ◦=t◦+T1, (2.51) also implies that
v(t)≤v(t◦+T1)e−(t−t◦−T1)+ t
t◦+T1
β(r, T˜ 1)e−(t−s)ds , ∀t≥t◦+T1.
This, in vue of (2.52), implies that
v(t) ≤ α¯2(r) + ˜β(r,0) e−(t−t◦−T1)+ε1
2 , ∀t≥t◦+T1.
It follows thatv(t)≤ε1for allt≥t◦+T withT :=T1+ ln 2[¯α2(r)+ ˜ε1β(r,0)] . Finally, definingε:= ¯α2(ε1) we conclude that |x1(t)| ≤ε for allt ≥t◦+T. The result follows observing that ε1 is arbitrary, ¯α2 ∈ K∞, and that Σ2 is UGAS by assumption.
(Necessity). By assumption there existsβ∈ KLsuch that|x(t)| ≤β(|x◦|, t− t◦). UGB follows observing that |x(t)| ≤ β(|x◦|,0). Also, notice that the solutions x(t) restricted to x2(t)≡ 0 satisfy |x1(t)| ≤ β(|x1◦|, t−t◦) which implies UGAS of (2.37). It is clear that (2.37) is UGAS only if (2.35b) is UGAS.
As discussed in the previous section, the next question is how to guarantee uniform global boundedness. This can be established by imposing extragrowth rateassumptions. In particular, in Section 2.3.5, under Assumptions 1 and 2, we shall consider the three previously mentioned cases according to the growth rates off1(t, x1) andg(t, x). For this, we make use of the following concepts.
Definition 2.9 (small order). Let : Rn → Rn, ϕ : R≥0×Rn → Rn be continuous functions of their arguments. We denote ϕ(t,ã) = o( (ã)) (and say that “phi is of small order rho”) if there exists a continuous function λ:R≥0→R≥0 such that|ϕ(t, x)| ≤λ(|x|)| (x)|for all(t, x)∈R≥0×Rn and
|x|→∞lim λ(|x|) = 0.
A direct consequence of this definition is that the following limit holds uni- formly int:
|x|→∞lim
|ϕ(t, x)|
| (x)| = 0.
Definition 2.10.Let (ã)andϕ(ã,ã)be continuous. We say that the function (x)majorises the functionϕ(t, x) if
lim sup
|x|→∞
|ϕ(t, x)|
| (x)| <+∞ ∀t≥0 .
Notice that, as a consequence of the definition above, it holds true that there exist finite positive constantsηandλsuch that, the following holds uniformly int:
|x| ≥η ⇒ |ϕ(t, x)|
| (x)| < λ . (2.53)
We may also refer to this property as “large order” or “order” and write ϕ(t,ã) =O( (ã)) to say that “phi is of large order rho”.