Quantitative Local Analysis of Nonlinear Systems - University of ...

where
[∇z(c(τ c ))f(c(τ c ))] T α < 0,
l 1 (c(τ c )) ≤ z(c(τ c )) T α, and z(c(0)) T α ≤ γ,
(3.11)
z(d(τ d )) T α ≥ γ + δ
for all c ∈ C, τ c ∈ T c , d ∈ D, and τ d ∈ T d . Note that z(c(0)) T α ≤ γ in (3.11) provides
necessary conditions for Ω p,β ⊆ Ω V,γ since c(0) ∈ Ω p,β for all c ∈ C. In practice, we
replace the strict inequality in (3.11) by [∇z(c(τ c ))f(c(τ c ))] T α ≤ −l 3 (c(τ c )), where l 3 is a
fixed, positive definite polynomial imposing a bound on the rate of decay of V along the
trajectories. More compactly and for future reference, express the inequalities represented
by (3.11) with this modification as Φ T α ≼ b.
The constraint that ∇V f be negative on a sublevel set of V implies that ∇V f is negative
on a neighborhood of the origin. While a large number of sample points from the trajectories
will approximately enforce this, in some cases (e.g. exponentially stable linearization) it
is easy to analytically express as a constraint on the low order terms of the polynomial
Lyapunov function. For instance, assume V has a positive-definite quadratic part, and that
separate eigenvalue analysis has established that the linearization of (3.1) at the origin, i.e.,
ẋ = ∇f(0)x, is asymptotically stable. Define
L(Q) := (∇f(0)) T Q + Q (∇f(0)) ,
where Q T = Q ≻ 0 is such that x T Qx is the quadratic part of V . Then, if (3.8) holds, it
must be that
L(Q) ≺ 0. (3.12)
Let
Y lin := {α ∈ R n b
: Q = Q T ≻ 0 and (3.12) holds}.
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