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

this outer bound set to obtain candidate V ’s, and then solve (3.9) for S, holding V fixed.
This “qualifies” V as a certificate, and then further BMI optimization (e.g. coordinate-wise
affine search) is executed, using this (V, S) pair as an initial seed.
3.2 Relaxation of the Bilinear SOS Problem Using Simulation
Data
The usefulness of simulation in understanding the ROA for a given system is undeniable.
Faced with the task of performing a stability analysis (e.g., “for a given p, is Ω p,β contained
in the ROA”), a pragmatic, fruitful and wise approach begins with a linearized analysis
and at least a modest amount of simulation runs. Certainly, just one divergent trajectory
starting in Ω p,β certifies that Ω p,β ⊄ R 0 . Conversely, a large collection of only convergent
trajectories hints to the likelihood that indeed Ω p,β ⊂ R 0 . Suppose this latter condition is
true, let C be the set of N conv trajectories c converging to the origin with initial conditions
in Ω p,β . In the course of simulation runs, divergent trajectories d whose initial conditions
are not in Ω p,β may also get discovered, so let the set of d’s be denoted by D and N div be
the number of elements of D. Although C and D depend on β and the manner in which
Ω p,β is sampled, this is not explicitly notated.
With β and γ fixed, the set of Lyapunov functions which certify that Ω p,β ⊂ R 0 , using
conditions (3.6)-(3.8), is simply
{V ∈ R[x] : (3.6) − (3.8) hold for some s i ∈ Σ[x]} .
Of course, this set could be empty, but it must be contained in the convex set
{V ∈ R[x] : (3.10) holds},
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