Linear Matrix Inequalities in Control

Convex functions
Definition
A function f : S → R is convex if
S is convex and
for all x1, x2 ∈ S, α ∈ (0, 1) there holds
We have
f(αx1 + (1 − α)x2) ≤ αf(x1) + (1 − α)f(x2)
f : S → R convex =⇒ {x ∈ S | f(x) ≤ γ}
Sublevel sets
Derives convex sets from convex functions
Converse ⇐= is not true!
f is strictly convex if < instead of ≤
convex for all γ ∈ R
Siep Weiland and Carsten Scherer (DISC) Linear Matrix Inequalities in Control Class 1 20 / 59