Linear Matrix Inequalities in Control

Recap: infimum and minimum of functions
Infimum of a function
Any f : S → R has infimum L ∈ R ∪ {−∞} denoted as infx∈S f(x)
defined by the properties
L ≤ f(x) for all x ∈ S
L finite: for all ε > 0 exists x ∈ S with f(x) < L + ε
L infinite: for all ε > 0 there exist x ∈ S with f(x) < −1/ε
Minimum of a function
If exists x0 ∈ S with f(x0) = infx∈S f(x) we say that f attains its
minimum on S and write L = minx∈S f(x).
Minimum is uniquely defined by the properties
L ≤ f(x) for all x ∈ S
There exists some x0 ∈ S with f(x0) = L
Siep Weiland and Carsten Scherer (DISC) Linear Matrix Inequalities in Control Class 1 13 / 59