Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Why is convexity <strong>in</strong>terest<strong>in</strong>g ???<br />
Reason 3: subgradients<br />
Def<strong>in</strong>ition<br />
A vector g = g(x0) ∈ R n is a subgradient of f at x0 if<br />
for all x ∈ S<br />
f(x) ≥ f(x0) + 〈g, x − x0〉<br />
Geometric idea: graph of aff<strong>in</strong>e function x ↦→ f(x0) + 〈g, x − x0〉 tangent<br />
to graph of f at (x0, f(x0)).<br />
Theorem<br />
A convex function f : S → R has a subgradient at every <strong>in</strong>terior po<strong>in</strong>t x0<br />
of S.<br />
Siep Weiland and Carsten Scherer (DISC) <strong>L<strong>in</strong>ear</strong> <strong>Matrix</strong> <strong>Inequalities</strong> <strong>in</strong> <strong>Control</strong> Class 1 27 / 59