  We study the problem of co-designing control barrier functions (CBF) and
linear state feedback controllers for continuous-time linear systems. We
achieve this by means of a single semi-definite optimization program. Our
formulation can handle mixed-relative degree problems without requiring an
explicit safe controller. Different L-norm based input limitations can be
introduced as convex constraints in the proposed program. We demonstrate our
results on an omni-directional car numerical example.
