  Funnel synthesis refers to a procedure for synthesizing a time-varying
controlled invariant set and an associated control law around a nominal
trajectory. The computation of the funnel involves solving a continuous-time
differential equation or inequality, ensuring the invariance of the funnel.
Previous approaches often compromise the invariance property of the funnel; for
example, they may enforce the equation or the inequality only at discrete
temporal nodes and not having formal guarantee of invariance at all times. This
paper proposes a computational funnel synthesis method that can satisfy the
invariance of the funnel without such compromises. We derive a finite number of
linear matrix inequalities (LMIs) that imply the satifaction of a
continuous-time differential linear matrix inequality guaranteeing the
invariance of the funnel at all times from the initial to the final time. To
this end, we utilize LMI conditions ensuring matrix copositivity, which then
imply continuous-time invariance. The primary contribution of the paper is to
prove that the resulting funnel is indeed invariant over a finite time horizon.
We validate the proposed method via a three-dimensional trajectory planning and
control problem with obstacle avoidance constraints.
