  In this paper we present a Learning Model Predictive Control (LMPC) strategy
for linear and nonlinear time optimal control problems. Our work builds on
existing LMPC methodologies and it guarantees finite time convergence
properties for the closed-loop system. We show how to construct a time varying
safe set and terminal cost function using closed-loop data. The resulting LMPC
policy is time varying and it guarantees recursive constraint satisfaction and
non-decreasing performance. Computational efficiency is obtained by convexifing
the safe set and terminal cost function. We demonstrate that, for a class of
nonlinear system and convex constraints, the convex LMPC formulation guarantees
recursive constraint satisfaction and non-decreasing performance. Finally, we
illustrate the effectiveness of the proposed strategies on minimum time
obstacle avoidance and racing examples.
