  In this work we provide provable regret guarantees for an online
meta-learning control algorithm in an iterative control setting, where in each
iteration the system to be controlled is a linear deterministic system that is
different and unknown, the cost for the controller in an iteration is a general
additive cost function and the control input is required to be constrained,
which if violated incurs an additional cost. We prove (i) that the algorithm
achieves a regret for the controller cost and constraint violation that are
$O(T^{3/4})$ for an episode of duration $T$ with respect to the best policy
that satisfies the control input control constraints and (ii) that the average
of the regret for the controller cost and constraint violation with respect to
the same policy vary as $O((1+\log(N)/N)T^{3/4})$ with the number of iterations
$N$, showing that the worst regret for the learning within an iteration
continuously improves with experience of more iterations.
