  We study predictive control in a setting where the dynamics are time-varying
and linear, and the costs are time-varying and well-conditioned. At each time
step, the controller receives the exact predictions of costs, dynamics, and
disturbances for the future $k$ time steps. We show that when the prediction
window $k$ is sufficiently large, predictive control is input-to-state stable
and achieves a dynamic regret of $O(\lambda^k T)$, where $\lambda < 1$ is a
positive constant. This is the first dynamic regret bound on the predictive
control of linear time-varying systems. Under more assumptions on the terminal
costs, we also show that predictive control obtains the first competitive bound
for the control of linear time-varying systems: $1 + O(\lambda^k)$. Our results
are derived using a novel proof framework based on a perturbation bound that
characterizes how a small change to the system parameters impacts the optimal
trajectory.
