  We investigate the problem of data-driven, on-the-fly control of systems with
unknown nonlinear dynamics where data from only a single finite-horizon
trajectory and possibly side information on the dynamics are available. Such
side information may include knowledge of the regularity of the dynamics,
monotonicity of the states, or decoupling in the dynamics between the states.
Specifically, we develop two algorithms, $\texttt{DaTaReach}$ and
$\texttt{DaTaControl}$, to over-approximate the reachable set and design
control signals for the system on the fly. $\texttt{DaTaReach}$ constructs a
differential inclusion that contains the unknown vector field. Then, it
computes an over-approximation of the reachable set based on interval
Taylor-based methods applied to systems with dynamics described as differential
inclusions. $\texttt{DaTaControl}$ enables convex-optimization-based,
near-optimal control using the computed over-approximation and the
receding-horizon control framework. We provide a bound on its suboptimality and
show that more data and side information enable $\texttt{DaTaControl}$ to
achieve tighter suboptimality bounds. Finally, we demonstrate the efficacy of
$\texttt{DaTaControl}$ over existing approaches on the problems of controlling
a unicycle and quadrotor systems.
