  We consider the problem of designing finite-horizon safe controllers for a
dynamical system for which no explicit analytical model exists and limited data
only along a single trajectory of the system are available. Given samples of
the states and inputs of the system, and additional side information in terms
of regularity of the evolution of the states and conservation laws, we
synthesize a controller such that the evolution of the states avoid some
pre-specified unsafe set over a given finite horizon. Motivated by recent
results on Whitney's extension theorem, we use piecewise-polynomial
approximations of the trajectories based on the data along with the regularity
side information to formulate a data-driven differential inclusion model that
can predict the evolution of the trajectories. For these classes of data-
driven differential inclusions, we propose a safety analysis theorem based on
barrier certificates. As a corollary of this theorem, we demonstrate that we
can design controllers ensuring safety of the solutions to the data-driven
differential inclusion over a finite horizon. From a computational standpoint,
our results are cast into a set of sum-of-squares programs whenever the
certificates are parametrized by polynomials of fixed degree and the sets are
semi-algebraic. This computational method allows incorporating side information
in terms of conservation laws and integral constraints using
Positivstellensatz-type arguments.
