  For data-driven control of nonlinear systems, the basis functions
characterizing the dynamics are usually essential. In existing works, the basis
functions are often carefully chosen based on pre-knowledge of the dynamics so
that the system can be expressed or well-approximated by the basis functions
and the experimental data. For a more general setting where explicit
information on the basis functions is not available, this paper presents a
data-driven approach for stabilizer design and closed-loop analysis via the
Lyapunov method. First, based on Taylor's expansion and using input-state data,
a stabilizer and a Lyapunov function are designed to render the known
equilibrium locally asymptotically stable. Then, data-driven conditions are
derived to check whether a given sublevel set of the found Lyapunov function is
an invariant subset of the region of attraction. One of the main challenges is
how to handle Taylor's remainder in the design of the local stabilizers and the
analysis of the closed-loop performance.
