  This survey presents recent research on determining control-theoretic
properties and designing controllers with rigorous guarantees using
semidefinite programming and for nonlinear systems for which no mathematical
models but measured trajectories are available. Data-driven control techniques
have been developed to circumvent a time-consuming modelling by first
principles and because of the increasing availability of data. Recently, this
research field has gained increased attention by the application of Willems'
fundamental lemma, which provides a fertile ground for the development of
data-driven control schemes with guarantees for linear time-invariant systems.
While the fundamental lemma can be generalized to further system classes, there
does not exist a comparable data-based system representation for nonlinear
systems. At the same time, nonlinear systems constitute the majority of
practical systems. Moreover, they include additional challenges such as
data-based surrogate models that prevent system analysis and controller design
by convex optimization. Therefore, a variety of data-driven control approaches
has been developed with different required prior insights into the system to
ensure a guaranteed inference. In this survey, we will discuss developments in
the context of data-driven control for nonlinear systems. In particular, we
will focus on methods based on system representations providing guarantees from
finite data, while the analysis and the controller design boil down to convex
optimization problems given as SDP. Thus, these approaches achieve reasonable
advances compared to the state-of-the-art system analysis and controller design
by models from system identification. Specifically, the paper covers system
representations based on extensions of Willems' fundamental lemma, set
membership, kernel techniques, the Koopman operator, and feedback
linearization.
