  This paper studies symbolic abstractions for nonlinear control systems using
logarithmic quantization. With a logarithmic quantizer, we approximate the
state and input sets, and then construct a novel discrete abstraction for
nonlinear control systems. A feedback refinement relation between the
constructed discrete abstraction and the original system is established. Using
the constructed discrete abstraction, the safety controller synthesis problem
is studied. With the discrete abstraction and the abstract specification, the
existence of a safety controller is investigated, and the algorithm is proposed
to compute the abstract controller. Finally, a numerical example is given to
illustrate the obtained results.
