  The control Barrier function approach has been widely used for safe
controller synthesis. By solving an online convex quadratic programming
problem, an optimal safe controller can be synthesized implicitly in
state-space. Since the solution is unique, the mapping from state-space to
control inputs is injective, thus enabling us to evaluate the underlying
relationship. In this paper we aim at explicitly synthesizing a safe control
law as a function of the state for nonlinear control-affine systems with
limited control ability. We propose to transform the online quadratic
programming problem into an offline parameterized optimisation problem which
considers states as parameters. The obtained explicit safe controller is shown
to be a piece-wise Lipschitz continuous function over the partitioned state
space if the program is feasible. We address the infeasible cases by solving a
parameterized adaptive control Barrier function-based quadratic programming
problem. Extensive simulation results show the state-space partition and the
controller properties.
