  This paper presents a method to stabilize state and input constrained
nonlinear systems using an offline optimization on variable triangulations of
the set of admissible states. For control-affine systems, by choosing a
continuous piecewise affine (CPA) controller structure, the non-convex
optimization is formulated as iterative semi-definite programming (SDP), which
can be solved efficiently using available software. The method has very general
assumptions on the system's dynamics and constraints. Unlike similar existing
methods, it avoids finding terminal invariant sets, solving non-convex
optimizations, and does not rely on knowing a control Lyapunov function (CLF),
as it finds a CPA Lyapunov function explicitly. The method enforces a desired
upper-bound on the decay rate of the state norm and finds the exact region of
attraction. Thus, it can be also viewed as a systematic approach for finding
Lipschitz CLFs in state and input constrained control-affine systems. Using the
CLF, a minimum norm controller is also formulated by quadratic programming for
online application.
