  This paper studies the problem of constructing control Lyapunov functions
(CLFs) and feedback stabilization strategies for deterministic nonlinear
control systems described by ordinary differential equations. Many numerical
methods for solving the Hamilton-Jacobi-Bellman partial differential equations
specifying CLFs typically require dense state space discretizations and
consequently suffer from the curse of dimensionality. A relevant direction of
attenuating the curse of dimensionality concerns reducing the computation of
the values of CLFs and associated feedbacks at any selected states to
finite-dimensional nonlinear programming problems. In this work, exit-time
optimal control is used for that purpose. First, we state an exit-time optimal
control problem with respect to a sublevel set of an appropriate local CLF and
establish that, under a number of reasonable conditions, the concatenation of
the corresponding value function and the local CLF is a global CLF in the whole
domain of asymptotic null-controllability. This leads to a
curse-of-dimensionality-free approach to feedback stabilization. We also
investigate the formulated optimal control problem. A modification of these
constructions for the case when one does not find a suitable local CLF is
provided as well. Supporting numerical simulation results that illustrate our
development are subsequently presented and discussed. Furthermore, it is
pointed out that the curse of complexity may cause significant issues in
practical implementation even if the curse of dimensionality is mitigated.
