Global extension of local control Lyapunov functions via exit-time optimal control

This paper studies the feedback stabilization problem for deterministic nonlinear control systems described by ODEs. Assuming that a local control Lyapunov function (CLF) with certain properties can be explicitly constructed, we propose a way of extending it to a global CLF. Under a number of reasonable conditions, it is proved that the concatenation of the local CLF and the value function for an appropriately formulated exit-time optimal control problem leads to a global CLF satisfying the sought-after infinitesimal decrease property. The exit-time problem is stated with respect to a sublevel set of the local CLF. We then develop the theoretical foundations of a curse-of-dimensionality-free characteristics based approach for approximating the related value function. Future research directions and possible applications are also discussed.