  Control Lyapunov Functions (CLFs) have been extensively used in the control
community. A well-known drawback is the absence of a systematic way to
construct CLFs for general nonlinear systems, and the problem can become more
complex with input or state constraints. Our preliminary work on constructing
Control Lyapunov Value Functions (CLVFs) using Hamilton-Jacobi (HJ)
reachability analysis provides a method for finding a non-smooth CLF. In this
paper, we extend our work on CLVFs to systems with bounded disturbance and
define the Robust CLVF (R-CLVF). The R-CLVF naturally inherits all properties
of the CLVF; i.e., it first identifies the "smallest robust control invariant
set (SRCIS)" and stabilizes the system to it with a user-specified exponential
rate. The region from which the exponential rate can be met is called the
"region of exponential stabilizability (ROES)." We provide clearer definitions
of the SRCIS and more rigorous proofs of several important theorems. Since the
computation of the R-CLVF suffers from the "curse of dimensionality," we also
provide two techniques (warmstart and system decomposition) that solve it,
along with necessary proofs. Three numerical examples are provided, validating
our definition of SRCIS, illustrating the trade-off between a faster decay rate
and a smaller ROES, and demonstrating the efficiency of computation using
warmstart and decomposition.
