  Control Lyapunov functions (CLFs) play a vital role in modern control
applications, but finding them remains a problem. Recently, the control
Lyapunov-value function (CLVF) and robust CLVF have been proposed as solutions
for nonlinear time-invariant systems with bounded control and disturbance.
However, the CLVF suffers from the ''curse of dimensionality,'' which hinders
its application to practical high-dimensional systems. In this paper, we
propose a method to decompose systems of a particular coupled nonlinear
structure, in order to solve for the CLVF in each low-dimensional subsystem. We
then reconstruct the full-dimensional CLVF and provide sufficient conditions
for when this reconstruction is exact. Moreover, a point-wise optimal
controller can be obtained using a quadratic program. We also show that when
the exact reconstruction is impossible, the subsystems' CLVFs and their
``admissible control sets'' can be used to generate a Lipschitz continuous CLF.
We provide several numerical examples to validate the theory and show
computational efficiency.
