  Merging two Control Lyapunov Functions (CLFs) means creating a single
"new-born" CLF by starting from two parents functions. Specifically, given a
"father" function, shaped by the state constraints, and a "mother" function,
designed with some optimality criterion, the merging CLF should be similar to
the father close to the constraints and similar to the mother close to the
origin. To successfully merge two CLFs, the control-sharing condition is
crucial: the two functions must have a common control law that makes both
Lyapunov derivatives simultaneously negative. Unfortunately, it is difficult to
guarantee this property a-priori, i.e., while computing the two parents
functions. In this paper, we propose a technique to create a constraint-shaped
"father" function that has the control-sharing property with the "mother"
function. To this end, we introduce a partial control-sharing, namely, the
control-sharing only in the regions where the constraints are active. We show
that imposing partial control-sharing is a convex optimization problem.
Finally, we show how to apply the partial control-sharing for merging
constraint-shaped functions and the Riccati-optimal functions, thus generating
a CLF with bounded complexity that solves the constrained linear-quadratic
stabilization problem with local optimality.
