  Positive systems naturally arise in situations where the model tracks
physical quantities. Although the linear case is well understood, analysis and
controller design for nonlinear positive systems remain challenging. Model
reduction methods can help tame this problem. Here we propose a notion of model
reduction for a class of positive bilinear systems with (bounded) matrix and
exogenous controls. Our reduction, called proper positive lumping, aggregates
the original system such that states of the corresponding reduced model
represent non-negative linear combinations of original state variables. We
prove a characterization result showing that the reductions by proper positive
lumping are exactly those preserving the optimality of a suitable class of
value functions. Moreover, we provide an efficient polynomial-time algorithm
for the computation of the minimal lumping. We numerically evaluate our
approach by applying it to a number of benchmark case studies.
