  We consider a class of monotone systems in which the control signal
multiplies the state. Among other applications, such bilinear systems can be
used to model the evolutionary dynamics of HIV in the presence of combination
drug therapy. For this class of systems, we formulate an infinite horizon
optimal control problem, prove that the optimal control signal is constant over
time, and show that it can be computed by solving a finite-dimensional
non-smooth convex optimization problem. We provide an explicit expression for
the subdifferential set of the objective function and use a subgradient
algorithm to design the optimal controller. We further extend our results to
characterize the optimal robust controller for systems with uncertain dynamics
and show that computing the robust controller is no harder than computing the
nominal controller. We illustrate our results with an example motivated by
combination drug therapy.
