Optimal Control of Interacting Systems with DNSS Property: The Case of Illicit Drug Use

In this paper we generalize a one-dimensional optimal control problem with DNSS property to a two-dimensional optimal control problem. This is done by taking the direct product of the model with itself, i.e. we combine two similar system dynamics under a joint objective functional that is separable in both states and controls. This framework can be applied to the construction of various optimal control problems, such as optimal marketing of related products, optimal growth of separate but interacting economies, or optimal control of related epidemics of infectious diseases. We study such a system for a particular case drawn from the domain of drug control. The main result of this paper is that in this domain even a modest amount of interaction can sometimes make a very big difference, and hence that drawing conclusions by simplifying the real world into two independent one-dimensional models may be problematic. Methodologically the combination of two systems with DNSS property leads to a fascinating series of situations with multiple optimal steady states and associated threshold behavior. These instances reflect some important recent developments in optimal dynamic control theory.