Convex, monotone systems are optimally operated at steady-state

Marius Schmitt, Chithrupa Ramesh, Paul J. Goulart, John Lygeros

Abstract: We consider a special class of monotone systems for which the system equations are also convex in both the state and the input. For such systems we study optimal infinite horizon operation with respect to an objective function that is also monotone and convex. The main results state that, under some technical assumptions, these systems are optimally operated at steady state, i.e. there does not exist any time-varying trajectory over an infinite horizon that outperforms stabilizing the system in the optimal equilibrium. We draw a connection to recent results on dissipative systems in the context of Economic Model Predictive Control, where systems that are optimally operated at steady state have already been studied. Finally, we apply the main result to a problem in traffic control, where we are able to disprove the existence of improving periodic trajectories involving the alternation of congestion and free flow for freeway ramp metering.