Optimal Control with $L^{\infty}$ cost: incorporating peak minimization

Abstract

Inventory and queueing systems are often designed by controlling weighted combination of some time-averaged performance metrics (like cumulative holding, shortage, server-utilization or congestion costs); but real-world constraints, like fixed storage or limited waiting space, require attention to peak levels reached during the operating period. This work formulates such control problems, which are any arbitrary weighted combination of some integral cost terms and an L-infinity(peak-level) term. The resultant control problem does not fall into standard control framework, nor does it have standard solution in terms of some partial differential equations. We introduce an auxiliary state variable to track the instantaneous peak-levels, enabling reformulation into the classical framework. We then propose a smooth approximation to handle the resultant discontinuities, and show the existence of unique value function that uniquely solves the corresponding Hamilton-Jacobi-Bellman equation. We apply this framework to two key applications to obtain an optimal design that includes controlling the peak-levels. Surprisingly, the numerical results show peak inventory can be minimized with negligible revenue loss (under 6%); without considering peak-control, the peak levels were significantly higher. The peak-optimal policies for queueing-system can reduce peak-congestion by up to 27%, however, at the expense of higher cumulative-congestion costs. Thus, for inventory-control, the performance of the average-terms did not degrade much, while the same is not true for queueing-system. Hence, one would require a judiciously chosen weighted design of all the costs involved including the peak-levels for any application and such a design can now be derived numerically using the proposed framework.