Optimal Control with $L^\infty$ and Integral Cost Functionals

Abstract

Many control problems are classically formulated using integral costs that capture the cumulative performance of a system. Peak or worst-case behavior is captured via supremum or $L^\infty$-costs in another variety of control problems. When both cumulative and peak performance are important, it is natural to consider objective functions that combine the two costs. Although each criterion is well studied, their combination has not been explored extensively and we precisely work on such control problems. Towards establishing the existence, we first consider the relaxed framework, where control is considered using probability distributions. Using the well-known compactness and convexity properties of such control spaces, we establish the existence of an optimal relaxed control---we eventually establish the existence of an $ε$-optimal pure (or classical) control, for every $ε> 0.$ Despite these existence results, computing optimal policies remains challenging due to the non-smoothness introduced by the supremum term, and it is not clear whether the dynamic programming principle holds for our combined problem. To address this, we introduce a family of smooth approximations that yield standard control problems with well-defined optimal (pure) solutions. Using Maximum Theorem, we establish that the solutions of the smooth problems among pure controls form $ε$-optimal for the original problem, with $ε$ tending to zero as the smoothing parameter converges to zero. Finally using the methods proposed in this paper, we study a queueing problem to illustrate (among others) that the required trade-off between peak congestion levels and cumulative performance can be achieved.