On the Structure of the Optimal Solution to a Periodic Drug-Delivery Problem

This paper examines the shaping of a drug's delivery—in this case, nicotine—to maximize its efficacy. Previous research: (i) furnishes a pharmacokinetic–pharmacodynamic (PKPD) model of this drug's metabolism; (ii) shows that the drug-delivery problem is proper, meaning that its optimal solution is periodic; (iii) shows that the underlying PKPD model is differentially flat; and (iv) exploits differential flatness to solve the problem by optimizing the coefficients of a truncated Fourier expansion of the flat output trajectory. In contrast, the work in this article provides insight into the structure of the theoretical solution to this optimal periodic control (OPC) problem. First, we argue for the existence of a bijection between feasible periodic input and state trajectories of the problem. Second, we exploit Pontryagin's maximum principle to show that the optimal periodic solution has a bang–singular–bang structure. Building on these insights, this article proposes two different numerical methods for solving this OPC problem. One method uses nonlinear programming (NLP) to optimize the states at which the optimal solution transitions between the different solution arcs. The second method approximates the control input trajectory as piecewise constant and optimizes the discrete values of the input sequence. The paper concludes by discussing the computational costs of these two algorithms as well as the importance of the associated insights into the structure of the optimal solution trajectory.