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research-article

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

[+] Author and Article Information
Mohammad Ghanaatpishe

Department of Mechanical Engineering The Pennsylvania State University University Park, PA 16802
mug198@psu.edu

Hosam K. Fathy

Department of Mechanical Engineering The Pennsylvania State University University Park, PA 16802
hkf2@psu.edu

1Corresponding author.

ASME doi:10.1115/1.4035459 History: Received February 17, 2016; Revised December 06, 2016

Abstract

This article 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, the article proposes two different numerical methods for solving this OPC problem. One method uses nonlinear programming 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 article 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.

Copyright (c) 2016 by ASME
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