Structured decentralized control of positive systems with applications to combination drug therapy and leader selection in directed networks

TL;DR

This paper develops convex optimization methods for structured control of positive systems, with applications to leader selection in networks and combination drug therapy, providing new algorithms and performance bounds.

Contribution

It proves convexity of $H_2$ and $H_$ control problems for positive systems and introduces customized algorithms for leader selection and drug therapy optimization.

Findings

Convexity of $H_2$ and $H_$ control for positive systems established.

New algorithms for optimal control and leader selection developed.

Performance bounds for leader selection and drug therapy optimization derived.

Abstract

We study a class of structured optimal control problems in which the main diagonal of the dynamic matrix is a linear function of the design variable. While such problems are in general challenging and nonconvex, for positive systems we prove convexity of the $H_2$ and $H_\infty$ optimal control formulations which allow for arbitrary convex constraints and regularization of the control input. Moreover, we establish differentiability of the $H_\infty$ norm when the graph associated with the dynamical generator is weakly connected and develop a customized algorithm for computing the optimal solution even in the absence of differentiability. We apply our results to the problems of leader selection in directed consensus networks and combination drug therapy for HIV treatment. In the context of leader selection, we address the combinatorial challenge by deriving upper and lower bounds on optimal performance. For combination drug therapy, we develop a customized subgradient method for efficient treatment of diseases whose mutation patterns are not connected.