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We study a class of structured optimal control problems in which the maindiagonal of the dynamic matrix is a linear function of the design variable.While such problems are in general challenging and nonconvex, for positivesystems we prove convexity of the $H_2$ and $H_\infty$ optimal controlformulations which allow for arbitrary convex constraints and regularization ofthe control input. Moreover, we establish differentiability of the $H_\infty$norm when the graph associated with the dynamical generator is weakly connectedand develop a customized algorithm for computing the optimal solution even inthe absence of differentiability. We apply our results to the problems ofleader selection in directed consensus networks and combination drug therapyfor HIV treatment. In the context of leader selection, we address thecombinatorial challenge by deriving upper and lower bounds on optimalperformance. For combination drug therapy, we develop a customized subgradientmethod for efficient treatment of diseases whose mutation patterns are notconnected.

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