On the convexity of a class of structured optimal control problems for positive systems

We study a class of structured optimal control problems for positive systems in which the design variable modifies the main diagonal of the dynamic matrix. For this class of systems, we establish convexity of both the H2 and H∞ optimal control formulations. In contrast to previous approaches, our formulation allows for arbitrary convex constraints and regularization of the design parameter. We provide expressions for the gradient and subgradient of the H2 and norms and establish graph-theoretic conditions under which the H∞ norm is continuously differentiable. Finally, we develop a customized proximal algorithm for computing the solution to the regularized optimal control problems and apply our results for HIV combination drug therapy design.

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