Convex reformulation of a robust optimal control problem for a class of positive systems

In this paper we consider the robust optimal control problem for a class of positive systems with an application to design of optimal drug dosage for HIV therapy. We consider uncertainty modeled as a Linear Fractional Transformation (LFT) and we show that, with a suitable change of variables, the structured singular value, μ, is a convex function of the control parameters. We provide graph theoretical conditions that guarantee μ to be a continuously differentiable function of the controller parameters and an expression of its gradient or subgradient. We illustrate the result with a numerical example where we compute the optimal drug dosages for HIV treatment in the presence of model uncertainty