Optimal control of a class of positive Markovian bilinear systems

In this paper a class of input-parametrized bilinear positive Markovian systems is considered. The class considered has switching control of the diagonal entries of the dynamical matrix, together with Markov jump dynamics. This class is relevant to a variety of dynamical models arising in system biology and compartmental systems. It is proven that the component of the expected value of the state vector is a convex functional of the input variables. If a linear cost of the final state is considered this implies that any Pontryagin solution of the associated optimal control problem is optimal and can be numerically computed by using standard gradient-type algorithms. Moreover, suboptimal strategies, both in open loop and closed loop are provided, mainly based on linear programming tools. An example is provided to illustrate the theory.