Convexity of the cost functional in an optimal control problem for a class of positive switched systems

This paper deals with the optimal control of a class of positive switched systems. The main feature of this class is that switching alters only the diagonal entries of the dynamic matrix. The control input is represented by the switching signal itself and the optimal control problem is that of minimizing a positive linear combination of the final state variable. First, the switched system is embedded in the class of bilinear systems with control variables living in a simplex, for each time point. The main result is that the cost is convex with respect to the control variables. This ensures that any Pontryagin solution is optimal. Algorithms to find the optimal solution are then presented and an example, taken from a simplified model for HIV mutation mitigation is discussed.