State-feedback stabilizability, optimality, and convexity in switched positive linear systems

The present paper is concerned with state-feedback stabilizability in discrete-time switched positive linear systems. Necessary and sufficient conditions for state-feedback exponential stabilizability, in this class of switched systems, are presented. It is shown that, a switched positive linear system is state-feedback exponentially stabilizable if and only if an associated sequence, whose elements are computable via linear programming, has an element smaller than one. Also, a switched positive linear system is state-feedback exponentially stabilizable if and only if there exits a product of their modes matrices whose spectral radius is smaller than one. Equivalently, the state-feedback exponential stabilizability of a switched positive linear system is shown to be equivalent to the solvability of an associated dynamic programming equation on a given convex cone. That associated dynamic programming equation it is shown to have at most one solution. This unique solution, of the associated dynamic programming equation, is shown to be concave, monotonic, positively homogeneous, and the optimal cost functional of a related optimal control problem (involving the switched positive linear system) whose complete solution is also presented in this communication.