State-feedback stabilizability in switched homogeneous systems

The present article is concerned with state-feedback stabilizability of discrete-time switched homogeneous systems. Necessary and sufficient conditions for state-feedback exponential stabilizability are presented. It is shown that, a switched homogeneous system is state-feedback exponentially stabilizable if and only if an associated sequence converges to zero. Equivalently, a switched homogeneous system is state-feedback exponentially stabilizable if and only if an associated dynamic programming equation admits a solution on a given convex set. This unique solution of that associated dynamic programming equation is shown to be the optimal cost functional of a related infinite-horizon quadratic regulator problem (for the switched homogeneous system) whose solution is also presented. A numerical example illustrates the results reported in the paper.