Feedback Diagonalization and Stabilization for Discrete Linear Ensemble Systems

We consider a countably infinite collection of linear, scalar control systems, where each system is represented by the pair $(a_n,b_n)$ with $a_n>0$ for $n\in \mathbb{N}$, and all systems are forced by a common scalar control input. We refer to this collection of systems as a discrete linear ensemble system. The ensemble system is feedback stabilizable if there exists a common feedback control input that asymptotically stabilizes every system simultaneously. Unlike finite-dimensional linear systems, an infinite-dimensional linear system is not guaranteed to be stable if its poles lie in the open left half-plane. Stability is guaranteed, however, if (i) its poles are contained in the closed left half-plane and (ii) its infinitesimal generator is diagonalizable. In this paper, we provide necessary and sufficient conditions for the existence of a static, linear feedback control law that ensures the closed-loop ensemble system satisfies (i) and (ii). In particular, we show that the exponential decay of the sequences $(|b_n|)_{n\in \mathbb{N}}$ and $(a_n/|b_n|)_{n\in \mathbb{N}}$ is necessary and, under an assumption on the desired poles, sufficient.