The set of controllable multi-input systems is generically convex

Abstract

In this paper, we investigate connectedness and convexity properties of the subspace $\mathbf {L}_{n,m}^c(\mathbb {R})$ of controllable input pairs $(A,B)\in \mathbf {L}_{n,m}(\mathbb {R}):= \mathbb {R}^{n\times n}\times \mathbb {R}^{n\times m}$. We introduce three restricted convexity properties (“dense”, “almost sure” and “generic” convexity). In order to prove that the space $\mathbf {L}_{n,m}^c(\mathbb {R})$ possesses these properties, we study the intersection of straight lines in $\mathbf {L}_{n,m}(\mathbb {R})$ with the algebraic variety of uncontrollable input pairs in $\mathbf {L}_{n,m}(\mathbb {R})$. While in the single-input case ($m=1$), the space $\mathbf {L}_{n,1}^c(\mathbb {R})$ consists of two connected components, we prove that the space $\mathbf {L}_{n,m}^c(\mathbb {R})$ is generically convex in the multi-input case. This is our main result. It directly implies Brockett’s theorem that $\mathbf {L}_{n,m}^c(\mathbb {R})$ is pathwise connected if $m\ge 2$. As another application, we derive the theorem of Hazewinkel and Kalman about the non-existence of continuous canonical forms for multi-input systems.