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.

在本文中，我们研究了可控输入对\（（A，B）\ in中子空间\（\ mathbf {L} _ {n，m} ^ c（\ mathbb {R}）\）的连通性和凸性\ mathbf {L} _ {n，m}（\ mathbb {R}）：= \ mathbb {R} ^ {n \ times n} \ times \ mathbb {R} ^ {n \ times} \）。我们介绍了三个受限的凸性（“密集”，“几乎确定”和“一般”凸性）。为了证明空间\（\ mathbf {L} _ {n，m} ^ c（\ mathbb {R}）\）具有这些性质，我们研究\（\ mathbf {L} _ {n，m}（\ mathbb {R}）\）中\（\ mathbf {L} _ {n，m}（\ mathbb {R}）\）中具有不可控制的输入对的代数形式。在单输入情况下（\（m = 1 \）），空格\（\ mathbf {L} _ {n，1} ^ c（\ mathbb {R}）\）由两个相连的组件组成，我们证明了空间\（\ mathbf {L} _ {n，m} ^ c （\ mathbb {R}）\）在多输入情况下通常是凸的。这是我们的主要结果。它直接暗示了Brockett定理，如果\（m \ ge 2 \）是\（\ mathbf {L} _ {n，m} ^ c（\ mathbb {R}）\）的路径连接。作为另一个应用程序，我们推导了Hazewinkel和Kalman关于多输入系统的连续规范形式不存在的定理。