Controllability for Distributed Bilinear Systems

This paper studies controllability of systems of the form ${{dw} / {dt}} = \mathcal {A}w + p(t)\mathcal {B}w$ where $\mathcal{A}$ is the infinitesimal generator of a $C^0$ semigroup of bounded linear operators $e^{\mathcal{A}t} $ on a Banach space $X$, $\mathcal{B}:X \to X$ is a $C^1$ map, and $p \in L^1 ([0,T];\mathbb{R})$ is a control. The paper (i) gives conditions for elements of $X$ to be accessible from a given initial state $w_0$ and (ii) shows that controllability to a full neighborhood in $X$ of $w_0$ is impossible for $\dim X = \infty $. Examples of hyperbolic partial differential equations are provided.