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This paper classifies continuous linear flows using concepts and techniques from topological dynamics. Specifically, the concepts of equivalence and conjugacy are adapted to flows on vector bundles, and the Lyapunov decomposition is characterized using the induced flows on the Grassmann and the flag bundles. These results are then applied to bilinear control systems, for which their behavior in $ \mathbb{R}^{d} $ , on the projective space $\mathbb{P}^{{d{\text{ - 1}}}} $ , and on the Grassmannians is characterized.

On Topological Equivalence of Linear Lows with Applications to Bilinear Control Systems