  Willems et al.'s fundamental lemma asserts that all trajectories of a linear
system can be obtained from a single given one, assuming that a persistency of
excitation condition holds. This result has profound implications for system
identification and data-driven control, and has seen a revival over the last
few years. The purpose of this paper is to extend Willems' lemma to the
situation where multiple (possibly short) system trajectories are given instead
of a single long one. To this end, we introduce a notion of collective
persistency of excitation. We will then show that all trajectories of a linear
system can be obtained from a given finite number of trajectories, as long as
these are collectively persistently exciting. We will demonstrate that this
result enables the identification of linear systems from data sets with missing
data samples. Additionally, we show that the result is of practical
significance in data-driven control of unstable systems.
