Platoons of vehicles relying on inter-vehicle communication and control mechanisms have a great potential, e.g. to reduce fuel consumption and increase capacity. Linear stability analysis can be performed to investigate the propagation of perturbations in such cooperative platoons. Existing work, however, does not consider the stability analysis of the linearised global cooperative platoon system. Our car-following model framework is based on the cooperative relation introduced by Wilson, R. E. [2008. “Mechanisms for Spatio-Temporal Pattern Formation in Highway Traffic Models.” Philosophical Transactions of the Royal Society A (2008) 366, 2017–2032]. We present the state-space representation of the linearised dynamical system. We prove analytically, and illustrate in simulations, that a platoon of vehicles is always linearly-stable provided that the rational relations [Wilson, R. E., and J. A. Ward. 2011. “Car-Following Models: Fifty Years of Linear Stability Analysis: A Mathematical Perspective.” Transportation Planning and Technology 13, 2167–2176] hold and the coefficients of cooperation are non-negative. We present a brief analysis of controllability, considering heterogeneity in the control inputs. Finally, we show that for a closed platoon, the conditions for linear stability are the same as the conditions for stability of a one-dimensional infinite traffic flow.