Literature has shown potentials of Connected/Cooperative Automated Vehicles (CAVs) in improving highway operations, especially on roadway capacity and flow stability. However, benefits were also shown to be negligible at low market penetration rates. This work develops a novel adaptive driving strategy for CAVs to stabilise heterogeneous vehicle strings by controlling one CAV under vehicle-to-infrastructure (V2I) communications. Assumed is a roadside system with V2I communications, which receives control parameters of the CAV in the string and estimates parameters imperfectly of non-connected automated vehicles. It determines the adaptive control parameters (e.g. desired time gap and feedback gains) of the CAV if a downstream disturbance is identified and sends them to the CAV. The CAV changes its behaviour based on the adaptive parameters commanded by the roadside system to suppress the disturbance. The proposed adaptive driving strategy is based on string stability analysis of heterogeneous vehicle strings. To this end, linearised vehicle dynamics model and control law are used in the controller parametrisation and Laplace transform of the speed and gap error dynamics in time domain to frequency domain enables the determination of sufficient string stability criteria of heterogeneous strings. The analytical string stability conditions give new insights into automated vehicular string stability properties in relation to the system properties of time delays and controller design parameters of feedback gains and desired time gap. It further allows the quantification of a stability margin, which is subsequently used to adapt the feedback control gains and desired time gap of the CAV to suppress the amplification of gap and speed errors through the string. Analytical results are verified via systematic simulation of both homogeneous and heterogeneous strings. Simulation demonstrates the predictive power of the analytical string stability conditions. The performance of the adaptive driving strategy under V2I cooperation is tested in simulation. Results show that even the estimation of control parameters of non-connected automated vehicles are imperfect and there is mismatch between the model used in analytical derivation and that in simulation, the proposed adaptive driving strategy suppresses disturbances in a wide range of situations.