Due to the inherent relevance of passive (physically representative) models for control, state-estimation, and motion simulation in the field of marine systems, in this paper, an optimisation-based approach to passivation of linear time-invariant (LTI) systems is proposed with application to physically consistent dynamical modelling of marine structures. In particular, the presented strategy is based upon the introduction of a suitably designed perturbation, computed via minimisation of a linear objective subject to a specific set of linear matrix inequalities (LMIs). The performance of the passivation technique is showcased in terms of two case studies: An offshore platform, with a frequency-domain response computed by means of hydrodynamic codes, and a 1:20 scale wave energy converter (WEC), is characterised in terms of real experimental data.