Sustainability-targeting attacks (STA) are a growing threat to cyber-physical system (CPS)-based infrastructure, as sustainability goals become an integral part of CPS objectives. STA can be especially disruptive if it impacts the long-term sustainability cost of CPS, while its performance goals remain within acceptable parameters. Thus, in this work, we propose a general mathematical framework for modeling such stealthy STA and derive the feasibility conditions for generating a minimum-effort maximum-impact STA on a linear CPS using a max-min formulation. A gradient ascent descent algorithm is used to construct this attack policy with an added constraint on stealthiness. An illustrative example has been simulated to demonstrate the impact of the generated attack on the sustainability cost of the CPS.