Cold start emissions are the most significant contributor to the accumulated emissions of a vehicle and poses a critical design limit for the design of clean and efficient vehicles. The core reasons for the emissions are the initial low temperature and the thermal inertia of the exhaust aftertreatment systems. Moreover, it also costs fuel to perform the heating of the catalyst. It is therefore of high interest to develop efficient control schemes that can reduce the time to light off. To facilitate this a model structure and a method, based on the explicit solution to the catalyst differential equations are developed, that can be used to analyze both time and fuel optimal heating control strategies. The method is developed to be applicable to both gasoline and diesel aftertreatment systems. A case study is performed on a Diesel engine and the results show that the solutions exhibit a structured and simple two-phase pattern. There is a first heating phase, where the catalyst is fed with a high temperature gas, building up a high inlet temperature. Then in a second phase the flow is kept high and the temperature is pushed through the catalyst. The strategy is easy to understand and realize in a real time control system. (C) 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.