Driver support systems aim to support or take over part of the driving task in order to drive vehicles in a safe, comfortable and efficient way. To this end, in literature, different controller designs have been put forward over the past two decades. This paper proposes an optimal control framework to model driver support systems. In this approach, the acceleration of vehicles in a platoon following a platoon leader is controlled in such a way that a generic objective function is optimized. To solve the optimization problem, a numerical solution approach is put forward applying Pontryagin's minimum principle. Both competitive and cooperative controllers are put forward, in which each vehicle either optimizes its own situation, or where all vehicles cooperate to optimize the overall performance of the platoon. We present several examples showing the difference between the strategies. Results show that the computational complexity of the proposed approach is sufficiently small to allow for real-time computation of car-following strategies, compared to previous approaches. An important property is that this complexity does not increase exponentially with the complexity of the prediction model, nor with the size of the control vector.