Transient stability controller design based on an optimality principle is an emerging approach for resolving high order contingencies that can otherwise lead to rotor angle instability. It is difficult to design for high order contingencies due to their complexity. A challenging issue is the nonlinearity of the system, which limits the selection approaches available and results in a potentially large search space to find a suitable control action. Because rotor angle instability evolves rapidly, a fast control response is required. This paper develops a cost-based controller under the constraint of bounded computation. A method to downsize the admissible control search space is developed in a manner that retains controls most likely to provide the best cost. Subsequently, a further reduction in computation is designed through model elimination. A set of common cost metrics is applied for all controllers. Tradeoffs between the model-based and model-free approaches are demonstrated. The effect of computational constraints on the competing goals of control cost and control timeliness for complex contingencies is tested with the New England 39-bus system.