Consider two different track profiles with the same terminal locations and elevations, labelled A and B. When a train is managed to operate with a Speed-Optimal-Strategy, which essentially means to arrive at the destination as fast as possible. It's found that: ① Profile A results in a quicker journey than Profile B, ② Profile A leads to lower energy consumption than Profile B, or ③ Profile A provides benefits in both time and energy efficiency.

My question is, under what specific conditions can we prove that, if the train operates with a fixed travel time T and adopts an Energy-Efficient-Strategy, profile A will still be more energy-efficient than profile B? Are there any studies or publications that have examined this or a similar topic?