Pontryagin Maximum Principle

The Pontryagin Maximum Principle is a mathematical tool used to find the best way to control a system to achieve a goal efficiently. Imagine steering a vehicle to minimize fuel use while reaching a destination; this principle helps determine the optimal controls—like speed or direction—over time. It does this by combining the system's current state, the goal, and how changing controls affects future outcomes, guiding you to the most effective strategy. Essentially, it provides a set of conditions to identify the best course of action in complex dynamic problems.