Pontryagin's Maximum Principle

Pontryagin's Maximum Principle is a mathematical method used to find the best way to control a system over time, such as minimizing costs or maximizing efficiency. It involves defining a goal and a set of rules for how the system can change, then determining the control actions that optimize the outcome. The principle introduces auxiliary variables called costates, which help evaluate how changes affect the goal, and guides you to choose controls that maximize or minimize a specific function at every moment. This approach provides a systematic way to solve complex decision-making problems in engineering, economics, and other fields.