Pontryagin's Minimum Principle

Pontryagin's Minimum Principle is a mathematical tool used to find the best way to control a system to achieve a goal efficiently. Imagine you're guiding a vehicle; the principle helps determine the optimal steering and acceleration to reach a destination in the shortest time or with the least fuel, considering system dynamics and constraints. It does this by introducing an auxiliary mathematical function (the Hamiltonian) that combines system behavior and objectives, and then finds the control actions that minimize this function over time, ensuring optimal performance based on the specified criteria.