Pontryagin’s Theorem

Pontryagin’s Theorem provides conditions for identifying optimal controls in dynamic systems. It states that to optimize a performance function—such as minimizing cost or maximizing efficiency—there must exist a set of auxiliary variables called adjoint variables. These, together with the system’s state variables, satisfy specific differential equations known as the Pontryagin’s Maximum Principle. Essentially, the theorem guides how to choose control actions over time by ensuring that the combined state and adjoint variables follow optimal trajectories, leading to the best possible outcome for the system under given constraints.