Derivation of Pontryagins Minimum Principle

Basically, we intend to perturb the optimal control along the control axis as well as change the final time to examine the effect on the Hamiltonian. 

The time intervals of both perturbations are small enough to ensure the accuracy of the first-order approximation of the resulting new state. We will now examine these perturbations in the optimal control and determine the resulting states. 

---