One of the most profound results of applied mathematics, Pontryagin’s minimum principle provides the necessary conditions for the minimum of an optimal control problem. The elegance of the principle lies in the simplicity of its application to a vast variety of optimal control problems. Boltyanskii et al. (1956) developed the principle originally as a maximum principle requiring the Hamiltonian to be maximized at the minimum.

In this chapter, we will present the proof of the minimum principle. The minimum principle uses a positive multiplier for the objective functional in the Hamiltonian formulation.* With this provision, the minimum principle concludes that the minimum of the problem requires minimization of the Hamiltonian in an optimal control problem whose minimum needs to be determined.

Some readers may first want to get the essence of the minimum principle and go cursorily over the derivation in Section 5.4. This section may be skipped during the initial reading.