The Pontryagin maximum principle from dynamic programming and viscosity solutions to first-order partial differential equations
Emmanuel Nicholas Barron, Robert R. Jensen · Transactions of the American Mathematical Society · 1986
We prove the Pontryagin Maximum Principle for the Lagrange problem of optimal control using the fact that the value function of the problem is the viscosity solution of the associated Hamilton-Jacobi-Bellman equation. The proof here makes rigorous the formal proof of Pontryagin’s principle known for at least three decades.