Can someone please explain the difference between dynamic optimization via the Bellman equation and dynamic optimization via Pontryagin's maximization principle?
Thanks
2026-04-04 16:12:56.1775319176
Difference between Bellman and Pontryagin dynamic optimization?
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The Bellman principle poses an optimization problem using a nonlinear 1st order partial differential equation - the object being optimized is a function. Pontryagin's maximum principle poses the same problem using a form of the calculus of variations - the optimized object is a curve.
This implies in technical differences and different conditions for existence of solutions, which are discussed in the literature. The optimal curve given by the maximum principle is a Cauchy characteristic of the Hamilton Jacobi Bellman partial differential equation, assuming technical conditions such that both exist.
Bellman's method was originally formulated for discrete-time systems, and extended to continuous time ones. The maximum principle is mostly studied in the continuous time framework.