Hi Javier, your text is interesting. To answer your question: everything depends on how who include state-space equation into Hamiltonian for Pontryagin principle. If your state-space equations happen to behave according to exponential process, it is not a surprise that maximization of entropy measure is Jaynes's one. Look at subsection 8.3.3. on exponential growth model in a book of Kapur and Kesavan  "Entropy Optimization Principles with Applications". As you use convex functional in Pontryagin as they are used in Jaynes case there is no difference whether it is Pontryagin or any other convex variational optimization.

Thank you, Renaldas.  I will take a look at your recommendation: "Entropy Optimization Principles with Applications" by Kapur and Kesavan.