On Hamiltonian as limiting gradient in infinite horizon problem

Necessary conditions of optimality in the form of the Pontryagin Maximum Principle are derived for the Bolza-type discounted problem with free right end. The optimality is understood in the sense of the uniformly overtaking optimality. Such process is assumed to exist, and the corresponding payoff of the optimal process (expressed in the form of improper integral) is assumed to converge in the Riemann sense. No other assumptions on the asymptotic behaviour of trajectories or adjoint variables are required. In this paper, we prove that there exists a corresponding limiting solution of the Pontryagin Maximum Principle that satisfies the Michel transversality condition; in particular, the stationarity condition of the maximized Hamiltonian and the fact that the maximized Hamiltonian vanishes at infinity are proved. The connection of this condition with the limiting subdifferentials of payoff function along the optimal process at infinity is showed. The case of payoff without discount multiplier is also considered.