  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.
