Necessary Conditions In Infinite-Horizon Control Problem That Need No Asymptotic Assumptions

We consider a Bolza-type infinite-horizon control problem with free right end. We propose a modification of Halkin's general construction of necessary conditions of optimality in which the transversality condition is obtained through the theorems on stability of subdifferentials. For the weakly overtaking criterion, a necessary boundary condition on co-state arc is deduced, regardless of any assumptions on the asymptotic behavior of trajectories, adjoint variables, or their derivatives. Regardless of any assumptions, the Pontryagin Maximum Principle with this boundary condition allows to educe some convex hull of co-state arcs, corresponding to the convex subdifferential of payoff function (fixing the optimal control) at infinity. In the case of smooth payoff function at infinity, this condition educes the unique co-state arc, and the corresponding co-state arc coincides with the solution of the Cauchy-type formula proposed by S.M.Aseev and A.V.Kryazhimskii. These results are illustrated with a pair of examples