Necessary conditions for infinite horizon control problems regardless of any asymptotic assumptions

The talk will consider control problems on infinite horizon with an weakly overtaking optimality criterion; As an optimality condition necessary for such problems, L. S. Pontryagin's maximum principle will be considered together with a boundary condition at zero, which plays the role of a transversality condition at infinity. //// In the absence of an asymptotic terminal constraints in the problem (a problem with a free right end), such a necessary boundary condition can be described in terms of asymptotic subdifferentials at the initial state from the payoff function where the control is fixed. If this subdifferential is a singleton, this condition is equivalent to the representation of the adjoint trajectory proposed by A. V. Kryazhimsky and S. M. Aseev in the form of a Cauchy type formula. //// In the first part of the talk, it is planned to obtain, within the framework of the Halkin scheme, the necessary boundary condition for the system of the maximum principle of L. S. Pontryagin, reducing it to estimates of the subdifferential of limits of Lipschitz scalar functions. Most of the examples, including for Ramsey-type problems, as well as some simplification of such an approach are planned to be postponed until the second part of the talk.