Necessary Conditions for Control Problems on Infinite Intervals Without Any Asymptotic Assumptions

This report will discuss control problems on infinite intervals with a weakly overtaking optimality criterion. As a necessary condition for such problems, we will derive the maximum principle of L.S. Pontryagin along with a boundary condition at zero, which plays the role of a transversality condition at infinity.
In the absence of asymptotic terminal constraints in the problem (free right endpoint problem), this necessary boundary condition can be described in terms of the asymptotic subdifferential at the initial point from the cost function while fixing the suspect control for optimality. If this subdifferential is single-valued, the obtained condition is equivalent to the representation of the adjoint variable proposed by A.V. Kryazhimskiy and S.M. Aseev in the form of a Cauchy-type formula.
In the first part of the report, we plan to obtain the necessary boundary condition within the framework of the Halkin scheme for the system of Pontryagin's maximum principle, reducing it to estimates of the subdifferential limits of scalar Lipschitz functions. Most examples, including Ramsey-type problems, as well as the development of such an approach that simplifies obtaining necessary boundary conditions under various assumptions, will be postponed to the second part of the report.