On an Infinite Horizon Problem of Bolza Type

The first necessary conditions of optimality for infinite-horizon control problems were proved [1] on the verge of 1950–60s by L.S. Pontryagin and his associates (for the problems with the right end fixed at infinity). Only later [2] was the Maximum Principle proved for a reasonably broad class of problems, and yet the transversality-type conditions at infinity were not provided. Thus, the Maximum Principle for infinite horizon was not complete, which means the set of prospective optimal solutions it determined had the cardinality of continuum.
The principal obstacle on the way to transversality conditions at infinity is the fact that it is necessary to find the asymptotic conditions on the adjoint equation (i.e., on the linear system), that would be satisfied by at least one but not by all of its solutions. It was first done in [3] for a system with linear dynamics and the free right-hand end through passing to a functional space that allowed to extend all necessary solutions to infinity in the unique way. For the adjoint variable, there was proved a formula that supplemented the Maximum Principle to make it a complete system. In the papers [4–6], a more general formula (the Aseev–Kryazhimskii formula) was proved for other certain classes of nonlinear control problems. It takes the form of an improper integral of a function, the summability of which on the whole half-line is provided by means of imposing the asymptotic conditions (similar to the dominating discount conditions) on the system.
Another way to decrease the number of prospective solutions of such an incomplete system of relations was proposed by Seierstad [7]. He considered a set of shortened problems, in each of which he obtained the adjoint variable in the form of a solution of the complete system of relations (the Maximum Principle system for shortened problem). Under sufficiently strong assumptions he made, the adjoint variable, obtained as a pointwise limit, satisfied the maximum principle. The author extended this approach onto the class of infinite horizon control problems with the free right-hand end to the case of at least when the optimality criterion is at least the uniformly weakly overtaking optimality [8]. In particular, the transversality condition obtained through this means may be represented in the form of an Aseev–Kryazhimskii-type formula.
The author plans to report on the application of this approach to the problem of Bolza type.