# On transversality condition for overtaking optimality in infinite horizon control problem

## TL;DR

This paper establishes necessary conditions for infinite-horizon optimal control problems with overtaking optimality, introducing a boundary condition on the co-state arc that becomes complete under certain asymptotic assumptions.

## Contribution

It develops a boundary condition for the co-state arc in infinite-horizon control problems, providing a complete system of relations under additional asymptotic assumptions.

## Key findings

- Derived a boundary condition necessary for optimality.
- Showed the boundary condition leads to a unique co-state arc.
- Provided an explicit formula for the co-state arc using the boundary condition.

## Abstract

In this paper we investigate necessary conditions of optimality for infinite-horizon optimal control problems with overtaking optimality as an optimality criterion. For the case of local Lipschitz continuity of the payoff function, we construct a boundary condition on the co-state arc that is necessary for the optimality. We also show that, under additional assumptions on the payoff function's asymptotic behavior, the Pontryagin Maximum Principle with this condition becomes a complete system of relations, and this boundary condition points out the unique co-state arc through a Cauchy-type formula. An example is given to clarify the application of this formula as an explicit expression of the co-state arc. The cornerstone of this paper is the theorem on convergence of subdifferentials.