On a Class of Optimal Control Problems Arising in Mathematical Economics

This paper is devoted to the study of the properties of the adjoint variable in the relations of the Pontryagin maximum principle for a class of optimal control problems that arise in mathematical economics. This class is characterized by an infinite time interval on which a control process is considered and by a special goal functional defined by an improper integral with a discounting factor. Under a dominating discount condition, we discuss a variant of the Pontryagin maximum principle that was obtained recently by the authors and contains a description of the adjoint variable by a formula analogous to the well-known Cauchy formula for the solutions of linear differential equations. In a number of important cases, this description of the adjoint variable leads to standard transversality conditions at infinity that are usually applied when solving optimal control problems in economics. As an illustration, we analyze a conventionalized model of optimal investment policy of an enterprise.