On Adequate Transversality Conditions for Infinite Horizon Optimal Control Problems—A Famous Example of Halkin

In this paper we apply a duality concept of Klötzler (Equadiff IV. Proceedings of the Czechoslovak conference on differential equations and their applications held in Prague, 22–26 August, 1977, Lecture notes in mathematics, vol. 703, pp. 189–196, Springer, Berlin, 1979) to infinite horizon optimal control problems. The key idea is the choice of weighted Sobolev spaces as state spaces. Different criteria of optimality are known for specific problems (Carlson et al., Infinite horizon optimal control, Springer, New York, Berlin, Heidelberg, 1991; Feichtinger and Hartl, Optimale Kontrolle ökonomischer Prozesse, de Gruyter, Berlin, New York, 1986), e.g. the overtaking criterion of von Weizsäcker (1965), the catching up criterion of Gale (1967) and the sporadically catching up criterion of Halkin (1974). Corresponding to the formulated optimality criteria we develop a duality theory and prove sufficient conditions for optimality. An example of Halkin is presented, where the solution is obtained in the framework of these weighted spaces. The obtained dual solution of the problem demonstrates which kind of transversality conditions are natural.