Existence of Solutions for a Class of Infinite Horizon Optimal Control Problems without Discounting Arising in Economic Dynamics

The study of the existence and the structure of solutions of optimal control problems defined on infinite intervals and on sufficiently large intervals has recently become a rapidly growing area of research. See, for example, [2], [4]-[9], [13, 14,17,18], [22]-[26], [30] and the references mentioned therein. These problems arise in engineering [1,12,15], in models of economic growth [11,19,21,27,28], [30]-[32], in infinite discrete models of solid-state physics related to dislocations in one-dimensional crystals [3,29] and in the theory of thermodynamical equilibrium for materials [10, 16, 20]. In this paper, we study a large class of nonconvex infinite horizon discrete-time optimal control problems. This class contains optimal control problems arising in economic dynamics which describe a general two-sector model without discounting and with nonconcave utility functions representing the preferences of the planner. Let R (R +) be a set of all real (nonnegative) numbers, R n be the n-dimensional Euclidean space with a non-negative orthant R + = {x ∈ R : x = (x1, . . . , xn), xi ≥ 0, i = 1, . . . , n},