Abstract
This chapter discusses recent developments in the infinite time optimal control problem. The usual optimal control problem is formulated by an evolution equation. The infinite time optimal control problem is obtained by letting T → ∞, where T is an arbitrary end-time. The importance of this problem for the applications is obvious; for example, a firm may want to maximize its profits in the whole future, and while to consider the infinite time interval [0, ∞] seems unrealistic, it is usually even more unrealistic to impose an arbitrary end-time T. Problems of this type also arise in the optimal management of renewable resources. Periodic solutions to periodic control problems should be considered naturally within the context of optimization in [0, ∞]. The chapter highlights the optimality in [0, ∞]. It also presents Halkin's necessary conditions and explains the determining set for the adjoint vector.
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