# The Special Case of the Bounded Processes

## Abstract

Infinite-horizon discrete-time optimal control problems in the set of bounded processes are examined. According to Chichilnisky (Soc. Choice Welfare **13**, 231–257 (1996)) and Chichilnisky and Kalman (J. Optim. Theor. Appl. **30**(1), 19–32 (1980)) the space of bounded sequences was first used in economics by Debreu (Proc Natl Acad Sci USA **40,**(7), 588–592 (1954)). It can also be found in Carlson et al. (*Infinite Horizon Optimal Control: Deterministic and Stochastic Systems* (1991)), for example. From a mathematical point of view it allows to use analysis in Banach spaces. We establish necessary conditions of optimality for infinite-horizon discrete-time optimal control problems with state equation or state inequation, for bounded processes. It necessitates to manipulate the dual of *ℓ* _{ ∞ }, to establish results on bounded solutions of difference equations of order one. We apply abstract optimization theorems in Banach spaces to obtain strong and weak Pontryagin principles in Sect. 3.2. In Sect. 3.3, for problems governed by inequations, we work in ordered Banach spaces and we treat the state inequation as an infinity of inequality constraints, by using abstract results of optimization theory in ordered Banach spaces in the spirit of the Karush–Kuhn–Tucker theorem. In Sect. 3.4, we provide links with unbounded problems and in Sect. 3.5 we give sufficient conditions of optimality. The mathematical tools used in this chapter belong to linear and nonlinear functional analysis: sequence spaces, Nemytskii’s operators, duality in topological vector spaces, and ordered Banach spaces.

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