On the properties of monotone functionals for generalizing Bellman Principle

Dynamic programming is classically used for solving optimal control problems. This approach is based on equating the value function of the problem under study as well as the Bellman Principle. However, despite a sharing common framework, each optimal control problem is addressed independently, depending on the characteristics of the cost function. The objective of this article is to establish a unified approach within the context of deterministic, autonomous dynamics with an infinite horizon. The ultimate purpose is to determine conditions from which an optimal control problem admits a Bellman principle . To achieve this, we introduce a typology composed of three so-called monotone functionals, which encompass typical cost functions met in the literature. Any functional that complies with the properties of one of this class can be used for tackling optimal control problems through Bellman principle. Then, we show that classical functionals used in the literature can be classified in one these classes. Finally, we show how to build a functional according to this typology through a simple problem of optimization.