# Analyzing a Maximum Principle for Finite Horizon State Constrained Problems via Parametric Examples. Part 2: Problems with Bilateral State Constraints

## TL;DR

This paper examines the maximum principle for finite horizon state constrained optimal control problems with bilateral constraints, using parametric examples inspired by economic growth models, and discusses solution existence and necessary conditions.

## Contribution

It extends the analysis of the maximum principle to problems with bilateral state constraints through parametric examples and solution existence proofs.

## Key findings

- Established solution existence via Filippov's theorem.
- Analyzed the maximum principle as a necessary condition.
- Provided insights into local optimal processes in economic models.

## Abstract

In the present paper, the maximum principle for finite horizon state constrained problems from the book by R. Vinter [\textit{Optimal Control}, Birkh\"auser, Boston, 2000; Theorem~9.3.1] is analyzed via parametric examples. The latter has origin in a recent paper by V.~Basco, P.~Cannarsa, and H.~Frankowska, and resembles the optimal growth problem in mathematical economics. The solution existence of these parametric examples is established by invoking Filippov's existence theorem for Mayer problems. Since the maximum principle is only a necessary condition for local optimal processes, a large amount of additional investigations is needed to obtain a comprehensive synthesis of finitely many processes suspected for being local minimizers. Our analysis not only helps to understand the principle in depth, but also serves as a sample of applying it to meaningful prototypes of economic optimal growth models. Problems with unilateral state constraints have been studied in Part 1 of the paper. Problems with bilateral state constraints are addressed in this Part 2.