Analyzing a Maximum Principle for Finite Horizon State Constrained Problems via Parametric Examples. Part 1: Problems with Unilateral State Constraints

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 are studied in Part 1 of the paper. Problems with bilateral state constraints will be addressed in Part 2.