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In the present paper, the maximum principle for finite horizon stateconstrained 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, andH.~Frankowska, and resembles the optimal growth problem in mathematicaleconomics. The solution existence of these parametric examples is establishedby invoking Filippov's existence theorem for Mayer problems. Since the maximumprinciple is only a necessary condition for local optimal processes, a largeamount of additional investigations is needed to obtain a comprehensivesynthesis of finitely many processes suspected for being local minimizers. Ouranalysis not only helps to understand the principle in depth, but also servesas a sample of applying it to meaningful prototypes of economic optimal growthmodels. Problems with unilateral state constraints are studied in Part 1 of thepaper. Problems with bilateral state constraints will be addressed in Part 2.

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