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Towards Geometric Time Minimal Control without Legendre Condition and with Multiple Singular Extremals for Chemical Networks

This article deals with the problem of maximizing the production of a species for a chemical network by controlling the temperature. Under the so-called mass kinetics assumption the system can be modeled as a single-input control system using the Feinberg-Horn-Jackson graph associated to the reactions network. Thanks to Pontryagin's Maximum Principle, the candidates as minimizers can be found among extremal curves, solutions of a (non smooth) Hamiltonian dynamics and the problem can be stated as a time minimal control problem with a terminal target of codimension one. Using geometric control and singularity theory the time minimal syntheses (closed loop optimal control) can be classified near the terminal manifold under generic conditions. In this article we focus to the case where the generalized Legendre-Clebsch condition is not satisfied, which paves the road to complicated syntheses with several singular arcs. In particular it is related to the situation for a weakly reversible network like the McKeithan scheme of two reactions.