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Investigation of Controllability and Regularity Conditions for State Constrained Problems * *This publication was supported by the Russian Foundation for Basic Research, project no. 16-31-60005, and by the Ministry of Education and Science of the Russian Federation (Agreement number 02.a03.21.0008 and project no. 1.962.2017). The results of Section 2 were obtained by the first author under the support of the Russian Science Foundation (project no. 17-11-01168). The third author also acknowledges the support of FCT R&D Unit SYSTEC ¿ POCI-01-0145-FEDER-006933/SYSTEC funded by ERDF, COM-PETE2020, FCT/MEC, PT2020, and Project STRIDE ¿ NORTE-01-0145-FEDER-000033, funded by ERDF, NORTE 2020.