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.