Geodesic
Weakening State Constraints in Optimal Control Problems
Informatics and Automation, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 31-44

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An optimal control problem is considered in which the integral term of the functional to be minimized contains the characteristic function of a given open set of undesirable system states. The statement of this problem can be viewed as a weakening of the statement of the standard optimal control problem with a state constraint. Conditions are obtained that guarantee the equivalence of these problems. Two illustrative examples are given.
Keywords: optimal control, state constraint, risk zone, differential inclusion. 
S. M. Aseev. Weakening State Constraints in Optimal Control Problems. Informatics and Automation, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 31-44. http://geodesic.mathdoc.fr/item/TRSPY_2023_321_a1/
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