## A problem of dynamic optimization in the presence of dangerous factors.(English)Zbl 1452.49012

Tarasyev, Alexander (ed.) et al., Stability, control and differential games. Proceedings of the international conference on stability, control and differential games (SCDG2019), Yekaterinburg, Russia, September 16–20, 2019. Cham: Springer. Lect. Notes Control Inf. Sci. – Proc., 273-281 (2020).
An optimal control problem with a mixed functional and free stopping time is considered. Dynamics of the system is given by means of a differential inclusion. The integral term of the functional contains the characteristic function of a given open set $$M \subset\mathbb{R}^n$$ which can be interpreted as a “risk” or “dangerous” zone. The statement of the problem (Problem P) can be treated as a weakening of the statement of the classical optimal control problem with state constraints. The goal of the present paper is to study relationships between Problem P and the optimal control problem for differential inclusion with state constraints. The main result states that, under suitable controllability assumptions, Problem P can provide an exact penalization of the corresponding problem with state constraints. An illustrative example is also provided.
For the entire collection see [Zbl 1444.93003].

### MSC:

 49K21 Optimality conditions for problems involving relations other than differential equations 34A60 Ordinary differential inclusions

### Keywords:

optimal control; differential inclusion; control problem
Full Text:

### References:

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