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].


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


[1] Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) · Zbl 0582.49001
[2] Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York (1962) · Zbl 0102.32001
[3] Pshenichnyi, B.N., Ochilov, S.: On the problem of optimal passage through a given domain. Kibern. Vychisl. Tekhn. 99, 3-8 (1993)
[4] Pshenichnyi, B.N., Ochilov, S.: A special problem of time-optimal control. Kibern. Vychisl. Tekhn. 101, 11-15 (1994) · Zbl 0881.49021
[5] Aseev, S.M., Smirnov, A.I.: The Pontryagin maximum principle for the problem of optimal crossing of a given domain. Dokl. Math. 69(2), 243-245 (2004) · Zbl 1282.49015
[6] Aseev, S.M., Smirnov, A.I.: Necessary first-order conditions for optimal crossing of a given region. Comput. Math. Model. 18(4), 397-419 (2007) · Zbl 1136.65064
[7] Smirnov, A.I.: Necessary optimality conditions for a class of optimal control problems with discontinuous integrand. Proc. Steklov Inst. Math. 262, 213-230 (2008) · Zbl 1160.49020
[8] Aseev, S.M.: An optimal control problem with a risk zone. In: 11th International Conference on Large-Scale Scientific Computing, LSSC 2017, Sozopol, Bulgaria, 5-9 June, 2017. Lecture Notes in Computer Science, vol. 10665. Springer, Cham, pp. 185-192 (2018) · Zbl 1443.49002
[9] Aseev, S.M.: On an optimal control problem with discontinuous integrand. Proc. Steklov Inst. Math. 24(Suppl. 1), S3-S13 (2019) · Zbl 1420.49029
[10] Arutyunov, A.V., Aseev, S.M.: Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints. SIAM J. Control Optim. 35(3), 930-952 (1997) · Zbl 0873.49014
[11] Filippov, A.F.: Differential Equations with Discontinuous Right-Hand Sides. Kluwer, Dordrecht (1988)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.