# On a Few Questions Regarding the Study of State-Constrained Problems in Optimal Control

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## Abstract

The article is focused on the investigation of the necessary optimality conditions in the form of Pontryagin’s maximum principle for optimal control problems with state constraints. A number of results on this topic, which refine the existing ones, are presented. These results concern the nondegenerate maximum principle under weakened controllability assumptions and also the continuity of the measure Lagrange multiplier.

## Keywords

Optimal control Maximum principle State constraints## Mathematics Subject Classification

49N25## Notes

### Acknowledgements

We kindly thank professor A.V. Arutyunov for useful comments and fruitful discussions. The support from the Russian Foundation for Basic Research during the projects 16-31-60005, 18-29-03061, and the support of FCT R&D Unit SYSTEC—POCI-01-0145-FEDER-006933/SYSTEC funded by ERDF | COMPETE2020 | FCT/MEC | PT2020 extension to 2018, Project STRIDE NORTE-01-0145-FEDER-000033 funded by ERDF | NORTE2020, and FCT Project POCI-01-01-0145-FEDER-032485 funded by ERDF | COMPETE2020 | POCI are also acknowledged.

## References

- 1.Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Interscience, New York (1962)zbMATHGoogle Scholar
- 2.Gamkrelidze, R.V.: Optimum-rate processes with bounded phase coordinates. Dokl. Akad. Nauk SSSR
**125**, 475–478 (1959)MathSciNetzbMATHGoogle Scholar - 3.Warga, J.: Minimizing variational curves restricted to a preassigned set. Trans. Am. Math. Soc.
**112**, 432–455 (1964)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Dubovitskii, A.Y., Milyutin, A.A.: Extremum problems in the presence of restrictions. Zh. Vychisl. Mat. Mat. Fiz.
**5**(3), 395–453 (1965); U.S.S.R. Comput. Math. Math. Phys.**5**(3), 1–80 (1965)Google Scholar - 5.Neustadt, L.W.: An abstract variational theory with applications to a broad class of optimization problems. II: Applications. SIAM J. Control
**5**, 90–137 (1967)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Arutyunov, A.V., Tynyanskiy, N.T.: The maximum principle in a problem with phase constraints. Sov. J. Comput. Syst. Sci.
**23**, 28–35 (1985)MathSciNetGoogle Scholar - 7.Arutyunov, A.V.: On necessary optimality conditions in a problem with phase constraints. Sov. Math. Dokl.
**31**, 1 (1985)zbMATHGoogle Scholar - 8.Dubovitskii, A.Y., Dubovitskii, V.A.: Necessary conditions for strong minimum in optimal control problems with degeneration of endpoint and phase constraints. Usp. Mat. Nauk
**40**, 2 (1985)Google Scholar - 9.Arutyunov, A.V.: Perturbations of extremal problems with constraints and necessary optimality conditions. J. Sov. Math.
**54**, 6 (1991)CrossRefzbMATHGoogle Scholar - 10.Arutyunov, A.V., Blagodatskikh, V.I.: Maximum-principle for differential inclusions with space constraints, Number theory, algebra, analysis and their applications. Collection of articles. Dedicated to the centenary of Ivan Matveevich Vinogradov, Trudy Mat. Inst. Steklov., vol. 200, Nauka, Moscow (1991); Proc. Steklov Inst. Math., 200 (1993)Google Scholar
- 11.Arutyunov, A.V., Aseev, S.M., Blagodatskikh, V.I.: First-order necessary conditions in the problem of optimal control of a differential inclusion with phase constraints. Math. Sb.
**184**, 6 (1993)zbMATHGoogle Scholar - 12.Vinter, R.B., Ferreira, M.M.A.: When is the maximum principle for state constrained problems nondegenerate? J. Math. Anal. Appl.
**187**, 438–467 (1994)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Arutyunov, A.V., Aseev, S.M.: State constraints in optimal control. The degeneracy phenomenon. Syst. Control Lett.
**26**, 267–273 (1995)MathSciNetCrossRefzbMATHGoogle Scholar - 14.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 (1997)MathSciNetCrossRefzbMATHGoogle Scholar - 15.Ferreira, M.M.A., Fontes, F.A.C.C., Vinter, R.B.: Non-degenerate necessary conditions for nonconvex optimal control problems with state constraints. J. Math. Anal. Appl.
**233**, 116–129 (1999)MathSciNetCrossRefzbMATHGoogle Scholar - 16.Hager, W.W.: Lipschitz continuity for constrained processes. SIAM J. Control Optim.
**17**, 321–338 (1979)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Maurer, H.: Differential stability in optimal control problems. Appl. Math. Optim.
**5**(1), 283–295 (1979)MathSciNetCrossRefzbMATHGoogle Scholar - 18.Afanas’ev, A.P., Dikusar, V.V., Milyutin, A.A., Chukanov, S.A.: Necessary condition in optimal control. Nauka, Moscow (1990).
**[in Russian]**Google Scholar - 19.Galbraith, G.N., Vinter, R.B.: Lipschitz continuity of optimal controls for state constrained problems. SIAM J. Control Optim.
**42**, 5 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Arutyunov, A.V.: Properties of the Lagrange multipliers in the Pontryagin maximum principle for optimal control problems with state constraints. Differ. Equ.
**48**, 12 (2012)MathSciNetCrossRefGoogle Scholar - 21.Arutyunov, A.V., Karamzin, D.Y.: On some continuity properties of the measure Lagrange multiplier from the maximum principle for state constrained problems. SIAM J. Control Optim.
**53**, 4 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Halkin, H.: A satisfactory treatment of equality and operator constraints in the Dubovitskii–Milyutin optimization formalism. J. Optim. Theory Appl.
**6**, 2 (1970)MathSciNetCrossRefzbMATHGoogle Scholar - 23.Ioffe, A.D., Tikhomirov, V.M.: Theory of Extremal Problems. North-Holland, Amsterdam (1979)Google Scholar
- 24.Arutyunov, A.V.: Optimality Conditions: Abnormal and Degenerate Problems Mathematics and Its Application. Kluwer Academic Publisher, Dordrecht (2000)Google Scholar
- 25.Vinter, R.B.: Optimal Control. Birkhauser, Boston (2000)zbMATHGoogle Scholar
- 26.Milyutin, A.A.: Maximum Principle in a General Optimal Control Problem. Fizmatlit, Moscow (2001). [in Russian]Google Scholar
- 27.Arutyunov, A.V., Karamzin, D.Y., Pereira, F.L.: The maximum principle for optimal control problems with state constraints by R.V. Gamkrelidze: revisited. J. Optim. Theory Appl.
**149**, 474–493 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 28.Vinter, R.B., Papas, G.: A maximum principle for nonsmooth optimal control problems with state constraints. J. Math. Anal. Appl.
**89**, 212–232 (1982)MathSciNetCrossRefzbMATHGoogle Scholar - 29.Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983)zbMATHGoogle Scholar
- 30.Ioffe, A.D.: Necessary conditions in nonsmooth optimization. Math. Oper. Res.
**9**, 159–189 (1984)MathSciNetCrossRefzbMATHGoogle Scholar - 31.Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Discrete approximations of a controlled sweeping process. Set-Valued Var. Anal.
**23**(1), 69–86 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 32.Colombo, G., Henrion, R., Nguyen, D.H., Mordukhovich, B.S.: Optimal control of the sweeping process over polyhedral controlled sets. J. Differ. Equ.
**260**, 4 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 33.Cao, T.H., Mordukhovich, B.S.: Optimality conditions for a controlled sweeping process with applications to the crowd motion model. Discret. Contin. Dyn. Syst. Ser. B
**22**, 2 (2017)MathSciNetzbMATHGoogle Scholar - 34.Bryson, E.R., Yu-Chi, Ho: Applied Optimal Control. Taylor & Francis, London (1969)Google Scholar
- 35.Betts, J.T., Huffman, W.P.: Path-constrained trajectory optimization using sparse sequential quadratic programming. J. Guid. Control Dyn.
**16**(1), 59–68 (1993)CrossRefzbMATHGoogle Scholar - 36.Buskens, C., Maurer, H.: SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control. J. Comput. Appl. Math.
**120**, 85–108 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 37.Haberkorn, T., Trelat, E.: Convergence results for smooth regularizations of hybrid nonlinear optimal control problems. SIAM J. Control Optim.
**49**, 1498–1522 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 38.Dang, T.P., Diveev, A.I., Sofronova, E.A.: A Problem of Identification Control Synthesis for Mobile Robot by the Network Operator Method. Proceedings of the 11th IEEE Conference on Industrial Electronics and Applications (ICIEA), pp. 2413–2418 (2016)Google Scholar
- 39.Zeiaee, A., Soltani-Zarrin, R., Fontes, F.A.C.C., Langari, R.: Constrained directions method for stabilization of mobile robots with input and state constraints. Proceedings of the American Control Conference, pp. 3706–3711 (2017)Google Scholar
- 40.Mordukhovich, B.S.: Maximum principle in the problem of time optimal response with nonsmooth constraints. J. Appl. Math. Mech.
**40**(6), 960–969 (1976)MathSciNetCrossRefzbMATHGoogle Scholar - 41.Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. Volume II. Applications. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2006)CrossRefGoogle Scholar
- 42.Arutyunov, A.V., Vinter, R.B.: A simple ’finite approximations’ proof of the Pontryagin maximum principle under reduced differentiability hypotheses. Set-Valued Anal.
**12**(1–2), 5–24 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 43.Arutyunov, A.V., Karamzin, D.Y., Pereira, F.L.: Investigation of controllability and regularity conditions for state constrained problems. IFAC-PapersOnLine
**50**(1), 6295–6302 (2017)CrossRefGoogle Scholar - 44.Arutyunov, A.V., Karamzin, D.Y.: Properties of extremals in optimal control problems with state constraints. Differ. Equ.
**52**, 11 (2016)MathSciNetCrossRefzbMATHGoogle Scholar