We consider an optimal control problem for an autonomous differential inclusion with free terminal time and a mixed functional which contains the characteristic function of a given open set $M\subset\mathbb{R}^n$ in the integral term. The statement of the problem weakens the statement of the classical optimal control problem with state constraints to the case when the presence of admissible trajectories of the system in the set $M$ is physically allowed but unfavorable due to safety or instability reasons. Using an approximation approach, necessary conditions for the optimality of an admissible trajectory are obtained in the form of Clarke's Hamiltonian inclusion. The result involves a nonstandard stationarity condition for the Hamiltonian. As in the case of the problem with a state constraint, the obtained necessary optimality conditions may degenerate.Conditions guaranteeing their nondegeneracy and pointwise nontriviality are presented. The results obtained extend the author's previous results to the case of a problem with free terminal time and more general functional.
Keywords: risk zone, state constraints, optimal control, Hamiltonian inclusion, Pontryagin maximum principle, nondegeneracy conditions.
Funding Agency: This work was supported by the Russian Science Foundation (project no.~14-50-00005).
The paper was received by the Editorial Office on October 10, 2017.
Sergey Mironovich Aseev, Dr. Phys.-Math. Sci., Corresponding Member of RAS, Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, 119991 Russia, e-mail: aseev@mi.ras.ru.
REFERENCES
1. Arutyunov A.V. Perturbations of extremal problems with constraints and necessary optimality conditions. J. Soviet Math., 1991, vol. 54, no. 6, pp. 1342–1400. doi: https://doi.org/10.1007/BF01373649 .
2. Arutyunov A.V. Optimality conditions. Abnormal and degenerate problems. Math. and its Appl. (Dordrecht), vol. 526, Dordrecht, Kluwer Acad. Publ., 2000, 299 p. ISBN: 0-7923-6655-7 . Original Russian text published in Arutyunov A.V. Usloviya ekstremuma. Anormal’nye i vyrozhdennye zadachi. Moscow, Faktorial Publ., 1997, 255 p. ISBN: 5-88688-015-1 .
3. Aseev S.M. Optimization of dynamics of a control system in the presence of risk factors. Tr. Inst. Mat. i Mekh. UrO RAN, 2017, vol. 23, no. 1, pp. 27–42 (in Russian). doi: https://doi.org/10.21538/0134-4889-2017-23-1-27-42 .
4. Aseev S.M., Smirnov A.I. The Pontryagin maximum principle for the problem of optimally crossing a given domain. Dokl. Math., 2004, vol. 69, no. 2, pp. 243–245.
5. Aseev S.M., Smirnov A.I. Necessary first-order conditions for optimal crossing of a given region. Comput. Math. Model., 2007, vol. 18, no. 4, pp. 397–419. doi: https://doi.org/10.1007/s10598-007-0034-8 .
6. Ioffe A.D., Tihomirov V.M. Theory of extremal problems. Amsterdam, Elsevier North-Holland, 1979, 460 p. Original Russian text published in Ioffe A.D., Tikhomirov V.M. Teoriya ekstremal’nykh zadach, Moscow, Nauka Publ., 1974, 481 p.
7. Arutyunov, A.V., Karamzin, D.Yu., Pereira, F.L. The maximum principle for optimal control problems with state constraints by R.V. Gamkrelidze: revisited. J. Optim. Theory Appl., 2011, vol. 149, no. 3, pp. 474–493. doi: https://doi.org/10.1007/s10957-011-9807-5 .
8. Clarke F.H. Optimization and nonsmooth analysis. N Y, Wiley, 1983, 308 p. Translated to Russian under the title Optimizatsiya i negladkii analiz. Moscow, Nauka Publ. 1988. 280 p.
9. Mordukhovich B.Sh. Maximum principle in the problem of time optimal response with nonsmooth constraints. J. Appl. Math. Mech., 1976, vol. 40, no. 6, pp. 960–969.
doi: https://doi.org/10.1016/0021-8928(76)90136-2 .
10. Mordukhovich B.Sh. Metody approksimatsii v zadachakh optimizatsii i upravleniya [Approximation methods in problems of optimization and control]. Moscow, Nauka Publ., 1988, 360 p.
11. Natanson I.P. Theory of functions of a real variable. N Y, Frederick Ungar Publishing Co., 1955, 277 p. (Translation of chapters I to IX of the author’s Teoriya funktsii veshchestvennoi peremennoi, Moscow, Gostehizdat Publ., 1950).
12. Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. The mathematical theory of optimal processes, ed. L.W. Neustadt, N Y, London, Interscience Publishers John Wiley & Sons, Inc., 1962, 360 p. Original Russian text published in Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. Matematicheskaya teoriya optimal’nykh protsessov, Moscow, Fizmatgiz Publ., 1961, 393 p.
13. Pshenichnyi B.N., Ochilov S. On the problem of the optimal passage through a given domain. Kibernet. i Vychisl. Tekhn. 1993, vol. 99, pp. 3–8 (in Russian).
14. Pshenichnyi B.N., Ochilov S. On a special time-optimality problem. Kibernet. i Vychisl. Tekhn., 1994, vol. 101, pp. 11–15.
15. Smirnov A.I. Necessary optimality conditions for a class of optimal control problems with discontinuous integrand. Proc. Steklov Inst. Math., 2008, vol. 262, no. 1, pp. 213–230. doi: https://doi.org/10.1134/S0081543808030176 .
16. 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., 1997, vol. 35, no. 3, pp. 930–952. doi: 10.1137/S036301299426996X .
17. Aseev S.M. Methods of regularization in nonsmooth problems of dynamic optimization. J. Math. Sci., 1999, vol. 94, no. 3, pp. 1366–1393. doi: https://doi.org/10.1007/BF02365018 .
18. Cesari L. Optimization – theory and applications. Problems with ordinary differential equations. N Y, Springer, 1983, 542 p. doi: https://doi.org/10.1007/978-1-4613-8165-5 .
19. Ferreira, M.M.A., Vinter, R.B. When is the maximum principle for state constrained problems nondegenerate? J. Math. Anal. Appl., 1994, vol. 187, no. 2, pp. 438–467. doi: https://doi.org/10.1006/jmaa.1994.1366 .
20. Fontes F.A.C.C., Frankowska H. Normality and nondegeneracy for optimal control problems with state constraints. J. Optim. Theory Appl., 2015, vol. 166, no. 1, pp. 115–136. doi: https://doi.org/10.1007/s10957-015-0704-1 .
