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Refined Euler–Lagrange Inclusion for an Optimal Control Problem with Discontinuous Integrand

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Abstract

We study a free-time optimal control problem for a differential inclusion with mixed-type functional in which the integral term contains the characteristic function of a given open set of “undesirable” states of the system. The statement of this problem can be viewed as a weakening of the statement of the classical optimal control problem with state constraints. Using the approximation method, we obtain first-order necessary optimality conditions in the form of the refined Euler–Lagrange inclusion. We also present sufficient conditions for their nondegeneracy and pointwise nontriviality and give an illustrative example.

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Notes

  1. Recall that a point \(t\in [0,T)\), \(T>0\), is called a point of right approximate continuity of a function \(\xi \colon\, [0,T]\to{\mathbb R}^1\) if there exists a Lebesgue measurable set \(E\subset [t,T]\) such that \(t\) is a density point of \(E\) and the function \(\xi (\kern.5pt\cdot\kern.5pt) \) is right-continuous at the point \(t\) along the set \(E\) (see [23, Ch. IX, §6]).

  2. Recall that the contingent cone to a set \(A\) at a point \(\xi\in \overline A\) is the set \(K_A(\xi)=\{v \colon\, \exists\,v_i\to v,\ \exists\,\alpha_i\to +0 \colon\, \, \xi+\alpha_iv_i\in A\}\) (see [21]). Accordingly, \(\Gamma_A(\xi)=K_A^*(\xi)=\{p \colon\, \langle p,v\rangle\leq 0\ \forall\,v\in K_A(\xi)\}\).

References

  1. A. V. Arutyunov, “Perturbations of extremal problems with constraints and necessary optimality conditions,” J. Sov. Math. 54 (6), 1342–1400 (1991) [transl. from Itogi Nauki Tekh., Ser. Mat. Anal. 27, 147–235 (1989)].

    Article  Google Scholar 

  2. A. V. Arutyunov, Optimality Conditions: Abnormal and Degenerate Problems (Faktorial, Moscow, 1997; Kluwer, Dordrecht, 2000).

    MATH  Google Scholar 

  3. A. V. Arutyunov and S. M. Aseev, “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).

    Article  MathSciNet  Google Scholar 

  4. A. V. Arutyunov, D. Yu. Karamzin, and F. L. Pereira, “The maximum principle for optimal control problems with state constraints by R. V. Gamkrelidze: Revisited,” J. Optim. Theory Appl. 149 (3), 474–493 (2011).

    Article  MathSciNet  Google Scholar 

  5. S. M. Aseev, “Methods of regularization in nonsmooth problems of dynamic optimization,” J. Math. Sci. 94 (3), 1366–1393 (1999).

    Article  MathSciNet  Google Scholar 

  6. S. M. Aseev, “Extremal problems for differential inclusions with state constraints,” Proc. Steklov Inst. Math. 233, 1–63 (2001) [transl. from Tr. Mat. Inst. Steklova 233, 5–70 (2001)].

    Google Scholar 

  7. S. M. Aseev, “Optimization of dynamics of a control system in the presence of risk factors,” Tr. Inst. Mat. Mekh. (Ekaterinburg) 23 (1), 27–42 (2017).

    MathSciNet  Google Scholar 

  8. S. M. Aseev, “On an optimal control problem with discontinuous integrand,” Proc. Steklov Inst. Math. 304 (Suppl. 1), S3–S13 (2019) [transl. from Tr. Inst. Mat. Mekh. (Ekaterinburg) 24 (1), 15–26 (2018)].

    Article  MathSciNet  Google Scholar 

  9. S. M. Aseev, “An optimal control problem with a risk zone,” in Large-Scale Scientific Computing: 11th Int. Conf., LSSC 2017, Sozopol, 2017 (Springer, Cham, 2018), Lect. Notes Comput. Sci. 10665, pp. 185–192.

    Chapter  Google Scholar 

  10. S. M. Aseev, “A problem of dynamic optimization in the presence of dangerous factors,” in Stability, Control and Differential Games: Proc. Int. Conf. SCDG2019, Yekaterinburg, 2019 (Springer, Cham, 2020), Lect. Notes Control Inf. Sci. – Proc., pp. 273–281.

    Chapter  Google Scholar 

  11. S. M. Aseev and A. I. Smirnov, “The Pontryagin maximum principle for the problem of optimally crossing a given domain,” Dokl. Math. 69 (2), 243–245 (2004) [transl. from Dokl. Akad. Nauk 395 (5), 583–585 (2004)].

    MATH  Google Scholar 

  12. S. M. Aseev and A. I. Smirnov, “Necessary first-order conditions for optimal crossing of a given region,” Comput. Math. Model. 18 (4), 397–419 (2007) [transl. from Nelinein. Din. Upr. 4, 179–204 (2004)].

    Article  MathSciNet  Google Scholar 

  13. L. Cesari, Optimization—Theory and Applications: Problems with Ordinary Differential Equations (Springer, New York, 1983).

    Book  Google Scholar 

  14. F. H. Clarke, “The Euler–Lagrange differential inclusion,” J. Diff. Eqns. 19, 80–90 (1975).

    Article  MathSciNet  Google Scholar 

  15. F. H. Clarke, Optimization and Nonsmooth Analysis (J. Wiley & Sons, New York, 1983).

    MATH  Google Scholar 

  16. F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control (Springer, London, 2013), Grad. Texts Math. 264.

    Book  Google Scholar 

  17. M. M. A. Ferreira and R. B. Vinter, “When is the maximum principle for state constrained problems nondegenerate?,” J. Math. Anal. Appl. 187 (2), 438–467 (1994).

    Article  MathSciNet  Google Scholar 

  18. F. A. C. C. Fontes and H. Frankowska, “Normality and nondegeneracy for optimal control problems with state constraints,” J. Optim. Theory Appl. 166 (1), 115–136 (2015).

    Article  MathSciNet  Google Scholar 

  19. A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems (Nauka, Moscow, 1974; North-Holland, Amsterdam, 2009).

    Google Scholar 

  20. B. Sh. Mordukhovich, “Maximum principle in the problem of time optimal response with nonsmooth constraints,” J. Appl. Math. Mech. 40, 960–969 (1976) [transl. from Prikl. Mat. Mekh. 40 (6), 1014–1023 (1976)].

    Article  MathSciNet  Google Scholar 

  21. B. Sh. Mordukhovich, Approximation Methods in Problems of Optimization and Control (Nauka, Moscow, 1988) [in Russian].

    MATH  Google Scholar 

  22. B. Sh. Mordukhovich, “Optimal control of difference, differential, and differential–difference inclusions,” J. Math. Sci. 100 (6), 2613–2632 (2000) [transl. from Itogi Nauki Tekh., Ser.: Sovrem. Mat. Prilozh., Temat. Obz. 61, 33–65 (1999)].

    Article  MathSciNet  Google Scholar 

  23. I. P. Natanson, Theory of Functions of a Real Variable (Nauka, Moscow, 1974; Dover Publ., Mineola, NY, 2016).

    Google Scholar 

  24. E. S. Polovinkin and G. V. Smirnov, “Time-optimal problem for differential inclusions,” Diff. Eqns. 22 (8), 940–952 (1986) [transl. from Diff. Uravn. 22 (8), 1351–1365 (1986)].

    Google Scholar 

  25. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes (Fizmatgiz, Moscow, 1961; Pergamon, Oxford, 1964).

    MATH  Google Scholar 

  26. B. N. Pshenichnyi and S. Ochilov, “On the problem of optimal passage through a given domain,” Kibern. Vychisl. Tekh., Kiev 99, 3–8 (1993).

    MathSciNet  Google Scholar 

  27. B. N. Pshenichnyi and S. Ochilov, “On a special time-optimal problem,” Kibern. Vychisl. Tekh., Kiev 101, 11–15 (1994).

    MATH  Google Scholar 

  28. R. Schneider, “Equivariant endomorphisms of the space of convex bodies,” Trans. Am. Math. Soc. 194, 53–78 (1974).

    Article  MathSciNet  Google Scholar 

  29. A. I. Smirnov, “Necessary optimality conditions for a class of optimal control problems with discontinuous integrand,” Proc. Steklov Inst. Math. 262, 213–230 (2008) [transl. from Tr. Mat. Inst. Steklova 262, 222–239 (2008)].

    Article  MathSciNet  Google Scholar 

  30. G. V. Smirnov, Introduction to the Theory of Differential Inclusions (Am. Math. Soc., Providence, RI, 2002), Grad. Stud. Math. 41.

    MATH  Google Scholar 

  31. R. Vinter, Optimal Control (Birkhäuser, Boston, 2000).

    MATH  Google Scholar 

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Acknowledgments

The author is grateful to K. O. Besov for a number of useful comments.

Funding

This work is supported by the Russian Science Foundation under grant 19-11-00223.

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    S. M. Aseev

  2. Lomonosov Moscow State University, Moscow, 119991, Russia

    S. M. Aseev

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Correspondence to S. M. Aseev.

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Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 315, pp. 34–63 https://doi.org/10.4213/tm4247.

Translated by K. Besov

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Aseev, S.M. Refined Euler–Lagrange Inclusion for an Optimal Control Problem with Discontinuous Integrand. Proc. Steklov Inst. Math. 315, 27–55 (2021). https://doi.org/10.1134/S0081543821050047

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