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Impulsive Control Problems with State Constraints

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Optimal Impulsive Control

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 477))

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Abstract

In this chapter, in the context of the impulsive extension of the optimal control problem, the state constraints are studied. That is, it is assumed that a certain closed subset of the state space is given while feasible arcs are not permitted to take values outside of it. This set is defined functionally in our considerations. It should be noted that the state constraints are in great demand in various engineering applications. For example, an iRobot cleaning a house should be able to avoid obstacles or objects that arise in its path. These obstacles are nothing but state constraints, while the task of avoiding the obstacle represents an important class of problems with state constraints. Evidently, there is a host of other engineering problems, in which the state constraints play an important role. The chapter deals with the same problem formulation as in the previous chapter; however, the state constraints of the above type are added. For this impulsive control problem, the Gamkrelidze-like maximum principle is obtained. Conditions for nondegeneracy of the maximum principle are presented. The chapter ends with eight exercises.

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References

  1. Arutyunov, A., Karamzin, D., Pereira, F.: A nondegenerate maximum principle for the impulse control problem with state constraints. SIAM J. Control Optim. 43(5), 1812–1843 (2005)

    Article  MathSciNet  Google Scholar 

  2. Gusev, M.: On optimal control of generalized processes under nonconvex state constraints. Differential Games and Control Problems [in Russian], UNTs, Akad. Nauk SSSR, Sverdlovsk 15, 64–112 (1975)

    Google Scholar 

  3. Miller, B.: Generalized solutions in nonlinear optimization problems with impulse controls. i. the solution existence problem. Avtomatika i Telemekhanika 4, 62–76 (1995)

    Google Scholar 

  4. Miller, B.: Generalized solutions in nonlinear optimization problems with impulse controls. ii. representation of solutions by differential equations with measure. Avtomatika i Telemekhanika 5, 56–70 (1995)

    MathSciNet  Google Scholar 

  5. Gamkrelidze, R.: Optimal control processes for bounded phase coordinates. Izv. Akad. Nauk SSSR. Ser. Mat. 24, 315–356 (1960)

    MathSciNet  Google Scholar 

  6. Arutyunov, A., Karamzin, D., Pereira, F.: 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 

  7. Neustadt, L.: An abstract variational theory with applications to a broad class of optimization problems. i. general theory. SIAM J. Control 4(3), 505–527 (1966)

    Article  MathSciNet  Google Scholar 

  8. Neustadt, L.: An abstract variational theory with applications to a broad class of optimization problems. ii. applications. SIAM J. Control 5(1), 90–137 (1967)

    Article  MathSciNet  Google Scholar 

  9. Pontryagin, L., Boltyanskii, V., Gamkrelidze, R., Mishchenko, E.: Mathematical theory of optimal processes. Translation from the Russian ed. by L.W. Neustadt. Interscience Publishers, Wiley, 1st edn (1962)

    Google Scholar 

  10. Russak, I.: On general problems with bounded state variables. J. Optim. Theory Appl. 6(6), 424–452 (1970)

    Article  MathSciNet  Google Scholar 

  11. Warga, J.: Minimizing variational curves restricted to a preassigned set. Trans. Amer. Math. Soc. 112, 432–455 (1964)

    Article  MathSciNet  Google Scholar 

  12. Dubovitskii, A., Milyutin, A.: Extremum problems with constraints. Sov. Math. Dokl. 4, 452–455 (1963)

    Google Scholar 

  13. Dubovitskii, A., Milyutin, A.: Extremum problems in the presence of restrictions. USSR Comput. Math. Math. Phys. 5(3), 1–80 (1965)

    Article  Google Scholar 

  14. Arutyunov, A.V.: Optimality conditions. Abnormal and degenerate problems. In: Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht (2000)

    Google Scholar 

  15. Halkin, H.: A satisfactory treatment of equality and operator constraints in the dubovitskii-milyutin optimization formalism. J. Optim. Theory Appl. 6(2), 138–149 (1970)

    Article  MathSciNet  Google Scholar 

  16. Ioffe, A., Tikhomirov, V.: Studies in Mathematics and its Applications, vol. 6. Elsevier Science, North-Holland, Amsterdam (1979)

    Google Scholar 

  17. Maurer, H.: Differential stability in optimal control problems. Appl. Math. Optim. 5(1), 283–295 (1979)

    Article  MathSciNet  Google Scholar 

  18. Milyutin, A.: Maximum Principle for General Optimal Control Problem. Fizmatlit, Moscow (2001). [in Russian]

    Google Scholar 

  19. Clarke, F.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983)

    MATH  Google Scholar 

  20. Ioffe, A.: Necessary conditions in nonsmooth optimization. Math. Op. Res. 9(2), 159–189 (1984)

    Article  MathSciNet  Google Scholar 

  21. Vinter, R., Pappas, G.: A maximum principle for nonsmooth optimal-control problems with state constraints. J. Math. Anal. Appl. 89(1), 212–232 (1982)

    Article  MathSciNet  Google Scholar 

  22. Arutyunov, A., Karamzin, D.: Non-degenerate necessary optimality conditions for the optimal control problem with equality-type state constraints. J. Glob. Optim. 64(4), 623–647 (2016)

    Article  MathSciNet  Google Scholar 

  23. Davydova, A., Karamzin, D.: On some properties of the shortest curve in a compound domain. Differ. Equ. 51(12), 1626–1636 (2015)

    Article  MathSciNet  Google Scholar 

  24. Arutyunov, A.: Properties of the Lagrange multipliers in the pontryagin maximum principle for optimal control problems with state constraints. Differ. Equ. 48(12), 1586–1595 (2012)

    Article  MathSciNet  Google Scholar 

  25. Arutyunov, A., Karamzin, D.: On some continuity properties of the measure lagrange multiplier from the maximum principle for state constrained problems. SIAM J. Control Optim. 53(4), 2514–2540 (2015)

    Article  MathSciNet  Google Scholar 

  26. Arutyunov, A., Karamzin, D., Pereira, F.: Conditions for the absence of jumps of the solution to the adjoint system of the maximum principle for optimal control problems with state constraints. Proc. Stekl. Inst. Math. 292, 27–35 (2016)

    Article  Google Scholar 

  27. Arutyunov, A., Karamzin, D.: Properties of extremals in optimal control problems with state constraints. Differ. Equ. 52(11), 1411–1422 (2016)

    Article  MathSciNet  Google Scholar 

  28. Arutyunov, A., Karamzin, D., Pereira, F.: Investigation of controllability and regularity conditions for state constrained problems. In: Proceedings of the 20-th IFAC Congress, Toulouse, France (2017)

    Google Scholar 

  29. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  MathSciNet  Google Scholar 

  30. Mordukhovich, B.: Variational Analysis and Generalized Differentiation II. Applications. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 331. Springer, Berlin (2006)

    Google Scholar 

  31. Mordukhovich, B.: Variational Analysis and Generalized Differentiation I. Basic Theory. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 330. Springer, Berlin (2006)

    Google Scholar 

  32. Arutyunov, A.: On the theory of the maximum principle for state constrained optimal control problems with state constraints. Dokl. Math. SSSR 304(1) (1989)

    Google Scholar 

  33. Arutyunov, A.: Perturbations of extremal problems with constraints and necessary optimality conditions. J. Sov. Math. 54(6), 1342–1400 (1991)

    Article  Google Scholar 

  34. Arutyunov, A., Aseev, S.: 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 

  35. Arutyunov, A., Aseev, S., Blagodatskikh, V.: First-order necessary conditions in the problem of optimal control of a differential inclusion with phase constraints. Sb. Math. 79(1), 117–139 (1994)

    Article  MathSciNet  Google Scholar 

  36. Arutyunov, A., Tynyanskij, N.: The maximum principle in a problem with phase constraints. Sov. J. Comput. Syst. Sci. 23(1), 28–35 (1984)

    MathSciNet  MATH  Google Scholar 

  37. Arutyunov, A., Tynyanskiy, N.: Maximum principle in a problem with phase constraints. Sov. J. Comput. Syst. Sci. 23(1), 28–35 (1985)

    MathSciNet  Google Scholar 

  38. Dubovitskij, A., Dubovitskij, V.: Necessary conditions for a strong minimum in optimal control problems with degeneration of final and phase constraints. Usp. Mat. Nauk 40(2(242)), 175–176 (1985)

    Google Scholar 

  39. Ferreira, M., Fontes, F., Vinter, R.: Nondegenerate necessary conditions for nonconvex optimal control problems with state constraints. J. Math. Anal. Appl. 233(1), 116–129 (1999)

    Article  MathSciNet  Google Scholar 

  40. Ferreira, M., Vinter, R.: When is the maximum principle for state constrained problems nondegenerate? J. Math. Anal. Appl. 187(2), 438–467 (1994)

    Article  MathSciNet  Google Scholar 

  41. Palladino, M., Vinter, R.: Regularity of the hamiltonian along optimal trajectories. SIAM J. Control Optim. 53(4), 1892–1919 (2015)

    Article  MathSciNet  Google Scholar 

  42. Vinter, R.: Optimal Control. Birkhäuser, Boston (2000)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    Aram Arutyunov

  2. Institute of Control Sciences of the Russian Academy of Sciences, Moscow, Russia

    Aram Arutyunov

  3. RUDN University, Moscow, Russia

    Aram Arutyunov

  4. Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Moscow, Russia

    Dmitry Karamzin

  5. FEUP/DEEC, Porto University, Porto, Portugal

    Fernando Lobo Pereira

Authors
  1. Aram Arutyunov
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  2. Dmitry Karamzin
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  3. Fernando Lobo Pereira
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Correspondence to Dmitry Karamzin .

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Arutyunov, A., Karamzin, D., Lobo Pereira, F. (2019). Impulsive Control Problems with State Constraints. In: Optimal Impulsive Control. Lecture Notes in Control and Information Sciences, vol 477. Springer, Cham. https://doi.org/10.1007/978-3-030-02260-0_5

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