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Normality and Nondegeneracy of the Maximum Principle in Optimal Impulsive Control Under State Constraints

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Abstract

We investigate nondegenerate and normal forms of the maximum principle for general, free end-time, impulsive optimal control problems with state and endpoint constraints. We introduce constraint qualifications sufficient to avoid degeneracy or abnormality phenomena, which do not require any convexity and impose the existence of an inward pointing velocity just on the subset of times, in which the extended optimal trajectory has an outward pointing velocity (w.r.t. the state constraint). These conditions extend to impulsive problems some conditions, recently proposed by F. Fontes and H. Frankowska, for conventional optimization problems. The nontriviality of this extension is illustrated through some examples.

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Notes

  1. The space-time system (7) is rate-independent, hence our using only controls such that \(w_0+\left| w\right| =1\) is an useful but arbitrary choice (see, e.g., [5] for more details).

  2. Since every \(L^1\)-equivalence class contains Borel measurable representatives, we are tacitly assuming that all \(L^1\)-maps we are considering are Borel measurable.

  3. Let us recall that (2) is called commutative if the Lie brackets \([g_i,g_j]\equiv 0\) for all i, j.

  4. We use \(|\varvec{\eta }_i|\) to denote the total variation function of \(\varvec{\eta }_i\).

  5. For any \(r\in \mathbb {R}\), \(\delta _{\{r\}}\) is the Dirac unit measure concentrated at r.

  6. For any \(w\in \mathbb {R}^m\), when \(w=0\) we mean that \(\frac{w}{|w|}=0\).

  7. A cone \({\mathcal K}\subseteq \mathbb {R}^k\) is pointed if it contains no line, i.e., if z, \(-z\in {\mathcal K}\) implies that \(z=0\).

  8. Since the scalar product is bilinear, \(N_\varOmega ({{x}^0})\) can be replaced by the Clarke normal cone, as in [15].

  9. Given an interval \(I\subset \mathbb {R}\), we denote by \(\chi _{_{I}}\) the characteristic function of I.

References

  1. Rishel, R.W.: An extended Pontryagin principle for control systems whose control laws contain measures. SIAM J. Control 3(2), 191–205 (1965)

    MathSciNet  MATH  Google Scholar 

  2. Warga, J.: Variational problems with unbounded controls. J. Soc. Ind. Appl. Math. Ser. A Control 3, 424–438 (1965)

    Article  MathSciNet  Google Scholar 

  3. Bressan, A., Rampazzo, F.: On differential systems with vector-valued impulsive controls. Boll. Un. Mat. Ital. B (7) 2(3), 641–656 (1988)

    MathSciNet  MATH  Google Scholar 

  4. Miller, B.M.: The method of discontinuous time substitution in problems of the optimal control of impulse and discrete-continuous systems. Avtomat. i Telemekh. 12, 3–32 (1993) (Russian) [Translation in Autom. Remote Control 54 (1993), no. 12, part 1, 1727–1750] (1994)

  5. Motta, M., Rampazzo, F.: Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls. Differ. Integral Equ. 8(2), 269–288 (1995)

    MathSciNet  MATH  Google Scholar 

  6. Karamzin, D., de Oliveira, V.A., Pereira, F., Silva, G.N.: On some extension of optimal control theory. Eur. J. Control 20(6), 284–291 (2014)

    Article  MathSciNet  Google Scholar 

  7. Karamzin, D., de Oliveira, V.A., Pereira, F., Silva, G.N.: On the properness of an impulsive control extension of dynamic optimization problems. ESAIM Control Optim. Calc. Var. 21(3), 857–875 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Ferreira, M.M.A., Fontes, F.A.C.C., 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 

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

    Article  MathSciNet  Google Scholar 

  11. Frankowska, H., Tonon, D.: Inward pointing trajectories, normality of the maximum principle and the non occurrence of the Lavrentieff phenomenon in optimal control under state constraints. J. Convex Anal. 20(4), 1147–1180 (2013)

    MathSciNet  MATH  Google Scholar 

  12. Lopes, S.O., Fontes, F.A.C.C., de Pinho, M.R.: On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete Contin. Dyn. Syst. 29(2), 559–575 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Rampazzo, F., Vinter, R.: A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control. IMA J. Math. Control Inform. 16(4), 335–351 (1999)

    Article  MathSciNet  Google Scholar 

  15. 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)

    Article  MathSciNet  Google Scholar 

  16. Arutyunov, A.V.: Optimality Conditions. Abnormal and Degenerate Problems. Mathematics and its Applications, vol. 526. Kluwer Academic Publishers, Dordrecht (2000)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Arutyunov, A.V., Karamzin, D.Y., Pereira, F.L.: Investigation of controllability and regularity conditions for state constrained problems, IFAC-PapersOnline. In: Proceedings of the IFAC Congress in Toulouse, France, pp. 6295–6302 (2017)

  19. 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 

  20. Arutyunov, A., Karamzin, D., Pereira, F.: State constraints in impulsive control problems: Gamkrelidze-like conditions of optimality. J. Optim. Theory Appl. 166(2), 440–459 (2015)

    Article  MathSciNet  Google Scholar 

  21. Karamzin, D.Y.: Necessary conditions for the minimum in an impulsive optimal control problem. J. Math. Sci. N.Y. 139(6), 7087–7150 (2006)

    Article  MathSciNet  Google Scholar 

  22. Motta, M., Rampazzo, F., Vinter, R.: Normality and gap phenomena in optimal unbounded control. ESAIM Control Optim. Calc. Var. 24(4), 1645–1673 (2018)

    Article  MathSciNet  Google Scholar 

  23. Motta, M.: Minimum time problem with impulsive and ordinary controls. Discrete Contin. Dyn. Syst. 38(11), 5781–5809 (2018)

    Article  MathSciNet  Google Scholar 

  24. Motta, M., Sartori, C.: Finite fuel problem in nonlinear singular stochastic control. SIAM J. Control Optim. 46(4), 1180–1210 (2007)

    Article  MathSciNet  Google Scholar 

  25. Motta, M., Sartori, C.: Weakly coercive problems in nonlinear stochastic control: existence of optimal controls. SIAM J. Control Optim. 48(5), 3532–3561 (2010)

    Article  MathSciNet  Google Scholar 

  26. Miller, B.M., Rubinovich, E.Y.: Impulsive Control in Continuous and Discrete–Continuous Systems. Kluwer Academic, New York (2003)

    Book  Google Scholar 

  27. Aronna M.S., Motta M., Rampazzo F.: Necessary conditions involving Lie brackets for impulsive optimal control problems. In: Proceedings of the 58th IEEE Conference on Decision and Control, CDC 2019, Nice. http://arxiv.org/abs/1903.06109

  28. Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, vol. 178. Springer, New York (1998)

    Google Scholar 

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

    MATH  Google Scholar 

  30. Aronna, M.S., Motta, M., Rampazzo, F.: Infimum gaps for limit solutions. Set Valued Var. Anal. 23(1), 3–22 (2015)

    Article  MathSciNet  Google Scholar 

  31. Motta, M., Rampazzo, F.: Nonlinear systems with unbounded controls and state constraints: a problem of proper extension. NoDEA Nonlinear Differ. Equ. Appl. 3(2), 191–216 (1996)

    Article  MathSciNet  Google Scholar 

  32. Motta, M., Sartori, C.: Discontinuous solutions to unbounded differential inclusions under state constraints. Applications to optimal control problems. Set Valued Anal. 7(4), 295–322 (1999)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This research is partially supported by the Padua University Grant SID 2018 “Controllability, stabilizability and infimun gaps for control systems”, prot. BIRD 187147, and by the Gruppo Nazionale per l’ Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), Italy. The authors are very grateful to the anonymous referees for their constructive criticism to improve the article and for the helpful bibliographic suggestions.

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  1. Department of Mathematics “Tullio Levi-Civita”, University of Padua, Padua, Italy

    Monica Motta & Caterina Sartori

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  1. Monica Motta
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  2. Caterina Sartori
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Correspondence to Monica Motta.

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Communicated by Roland Herzog.

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Motta, M., Sartori, C. Normality and Nondegeneracy of the Maximum Principle in Optimal Impulsive Control Under State Constraints. J Optim Theory Appl 185, 44–71 (2020). https://doi.org/10.1007/s10957-020-01641-w

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