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Essential Boundedness and Singularity in Optimal Control

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Abstract

Sufficient optimality conditions for optimal control problems involving isoperimetric and mixed inequality and equality constraints are derived. The main novelty of our approach is the fact that, for such problems, discontinuous and singular solutions can be detected. In other words, our result can deal with solutions for which the proposed optimal control is not continuous, but only essentially bounded, and the classical, crucial strengthened Legendre-Clebsch condition is no longer imposed.

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References

  1. Agrachev A, Stefani G, Zezza PL. A Hamiltonian approach to strong minima in optimal control. AMS proceedings of differential geometry and control, Boulder, CO, 1997, AMS, Providence, RI; 1999. p. 11–22.

  2. Agrachev A, Stefani G, Zezza PL. Strong optimality for a bang-bang trajectory. SIAM J Control Optim 2002;41:991–1014.

    Article  MathSciNet  Google Scholar 

  3. Alt W, Felgenhauer U, Seydenschwanz M. Euler discretization for a class of nonlinear optimal control problems with control appearing linearly. Comput Optim Appl 2018;69:825–856.

    Article  MathSciNet  Google Scholar 

  4. Aronna MS, Bonnans JF, Dmitruk AV, Lotito PA. Quadratic order conditions for bang-singular extremals. Numer Algebra, Control Optim 2012;2:511–546.

    Article  MathSciNet  Google Scholar 

  5. Aronna MS, Bonnans JF, Goh BS. Second order analysis of control-affine problems with scalar state constraint. Math Program Ser A 2016;160:115–147.

    Article  MathSciNet  Google Scholar 

  6. Clarke FH. Functional analysis, calculus of variations and optimal control. London: Springer-Verlag; 2013.

    Book  Google Scholar 

  7. Dmitruk AV. Quadratic conditions for the Pontryagin minimum in an optimal control problem linear with respect to control. I. Decoding theorem. Mathematics of the USSR-Izvestiya 1987;28:275–303.

    Article  Google Scholar 

  8. Goh BS. Necessary conditions for singular extremals involving multiple control variables. SIAM J Control 1966;4:716–731.

    Article  MathSciNet  Google Scholar 

  9. Hestenes MR. Calculus of variations and optimal control theory. New York: John Wiley; 1966.

    MATH  Google Scholar 

  10. Loewen PD. Second-order sufficiency criteria and local convexity for equivalent problems in the calculus of variations. J Math Anal Appl 1990;146:512–522.

    Article  MathSciNet  Google Scholar 

  11. Malanowski K. Sufficient optimality conditions for optimal control subject to state constraints. SIAM J Control Optim 1997;35:205–227.

    Article  MathSciNet  Google Scholar 

  12. Malanowski K, Maurer H, Pickenhain S. Second order sufficient conditions for state-constrained optimal control problems. J Optim Theory Appl 2004;123:595–617.

    Article  MathSciNet  Google Scholar 

  13. Maurer H. First and second order sufficient optimality conditions in mathematical programming and optimal control. Mathematical Programming at Oberwolffach, Springer 1981;14:163–177.

    Article  MathSciNet  Google Scholar 

  14. Maurer H, Oberle HJ. Second order sufficient conditions for optimal control problems with free final time: the Riccati approach. SIAM J Control Optim 2002;41: 380–403.

    Article  MathSciNet  Google Scholar 

  15. Maurer H, Osmolovskii NP. Second order optimality conditions for bang-bang control problems. Control Cybern 2003;32:555–584.

    MATH  Google Scholar 

  16. Maurer H, Osmolovskii NP. Second order sufficient conditions for time optimal bang-bang control problems. SIAM J Control Optim 2004;42:2239–2263.

    Article  MathSciNet  Google Scholar 

  17. Maurer H, Osmolovskii NP. Second order sufficient optimality conditions for a control problem with continuous and bang-bang control components: Riccati approach. IFIP Adv Inf Commun Technol 2007;312:411–429.

    MathSciNet  MATH  Google Scholar 

  18. Maurer H, Pickenhain S. Second order sufficient conditions for control problems with mixed control-state constraints. J Optim Theory Appl 1995;86:649–667.

    Article  MathSciNet  Google Scholar 

  19. Maurer H, Pickenhain S. Sufficient conditions and sensitivity analysis for economic control problems. Ann Oper Res 1999;88:3–14.

    Article  MathSciNet  Google Scholar 

  20. McShane EJ. Sufficient conditions for a weak relative minimum in the problem of Bolza. Trans Am Math Soc 1942;52:344–379.

    Article  MathSciNet  Google Scholar 

  21. Milyutin AA, Osmolovskii NP. Calculus of Variations and Optimal Control. American Mathematical Society. Rhode Island: Providence; 1998.

    Book  Google Scholar 

  22. Moyer HG. Sufficient conditions for a strong minimum in singular control problems. SIAM J Control 1973;11:620–636.

    Article  MathSciNet  Google Scholar 

  23. Osmolovskii NP. Second order sufficient conditions for an extremum in optimal control. Control Cybern 2002;31:803–831.

    MathSciNet  MATH  Google Scholar 

  24. Osmolovskii NP. Sufficient quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints. J Math Sci 2011;173:1–106.

    Article  MathSciNet  Google Scholar 

  25. Osmolovskii NP. Second-order sufficient optimality conditions for control problems with linearly independent gradients of control constraints. ESAIM: Control, Optim Calc Variations 2012;18:452–482.

    MathSciNet  MATH  Google Scholar 

  26. Osmolovskii NP, Veliov VM. 2019. Metric sub-regularity in optimal control of affine problems with free end state, ESAIM: Control, Optim Calc Variations. https://doi.org/10.1051/cocv/2019046.

  27. Poggiolini L, Stefani G. Bang-singular-bang extremals: sufficient optimality conditions. J Dyn Control Syst 2011;17:469–514.

    Article  MathSciNet  Google Scholar 

  28. Poggiolini L, Stefani G. Strong local optimality for a bang-bang-singular extremal: the fixed-free case. SIAM J Control Optim 2018;56:2274–2294.

    Article  MathSciNet  Google Scholar 

  29. Rosenblueth JF. Systems with time delays in the calculus of variations: a variational approach. IMA J Math Control Inf 1988;5:125–145.

    Article  MathSciNet  Google Scholar 

  30. Rosenblueth JF. Variational conditions and conjugate points for the fixed-endpoint control problem. IMA J Math Control Inf 1999;16:147–163.

    Article  MathSciNet  Google Scholar 

  31. Rosenblueth JF, Sánchez Licea G. A direct sufficiency proof for a weak minimum in optimal control. Appl Math Sci 2010;4:253–269.

    MathSciNet  MATH  Google Scholar 

  32. Rosenblueth JF, Sánchez Licea G. Sufficiency and singularity in optimal control. IMA J Math Control Inf 2013;30:37–65.

    Article  MathSciNet  Google Scholar 

  33. Sánchez Licea G. Weakening the strengthened condition of Weierstrass for the isoperimetric problem in the calculus of variations. IMA J Math Control Inf 2008;25: 59–74.

    Article  MathSciNet  Google Scholar 

  34. Sánchez Licea G. Sufficiency by a direct method in the variable state problem of calculus of variations: singular extremals. IMA J Math Control Inf 2009;26:257–279.

    Article  MathSciNet  Google Scholar 

  35. Sánchez Licea G. Relaxing strengthened Legendre-Clebsch condition. SIAM J Control Optim 2013;51:3886–3902.

    Article  MathSciNet  Google Scholar 

  36. Schättler H. A local feedback synthesis of time-optimal stabilizing controls in dimension three. Math Control Signals Syst 1991;4:293–313.

    Article  MathSciNet  Google Scholar 

  37. Stefani G, Zezza PL. Optimality conditions for a constrained optimal control problem. SIAM J Control Optim 1996;34:635–659.

    Article  MathSciNet  Google Scholar 

  38. Tröltzsch F. 2010. Optimal control of partial differential equations. theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels. Graduate studies in mathematics, 112. American Mathematical Society, Providence, RI.

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Funding

The authors are grateful to Dirección General de Asuntos del Personal Académico, from Universidad Nacional Autónoma de México, for the support given. The first author as part of the PASPA program during a sabbatical stay at the Department of Mathematical Sciences, University of Bath, UK, and the second author for the project PAPIIT-IN113318.

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Authors and Affiliations

  1. IIMAS, Universidad Nacional Autónoma de México, Apartado Postal 20-126, 01000, Mexico City, Mexico

    Javier F. Rosenblueth

  2. Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, 04510, Mexico City, Mexico

    Gerardo Sánchez Licea

Authors
  1. Javier F. Rosenblueth
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  2. Gerardo Sánchez Licea
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Correspondence to Javier F. Rosenblueth.

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Rosenblueth, J.F., Licea, G.S. Essential Boundedness and Singularity in Optimal Control. J Dyn Control Syst 27, 87–105 (2021). https://doi.org/10.1007/s10883-020-09482-6

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  • DOI: https://doi.org/10.1007/s10883-020-09482-6

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