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Well-posedness of the shooting algorithm for control-affine problems with a scalar state constraint

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Abstract

We deal with a control-affine problem with scalar control subject to bounds, a scalar state constraint and endpoint constraints of equality type. For the numerical solution of this problem, we propose a shooting algorithm and provide a sufficient condition for its local convergence. We exhibit an example that illustrates the theory.

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References

  • Adams RA (1975) Sobolev spaces. Academic Press, New York

    MATH  Google Scholar 

  • Aftalion A, Bonnans J (2014) Optimization of running strategies based on anaerobic energy and variations of velocity. SIAM J Appl Math 74(5):1615–1636

    Article  MathSciNet  MATH  Google Scholar 

  • Aronna MS, Bonnans JF, Dmitruk AV, Lotito PA (2012) Quadratic order conditions for bang-singular extremals. Numer Algebra Control Optim AIMS J 2(3):511–546 (special issue dedicated to Professor Helmut Maurer on the occasion of his 65th birthday)

  • Aronna MS, Bonnans JF, Martinon P (2013) A shooting algorithm for optimal control problems with singular arcs. J Optim Theory Appl 158(2):419–459

    Article  MathSciNet  MATH  Google Scholar 

  • Aronna MS, Bonnans JF, BS G (2016) Second order analysis of control-affine problems with scalar state constraint. Math Program 160(1):115–147

    Article  MathSciNet  MATH  Google Scholar 

  • Arutyunov A, Karamzin D (2020) A survey on regularity conditions for state-constrained optimal control problems and the non-degenerate maximum principle. J Optim Theory Appl 184:697–723

    Article  MathSciNet  MATH  Google Scholar 

  • Beal L, Hill D, Martin R, Hedengren J (2018) Gekko optimization suite. Processes 6(8):106

    Article  Google Scholar 

  • Bock HG, Plitt KJ (1984) A multiple shooting algorithm for direct solution of optimal control problems*. In: IFAC proceedings, volumes 17, no (2), pp 1603–1608. 9th IFAC World Congress: a bridge between control science and technology, Budapest, Hungary, 2–6 July 1984

  • Bonnans JF (2019) Convex and stochastic optimization. Springer, Berlin

    Book  MATH  Google Scholar 

  • Bonnans JF, Shapiro A (2000) Perturbation analysis of optimization problems. Springer, New York

    Book  MATH  Google Scholar 

  • Bonnans JF, Gilbert JC, Lemaréchal C, Sagastizábal CA (2006) Numerical optimization: theoretical and practical aspects. Springer, Berlin

    MATH  Google Scholar 

  • Bonnard B, Faubourg L, Launay G, Trélat E (2003) Optimal control with state constraints and the space shuttle re-entry problem. J Dyn Control Syst 9(2):155–199

    Article  MathSciNet  MATH  Google Scholar 

  • Bulirsch R (1971) Die Mehrzielmethode zur numerischen Losung von nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung. Report der Carl-Cranz Gesellschaft

  • Cots O (2017) Geometric and numerical methods for a state constrained minimum time control problem of an electric vehicle. ESAIM Control Optim Calc Var 23(4):1715–1749

    Article  MathSciNet  MATH  Google Scholar 

  • Cots O, Gergaud J, Goubinat D, Wembe B (2022) Singular versus boundary arcs for aircraft trajectory optimization in climbing phase. ESAIM: M2AN (Online first)

  • de Pinho MR, Ferreira MM, Ledzewicz U, Schaettler H (2005) A model for cancer chemotherapy with state-space constraints. Nonlinear Anal Theory Methods Appl 63(5):2591–2602

    Article  MATH  Google Scholar 

  • Dmitruk AV (1977) Quadratic conditions for a weak minimum for singular regimes in optimal control problems. Sov Math Dokl 18(2):418–422

  • Dmitruk AV (1987) Quadratic order conditions for a Pontryagin minimum in an optimal control problem linear in the control. Math USSR Izv 28:275–303

    Article  MATH  Google Scholar 

  • Fletcher R (2013) Practical methods of optimization. Wiley, New Jersey

    MATH  Google Scholar 

  • Frankowska H, Tonon D (2013) Pointwise second-order necessary optimality conditions for the Mayer problem with control constraints. SIAM J Control Optim 51(5):3814–3843

    Article  MathSciNet  MATH  Google Scholar 

  • Fraser-Andrews G (1989) Numerical methods for singular optimal control. J Optim Theory Appl 61(3):377–401

    Article  MathSciNet  MATH  Google Scholar 

  • Fuller AT (1963) Study of an optimum non-linear control system. J Electron Control 1(15):63–71

    Article  MathSciNet  Google Scholar 

  • Goh BS (1966a) Necessary conditions for singular extremals involving multiple control variables. J SIAM Control 4:716–731

  • Goh BS (1966b) The second variation for the singular Bolza problem. J SIAM Control 4(2):309–325

  • Goh BS, Leitmann G, Vincent TL (1974) Optimal control of a prey–predator system. Math Biosci 19:263–286

    Article  MathSciNet  MATH  Google Scholar 

  • Graichen K, Petit N (2008) Solving the Goddard problem with thrust and dynamic pressure constraints using saturation functions. In: 17th World Congress of the International Federation of Automatic Control, vol. Proceedings of the 2008 IFAC World Congress. IFAC, Seoul, pp 14301–14306

  • Leparoux C, Hérissé B, Jean F (2022a) Structure of optimal control for planetary landing with control and state constraints. ESAIM COCV 28:67

  • Leparoux C, Hérissé B, Jean F (2022b) Optimal planetary landing with pointing and glide-slope constraints. In: 2022 IEEE 61st conference on decision and control (CDC). IEEE, pp 4357–4362

  • Levitin ES, Milyutin AA, Osmolovskiĭ NP (1978) Higher order conditions for local minima in problems with constraints. Uspekhi Mat Nauk 33(6(204)):85–148272

    MathSciNet  Google Scholar 

  • Maurer H (1976) Numerical solution of singular control problems using multiple shooting techniques. J Optim Theory Appl 18(2):235–257

    Article  MathSciNet  MATH  Google Scholar 

  • Maurer H, Gillessen W (1975) Application of multiple shooting to the numerical solution of optimal control problems with bounded state variables. Computing 15(2):105–126

    Article  MathSciNet  MATH  Google Scholar 

  • Maurer H, De Pinho MR (2016) Optimal control of epidemiological seir models with L1-objectives and control-state constraints. Pac J Optim 12(2):415–436

  • Maurer H, Kim J-HR, Vossen G (2005) On a state-constrained control problem in optimal production and maintenance. In: Deissenberg C, Hartl RF (eds) Optimal control and dynamic games, vol 7. Advances in computational management science. Springer, New York, pp 289–308

    Chapter  Google Scholar 

  • Morrison DD, Riley JD, Zancanaro JF (1962) Multiple shooting method for two-point boundary value problems. Commun ACM 5:613–614

    Article  MathSciNet  MATH  Google Scholar 

  • Oberle HJ (1979) Numerical computation of singular control problems with application to optimal heating and cooling by solar energy. Appl Math Optim 5(4):297–314

    Article  MathSciNet  MATH  Google Scholar 

  • Pesch HJ (1994) A practical guide to the solution of real-life optimal control problems. Control Cybern 23(1–2):7–60 (Parametric optimization)

    MathSciNet  MATH  Google Scholar 

  • Robbins HM (1967) A generalized Legendre–Clebsch condition for the singular case of optimal control. IBM J Res Dev 11:361–372

    Article  MATH  Google Scholar 

  • Schättler H (2006) Local fields of extremals for optimal control problems with state constraints of relative degree 1. J Dyn Control Syst 12(4):563–599

    Article  MathSciNet  MATH  Google Scholar 

  • Seywald H, Cliff EM (1993) Goddard problem in presence of a dynamic pressure limit. J. Guidance Control Dyn 16(4):776–781

    Article  MATH  Google Scholar 

  • Team Commands IS (2017) Bocop: an open source toolbox for optimal control. http://bocop.org

  • Trélat E (2012) Optimal control and applications to aerospace: some results and challenges. J Optim Theory Appl 154:713–758

    Article  MathSciNet  MATH  Google Scholar 

  • Vinter R (2000) Optimal control. Systems & control: foundations & applications. Birkhäuser Boston, Inc., Boston

    MATH  Google Scholar 

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Acknowledgements

The first author was supported by FAPERJ (Brazil) through the Jovem Cientista do Nosso Estado Program and by CNPq (Brazil) through the Universal Program and the Productivity in Research Scholarship. The second author was supported by the FiME Lab Research Initiative (Institut Europlace de Finance), and by the PGMO program. We acknowledge the anonymous reviewers for their careful reading and comments that helped us improve this manuscript. The first author thanks Gabriel de Lima Monteiro for his suggestions on the numerical implementations.

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Authors and Affiliations

  1. Escola de Matemática Aplicada, FGV EMAp, Praia de Botafogo 190, Rio de Janeiro, 22250-900, Brazil

    M. Soledad Aronna

  2. INRIA Saclay, L2S, Centrale Supelec, 91190, Gif-sur-Yvette, France

    J. Frédéric Bonnans

  3. School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, Perth, 6845, Australia

    Bean San Goh

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  1. M. Soledad Aronna
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  2. J. Frédéric Bonnans
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  3. Bean San Goh
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Aronna, M.S., Bonnans, J.F. & Goh, B.S. Well-posedness of the shooting algorithm for control-affine problems with a scalar state constraint. Comp. Appl. Math. 42, 217 (2023). https://doi.org/10.1007/s40314-023-02346-2

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  • DOI: https://doi.org/10.1007/s40314-023-02346-2

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