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Tangency Property and Prior-Saturation Points in Minimal Time Problems in the Plane

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Abstract

In this paper, we consider minimal time problems governed by control-affine-systems in the plane, and we focus on the synthesis problem in presence of a singular locus that involves a saturation point for the singular control. After giving sufficient conditions on the data ensuring occurrence of a prior-saturation point and a switching curve, we show that the bridge (i.e., the optimal bang arc issued from the singular locus at this point) is tangent to the switching curve at the prior-saturation point. This property is proved using the Pontryagin Maximum Principle that also provides a set of non-linear equations that can be used to compute the prior-saturation point. These issues are illustrated on a fed-batch model in bioprocesses and on a Magnetic Resonance Imaging (MRI) model for which minimal time syntheses for the point-to-point problem are discussed.

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Notes

  1. If the target can be reached from \(x_{0}\) and if \(f\), \(g\) have linear growth, then (2.2) admits an optimal solution, thanks to Filippov’s Existence Theorem, see, e.g., [28].

  2. Here, \(N_{\mathcal{T}}(x)\) stands for the (Mordukovitch) limiting normal cone to \(\mathcal{T}\) at point \(x\in \mathcal{T}\), see [28]. It coincides with the normal cone in the sense of convex analysis when \(\mathcal{T}\) is convex.

  3. By optimally invariant subset, we mean a subset \(\Omega \subset \mathbb{R}^{2}\) such that for any initial condition \(x_{0}\in \Omega \), an optimal trajectory stays in \(\Omega \).

  4. Following for instance [24, Sect. 2.8.4], chattering occurs for singular arcs of higher order (at least 2) for which the Legendre-Clebsch condition is not fulfilled.

  5. Since \(\mathcal{T} :=\{x_{f}\}\) is a point, there is no transversality condition at the terminal time.

  6. In contrast with the previous sections in which state variables are \((x_{1},x_{2})\), we chose to adopt the notation \((s,v)\) that is commonly used in the bioprocesses literature for fed-batch operations.

  7. Micro-organism concentration \(X>0\) can be expressed as a simple function of \((s,v)\), namely \(X=M/v+s_{\mathit{in}}-s\), thus (5.1) is enough to describe a bioreactor operated in fed-batch mode.

References

  1. Bayen, T., Gajardo, P., Mairet, F.: Optimal synthesis for the minimum time control problems of fed-batch bioprocesses for growth functions with two maxima. J. Optim. Theory Appl. 158, 521–553 (2013)

    Article  MathSciNet  Google Scholar 

  2. Bayen, T., Harmand, J., Sebbah, M.: Time-optimal control of concentration changes in the chemostat with one single species. Appl. Math. Model. 50, 257–278 (2017)

    Article  MathSciNet  Google Scholar 

  3. Bayen, T., Mazade, M., Mairet, F.: Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc. Discrete Contin. Dyn. Syst., Ser. B 20, 39–58 (2015)

    MathSciNet  MATH  Google Scholar 

  4. Bayen, T., Rapaport, A., Sebbah, M.: Minimal time control of the two tanks gradostat model under a cascade input constraint. SIAM J. Control Optim. 52, 2568–2594 (2014)

    Article  MathSciNet  Google Scholar 

  5. Bernard, O., Hadj-Sadok, Z., Dochain, D., Genovesi, A., Steyer, J.-P.: Dynamical model development and parameter identification for an anaerobic wastewater treatment process. Biotechnol. Bioeng. 75, 424–438 (2001)

    Article  Google Scholar 

  6. Bonnard, B., Chyba, M.: Singular Trajectories and Their Role in Control Theory. Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 40. Springer, Berlin (2003)

    MATH  Google Scholar 

  7. Bonnard, B., Drouot, J.: Towards Geometric Time Minimal Control without Legendre Condition and with Multiple Singular Extremals for Chemical Networks. arXiv:2001.04126

  8. Bonnard, B., Kupka, I.: Generic properties of singular trajectories. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14(2), 167–186 (1997)

    Article  MathSciNet  Google Scholar 

  9. Bonnard, B., Claeys, M., Cots, O., Martinon, P.: Geometric and numerical methods in the contrast imaging problem in nuclear magnetic resonance. Acta Appl. Math. 135, 5–45 (2015)

    Article  MathSciNet  Google Scholar 

  10. Bonnard, B., Cots, O., Rouot, J., Verron, T.: Time minimal saturation of a pair of spins and application in magnetic resonance imaging. In: Math. Control Related Fields (2019)

    Google Scholar 

  11. Bonnard, B., Cots, O., Glaser, S., Lapert, M., Sugny, D., Zhang, Y.: Geometric optimal control of the contrast imaging problem in nuclear magnetic resonance. IEEE Trans. Autom. Control 57, 1957–1969 (2012)

    Article  MathSciNet  Google Scholar 

  12. Bonnard, B., De Morant, J.: Toward a geometric theory in the time-minimal control of chemical batch reactors. SIAM J. Control Optim. 33, 1279–1311 (1995)

    Article  MathSciNet  Google Scholar 

  13. Bonnard, B., Pelletier, M.: Time minimal synthesis for planar systems in the neighborhood of a terminal manifold of codimension one. J. Math. Syst. Estim. Control 5, 22 (1995)

    MathSciNet  MATH  Google Scholar 

  14. Boscain, U., Piccoli, B.: Optimal Syntheses for Control Systems on 2-D Manifolds. Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 43. Springer, Berlin (2004)

    MATH  Google Scholar 

  15. Hermes, H., LaSalle, J.: Functional Analysis and Time Optimal Control. Mathematics in Science and Engineering, vol. 56. Academic Press, New York/London (1969)

    Book  Google Scholar 

  16. Kalboussi, N., Rapaport, A., Bayen, T., Amar, N.B., Ellouze, F., Harmand, J.: Optimal control of membrane-filtration systems. IEEE Trans. Autom. Control 64, 2128–2134 (2019)

    Article  MathSciNet  Google Scholar 

  17. Ledzewicz, U., Schättler, H.: Antiangiogenic therapy in cancer treatment as an optimal control problem. SIAM J. Control Optim. 46, 1052–1079 (2007)

    Article  MathSciNet  Google Scholar 

  18. Levitt, M.: Spin Dynamics: Basics of Nuclear Magnetic Resonance. Wiley, New York (2008)

    Google Scholar 

  19. Moreno, J.: Optimal time control of bioreactors for the wastewater treatment. Optim. Control Appl. Methods 20, 145–164 (1999)

    Article  MathSciNet  Google Scholar 

  20. Piccoli, B.: Classification of generic singularities for the planar time-optimal synthesis. SIAM J. Control Optim. 34, 1914–1946 (1996)

    Article  MathSciNet  Google Scholar 

  21. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. A Pergamon Press Book. The Macmillan Co., New York (1964). Translated by D.E. Brown

    MATH  Google Scholar 

  22. Rapaport, A., Bayen, T., Sebbah, M., Donoso-Bravo, A., Torrico, A.: Dynamical modeling and optimal control of landfills. Math. Models Methods Appl. Sci. 26, 901–929 (2016)

    Article  MathSciNet  Google Scholar 

  23. Schättler, H., Jankovic, M.: A synthesis of time-optimal controls in the presence of saturated singular arcs. Forum Math. 5, 203–241 (1993)

    Article  MathSciNet  Google Scholar 

  24. Schättler, H., Ledzewicz, U.: Geometric Optimal Control. Interdisciplinary Applied Mathematics, vol. 38. Springer, New York (2012)

    Book  Google Scholar 

  25. Sussmann, H.J.: Regular synthesis for time-optimal control of single-input real analytic systems in the plane. SIAM J. Control Optim. 25, 1145–1162 (1987)

    Article  MathSciNet  Google Scholar 

  26. Sussmann, H.J.: The structure of time-optimal trajectories for single-input systems in the plane: the \(C^{\infty }\) nonsingular case. SIAM J. Control Optim. 25, 433–465 (1987)

    Article  MathSciNet  Google Scholar 

  27. Sussmann, H.J.: The structure of time-optimal trajectories for single-input systems in the plane: the general real analytic case. SIAM J. Control Optim. 25, 868–904 (1987)

    Article  MathSciNet  Google Scholar 

  28. Vinter, R.: Optimal Control, Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston (2000)

    MATH  Google Scholar 

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Acknowledgements

We are very grateful to E. Trélat for helpful discussions and suggestions about the tangency property at the prior-saturation point.

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  1. Laboratoire de Mathématiques d’Avignon (EA 2151), Avignon Université, 84018, Avignon, France

    T. Bayen

  2. Toulouse Univ., INP-ENSEEIHT, IRIT and CNRS, 2 rue Camichel, 31071, Toulouse, France

    O. Cots

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Bayen, T., Cots, O. Tangency Property and Prior-Saturation Points in Minimal Time Problems in the Plane. Acta Appl Math 170, 515–537 (2020). https://doi.org/10.1007/s10440-020-00344-8

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