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Locally convex topologies and control theory

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Abstract

Using recent characterisations of topologies of spaces of vector fields for general regularity classes—e.g., Lipschitz, finitely differentiable, smooth, and real analytic—characterisations are provided of geometric control systems that utilise these topologies. These characterisations can be expressed as joint regularity properties of the system as a function of state and control. It is shown that the common characterisations of control systems in terms of their joint dependence on state and control are, in fact, representations of the fact that the natural mapping from the control set to the space of vector fields is continuous. The classes of control systems defined are new, even in the smooth category. However, in the real analytic category, the class of systems defined is new and deep. What are called “real analytic control systems” in this article incorporate the real analytic topology in a way that has hitherto been unexplored. Using this structure, it is proved, for example, that the trajectories of a real analytic control system corresponding to a fixed open-loop control depend on initial condition in a real analytic manner. It is also proved that control-affine systems always have the appropriate joint dependence on state and control. This shows, for example, that the trajectories of a control-affine system corresponding to a fixed open-loop control depend on initial condition in the manner prescribed by the regularity of the vector fields.

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Notes

  1. The condition on page 35 of this text is not sufficient to ensure uniqueness of trajectories.

  2. This is actually not a very good name. A better name, and the name used by Jafarpour and Lewis [35], would be the “smooth compact-open topology.” However, we wish to keep things simple here, and also use notation that is common between regularity classes.

  3. There is a potential confusion about “boundedness” in this paper. In Sect. 1.6 we have defined a notion of “essentially bounded” that is different, in general, from the notion of “essentially von Neumann bounded” that we use here. We will not quite encroach on areas where this confusion causes problems, but it is something to bear in mind. Jafarpour and Lewis [35] are a little more careful about this, explicitly making use of “bornologies.”

  4. Such complexifications exist, as shown in [35, §5.1.1].

References

  1. Abraham R, Marsden JE, Ratiu TS (1988) Manifolds, tensor analysis, and applications, 2nd edn. Applied Mathematical Sciences, vol 75. Springer, New York

  2. Agrachev AA, Gamkrelidze RV (1978) The exponential representation of flows and the chronological calculus. Math USSR Sbornik 107(4):467–532

    MathSciNet  MATH  Google Scholar 

  3. Agrachev AA, Sachkov Y (2004) Control theory from the geometric viewpoint. Encyclopedia of Mathematical Sciences, vol 87. Springer, New York

  4. Barbero-Liñán M, Muñoz-Lecanda MC (2009) Geometric approach to Pontryagin’s maximum principle. Acta Appl Math 108(2):429–485

    Article  MathSciNet  MATH  Google Scholar 

  5. Beckmann R, Deitmar A (2011) Strong vector valued integrals. arXiv:1102.1246v1 [math.FA]

  6. Bianchini RM, Stefani G (1993) Controllability along a trajectory: a variational approach. SIAM J Control Optim 31(4):900–927

    Article  MathSciNet  MATH  Google Scholar 

  7. Bierstedt KD (2007) Introduction to topological tensor products. Course notes, Mathematical Institute, University of Paderborn. http://tinyurl.com/mo64rdk

  8. Bonnard B, Chyba M (2003) Singular trajectories and their role in control theory. Mathematics and Applications, vol 40. Springer, New York

  9. Bourbaki N (1989) Algebra I. Elements of mathematics. Springer, New York

    MATH  Google Scholar 

  10. Bourbaki N (1990) Algebra II. Elements of mathematics. Springer, New York

    MATH  Google Scholar 

  11. Bressan A, Piccoli B (2007) Introduction to the mathematical theory of control. Applied Mathematics, vol 2. American Institute of Mathematical Sciences, Wilmington

  12. Brockett RW (1977) Control theory and analytical mechanics. In: Martin C, Hermann R (eds) Geometric control theory. Math Sci Press, Brookline, pp 1–48

    Google Scholar 

  13. Bus JCP (1984) The Lagrange multiplier rule on manifolds and optimal control of nonlinear systems. SIAM J Control Optim 22(5):740–757

    Article  MathSciNet  MATH  Google Scholar 

  14. Cieliebak K, Eliashberg Y (2012) From Stein to Weinstein and back: symplectic geometry of affine complex manifolds. American Mathematical Society Colloquium Publications, vol 59. American Mathematical Society, Providence

  15. Coddington EE, Levinson N (1955) Theory of ordinary differential equations. McGraw-Hill, New York. New edition: [16]

  16. Coddington EE, Levinson N (1984) Theory of ordinary differential equations, 8th edn. Robert E. Krieger Publishing Company, Huntington. First edition: [15]

  17. Conway JB (1985) A course in functional analysis, 2nd edn. Graduate Texts in Mathematics, vol 96. Springer, New York

  18. Coron JM (1994) Linearized control systems and applications to smooth stablization. SIAM J Control Optim 32(2):358–386

    Article  MathSciNet  MATH  Google Scholar 

  19. Delgado-Téllez M, Ibort A (2003) A panorama of geometric optimal control theory. Extracta Math 18(2):129–151

    MathSciNet  MATH  Google Scholar 

  20. Demailly JP (2012) Complex analytic and differential geometry. Unpublished manuscript made publicly available . http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook

  21. Diestel J, Fourie JH, Swart J (2008) The metric theory of tensor products: Grothendieck’s Résumé Revisited. American Mathematical Society, Providence

  22. Diestel J, Uhl Jr, JJ (1977) Vector measures. American Mathematical Society Mathematical Surveys and Monographs, vol 15. American Mathematical Society, Providence

  23. Domański P (2010) Notes on real analytic functions and classical operators. In: Blasco O, Bonet J, Calabuig J, Jornet D (eds) Proceedings of the third winter school in complex analysis and operator theory, Contemporary Mathematics, vol 561. American Mathematical Society, Providence, pp 3–47

  24. Domański P, Vogt D (2000) The space of real-analytic functions has no basis. Polska Akademia Nauk. Instytut Matematyczny. Studia Math 142(2):187–200

    MathSciNet  Google Scholar 

  25. Federer H (1996) Geometric measure theory. Classics in mathematics. Springer, New York. Reprint of 1969 edition

  26. Fritzsche K, Grauert H (2002) From holomorphic functions to complex manifolds. Graduate Texts in Mathematics, vol 213. Springer, New York

  27. Gamkrelidze RV (1978) Principles of optimal control theory. Mathematical Concepts and Methods in Science and Engineering, vol 7. Plenum Press, New York

  28. Gasquet C, Witomski P (1999) Fourier analysis and applications. Texts in Applied Mathematics, vol 30. Springer, New York. Translated from the French by Robert Ryan

  29. Groethendieck A (1973) Topological vector spaces. Notes on Mathematics and its Applications. Gordon & Breach Science Publishers, New York

  30. Gunning RC, Rossi H (1965) Analytic functions of several complex variables. American Mathematical Society, Providence. 2009 reprint by AMS

  31. Hirsch MW (1976) Differential topology. Graduate Texts in Mathematics, vol 33. Springer, New York

  32. Hörmander L (1973) An introduction to complex analysis in several variables, 2nd edn. North-Holland, Amsterdam

  33. Horváth J (1966) topological vector spaces and distributions, vol I. Addison Wesley, Reading

  34. Hungerford TW (1980) Algebra. Graduate Texts in Mathematics, vol 73. Springer, New York

  35. Jafarpour S, Lewis AD (2014) Time-varying vector fields and their flows. Springer Briefs in Mathematics. Springer, New York

  36. Jarchow H (1981) Locally convex spaces. Mathematical textbooks. Teubner, Leipzig

  37. Kawski M (1990) High-order small-time local controllability. Nonlinear Controllability and Optimal Control, Monographs and Textbooks in Pure and Applied Mathematics, vol 133. Dekker Marcel Dekker, New York, pp 431–467

  38. Kolář I, Michor PW, Slovák J (1993) Natural operations in differential geometry. Springer, New York

    Book  MATH  Google Scholar 

  39. Krantz SG, Parks HR (2002) A primer of real analytic functions. Birkhäuser Advanced Texts, 2nd edn. Birkhäuser, Boston

    Book  Google Scholar 

  40. Krener AJ (1977) The higher order maximum principle and its applications to singular extremals. SIAM J Control Optim 15(2):256–293

    Article  MathSciNet  MATH  Google Scholar 

  41. Langerock B (2003) Geometric aspects of the maximum principle and lifts over a bundle map. Acta Appl Math 77(1):71–104

    Article  MathSciNet  MATH  Google Scholar 

  42. Lee EB, Markus L (1967) Foundations of optimal control theory. Wiley, New York

    MATH  Google Scholar 

  43. Lewis AD (2014) Tautological control systems. Springer Briefs in Electrical and Computer Engineering—Control, Automation and Robotics. Springer, New York

  44. Lions JL (1971) Optimal control of systems governed by partial differential equations. Grundlehren der Mathematischen Wissenschaften, vol 170. Springer, New York

  45. Mangino EM (1997) (LF)-spaces and tensor products. Math Nachr 185:149–162

    Article  MathSciNet  MATH  Google Scholar 

  46. Michor PW (1980) Manifolds of differentiable mappings. Shiva Mathematics Series, vol 3. Shiva Publishing Limited, Orpington (1980)

  47. Nagano T (1966) Linear differential systems with singularities and an application to transitive Lie algebras. J Math Soc Japan 18:398–404

    Article  MathSciNet  MATH  Google Scholar 

  48. Nagano T (1966) Linear differential systems with singularities and an application to transitive Lie algebras. J Math Soc Japan 18:398–404

    Article  MathSciNet  MATH  Google Scholar 

  49. Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF (1961) Matematicheskaya teoriya optimal’ nykh protsessov. Gosudarstvennoe izdatelstvo fiziko-matematicheskoi literatury, Moscow. Reprint of translation: [50]

  50. Nagano T (1966) Linear differential systems with singularities and an application to transitive Lie algebras. J Math Soc Japan 18:398–404

    Article  MathSciNet  MATH  Google Scholar 

  51. Retakh VS (1970) The subspaces of a countable inductive limit. Soviet Mathematics. Doklady. A translation of the mathematics section of. Doklady Akademii Nauk SSSR 11:1384–1386

    MATH  Google Scholar 

  52. Rudin W (1991) Functional analysis, 2nd edn. International Series in Pure and Applied Mathematics. McGraw-Hill, New York

  53. Ryan RA (2002) Introduction to tensor products of Banach spaces. Springer Monographs in Mathematics. Springer, New York

  54. Saunders DJ (1989) The geometry of jet bundles. London Mathematical Society Lecture Note Series, vol 142. Cambridge University Press, New York

  55. Schaefer HH, Wolff MP (1999) Topological vector spaces, 2nd edn. Graduate Texts in Mathematics, vol 3. Springer, New York

  56. Schwartz L (1950–1951) Théorie des distributions. Publications de l’Institut de Mathématique de l’Université de Strasbourg. Hermann, Paris. Reprint: [57]

  57. Schwartz L (1997) Théorie des distributions. Publications de l’Institut de Mathématique de l’Université de Strasbourg. Hermann, Paris. Original: [56]

  58. Sontag ED (1992) Universal nonsingular controls. Syst Control Lett 19(3):221–224

    Article  MathSciNet  MATH  Google Scholar 

  59. Sontag ED (1998) Mathematical control theory: deterministic finite dimensional systems, 2nd edn. Texts in Applied Mathematics, vol 6. Springer, New York

  60. Sussmann HJ (1973) Orbits of families of vector fields and integrability of distributions. Trans Am Math Soc 180:171–188

    Article  MathSciNet  MATH  Google Scholar 

  61. Sussmann HJ (1983) Lie brackets, real analyticity and geometric control. In: Differential geometric control theory, Progress in Mathematics, vol 27. Birkhäuser, Boston

  62. Sussmann HJ (1990) Why real analyticity is important in control theory. Perspectives in Control Theory (Sielpia), Progress in Systems and Control Theory, vol 2. Birkhäuser, Boston, pp 315–340

  63. Sussmann HJ (1997) An introduction to the coordinate-free maximum principle. In: Jakubczyk B, Respondek W (eds) Geometry of feedback and optimal control. Dekker Marcel Dekker, New York, pp 463–557

  64. Sussmann HJ (1998) Some optimal control applications of real-analytic stratifications and desingularization. Singularities Symposium-Łojasiewicz 70, Banach Center Publications, vol 44. Polish Academy of Sciences, Institute for Mathematics, Warsaw, pp 211–232

  65. Sussmann HJ, Jurdjevic V (1972) Controllability of nonlinear systems. J Differ Equ 12:95–116

    Article  MathSciNet  MATH  Google Scholar 

  66. Tröltzsch F (2010) Optimal control of partial differential equations. Graduate Studies in Mathematics, vol 112. American Mathematical Society, Providence

  67. Vogt D (2013) A fundamental system of seminorms for \(A(K)\) (2013). arXiv:1309.6292v1 [math.FA]

  68. Wengenroth J (1995) Retractive (LF)-spaces. Ph.D. thesis, Universität Trier, Trier

  69. Willems JC (1979) System theoretic models for the analysis of physical systems. Ricerche di Automatica 10(2):71–106

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The second author was a Visiting Professor in the Department of Mathematics at University of Hawaii, Manoa, when the paper was written, and would like to acknowledge the hospitality of the department, particularly that of Monique Chyba and George Wilkens. The second author would also like to thank his departmental colleague Mike Roth for numerous useful conversations over the years. While conversations with Mike did not lead directly to results in this paper, Mike’s willingness to chat about complex geometry and to answer ill-informed questions was always appreciated, and ultimately very helpful.

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  1. Department of Mathematics and Statistics, Queen’s University, Kingston, ON, K7L 3N6, Canada

    Saber Jafarpour & Andrew D. Lewis

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  1. Saber Jafarpour
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Correspondence to Andrew D. Lewis.

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Research supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.

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Jafarpour, S., Lewis, A.D. Locally convex topologies and control theory. Math. Control Signals Syst. 28, 29 (2016). https://doi.org/10.1007/s00498-016-0179-0

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