Skip to main content
SpringerLink
Log in

Ongoing and Future Work

  • Chapter
  • First Online:
Tautological Control Systems

Part of the book series: SpringerBriefs in Electrical and Computer Engineering ((BRIEFSCONTROL))

  • 763 Accesses

Abstract

In this chapter we will give a sketchy, but we hope compelling, idea of how the tautological control system framework can be used to say new things about control systems. This will also provide an illustration of how, in practice, one can do control theory within the confines of the tautological control system framework, without reverting to the comforting control parameterisations with which one is familiar. We will emphasise that some of these ideas are in the preliminary stages of investigation, so the final word on what results will look like has yet to be uttered. Nonetheless, we believe that even the clear problem formulations we give make it apparent that there is something “going on” here.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Agrachev AA (1999) Is it possible to recognize local controllability in a finite number of differentiations? Open problems in mathematical systems and control theory, Communications and Control Engineering Series. Springer, Heidelberg, pp 15–18

    Google Scholar 

  2. Agrachev AA, Gamkrelidze RV (1993) Local controllability and semigroups of diffeomorphisms. Acta Appl Math 32(1):1–57

    Google Scholar 

  3. Aguilar CO (2010) Local controllability of affine distributions. PhD thesis. Queen’s University, Kingston, Kingston, ON, Canada

    Google Scholar 

  4. Bacciotti A, Stefani G (1983) On the relationship between global and local controllability. Math Syst Theor 16(1):79–91

    Article  MATH  MathSciNet  Google Scholar 

  5. Basto-Gonçalves J (1998) Second-order conditions for local controllability. Syst Control Lett 35(5):287–290

    Article  MATH  Google Scholar 

  6. Bianchini RM, Kawski M (2003) Needle variations that cannot be summed. SIAM J Control Optim 42(1):218–238

    Article  MATH  MathSciNet  Google Scholar 

  7. Bianchini RM, Stefani G (1984) Normal local controllability of order one. Int J Control 39(4):701–714

    Article  MATH  MathSciNet  Google Scholar 

  8. Bianchini RM, Stefani G (1986) Local controllability along a reference trajectory. Analysis and optimization of systems, vol 83. Lecture Notes in Control and Information Sciences. Springer, Berlin, pp 342–353

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Brockett RW (1983) Asymptotic stability and feedback stabilization. In: Brockett RW, Millman RS, Sussmann HJ (eds) Differential geometric control theory, No. 27 in Progress in Mathematics, pp 181–191. Birkhäuser, Boston

    Google Scholar 

  11. Cartan H (1957) Variétés analytiques réelles et variétés analytiques complexes. Bull Soc Math France 85:77–99

    MATH  MathSciNet  Google Scholar 

  12. Clarke FH, Ledyaev YS, Sontag ED, Subotin AI (1997) Asymptotic controllability implies feedback stabilization. Institute of Electrical and Electronics Engineers. Trans Automat Control 42(10):1394–1407

    Google Scholar 

  13. Coron JM (1990) A necessary condition for feedback stabilization. Syst Control Lett 14(3): 227–232

    Google Scholar 

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

  15. Elkin VI (1999) Reduction of nonlinear control systems. A differential geometric approach. No. 472 in Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht (Translated from the 1997 Russian original by Naidu PSV)

    Google Scholar 

  16. Gunning RC (1990) Introduction to holomorphic functions of several variables, vol II: local theory. Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole, Belmont

    Google Scholar 

  17. Haynes GW, Hermes H (1970) Nonlinear controllability via Lie theory. J Soc Indust Appl Math Series A Control 8:450–460

    Google Scholar 

  18. Hermes H (1974) On local and global controllability. J Soc Indust Appl Math Series A Control 12:252–261

    Google Scholar 

  19. Hermes H (1976) High order conditions for local controllability and controlled stability. In: Proceedings of the 1976 IEEE Conference on Decision and control. Institute of Electrical and Electronics Engineers, Clearwater, pp 836–840

    Google Scholar 

  20. Hermes H (1976) Local controllability and sufficient conditions in singular problems. J Differ Equ 20(1):213–232

    Article  MATH  MathSciNet  Google Scholar 

  21. Hermes H (1977) High order controlled stability and controllability. Dynamical systems. Academic Press, New York, pp 89–99 (Proceedings of International Symposium, Gainesville, FL)

    Google Scholar 

  22. Hermes H (1982) On local controllability. SIAM J Control Optim 20(2):211–220

    Article  MATH  MathSciNet  Google Scholar 

  23. Hermes H, Kawski M (1987) Local controllability of a single input, affine system. In: Lakshmikantham V (ed) Nonlinear analysis and applications, vol 109. Lecture Notes in Pure and Applied Mathematics. Dekker Marcel Dekker, New York, pp 235–248

    Google Scholar 

  24. Hirschorn RM, Lewis AD (2002) Geometric local controllability: second-order conditions. In: Proceedings of the 41st IEEE Conference on decision and control. Institute of Electrical and Electronics Engineers, Las Vegas, pp 368–369

    Google Scholar 

  25. Isaiah P (2012) Feedback stabilisation of locally controllable systems. PhD thesis. Queen’s University, Kingston, ON, Canada

    Google Scholar 

  26. Jafarpour S, Lewis AD (2014) Locally convex topologies and control theory. Submitted to SIAM J Control Optim

    Google Scholar 

  27. Kawski M (1987) A necessary condition for local controllability. Differential geometry: the interface between pure and applied mathematics, vol 68. Contemporary Mathematics. American Mathematical Society, Providence, RI, pp 143–155

    Google Scholar 

  28. Kawski M (1988) Control variations with an increasing number of switchings. Bull Amer Math Soc (N.S.) 18(2):149–152

    Google Scholar 

  29. Kawski M (1990) The complexity of deciding controllability. Syst Control Lett 15(1):9–14

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  31. Kawski M (1992) High-order conditions for local controllability in practice. Recent advances in mathematical theory of systems, control, networks and signal processing, II. Mita, Tokya, pp 271–276

    Google Scholar 

  32. Kawski M (1998) Nonlinear control and combinatorics of words. In: Jakubczyk B, Respondek W (eds) Geometry of feedback and optimal control. Dekker Marcel Dekker, New York, pp 305–346

    Google Scholar 

  33. Kawski M (1999) Controllability via chronological calculus. In: Proceedings of the 38th IEEE Conference on decision and control. Institute of Electrical and Electronics Engineers, Phoenix, AZ, pp 2920–2926

    Google Scholar 

  34. Kawski M (2006) On the problem whether controllability is finitely determined. In: Proceedings of MTNS ’06

    Google Scholar 

  35. Lewis AD (2012) Fundamental problems of geometric control theory. In: Proceedings of the 51st IEEE Conference on decision and control. Institute of Electrical and Electronics Engineers, Maui, HI, pp 7511–7516

    Google Scholar 

  36. Lewis AD (2014) Linearisation of tautological control systems. Submitted to J Geom Mech

    Google Scholar 

  37. Montgomery R (20032) A tour of subriemannian geometries, their geodesics and applications, No. 91 in American Mathematical Society Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI

    Google Scholar 

  38. Orsi R, Praly L, Mareels IMY (2003) Necessary conditions for stability and attractivity of continuous systems. Int J Control 76(11):1070–1077

    Article  MATH  MathSciNet  Google Scholar 

  39. Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF (1961) Matematicheskaya teoriya optimal’ nykh protsessov. Gosudarstvennoe izdatelstvo fiziko-matematicheskoi literatury, Moscow (Reprint of translation: [40])

    Google Scholar 

  40. Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF (1986) The mathematical theory of optimal processes. Classics of Soviet Mathematics. Gordon and Breach Science Publishers, New York (Reprint of 1962 translation from the Russian by Trirogoff KN)

    Google Scholar 

  41. Ramanan S (2005) Global Calculus, No. 65 in Graduate Studies in Mathematics. American Mathematical Society, Providence, RI

    Google Scholar 

  42. Sontag ED (1988) Controllability is harder to decide than accessibility. SIAM J Control Optim 26(5):1106–1118

    Article  MATH  MathSciNet  Google Scholar 

  43. Sontag ED (1989) A “universal” construction of Artstein’s theorem on nonlinear stabilization. Syst Control Lett 13(2):117–123

    Article  MATH  MathSciNet  Google Scholar 

  44. Sontag ED (1998) Mathematical control theory: deterministic finite dimensional systems, 2 edn, No. 6 in Texts in Applied Mathematics. Springer, Heidelberg

    Google Scholar 

  45. Stefan P (1974) Accessible sets, orbits and foliations with singularities. Proc London Math Soc Third Series 29:699–713

    Article  MATH  MathSciNet  Google Scholar 

  46. Stefani G (1986) On the local controllability of a scalar-input control system. In: Theory and applications of nonlinear control systems. North-Holland, pp 167–179

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  48. Sussmann HJ (1978) A sufficient condition for local controllability. SIAM J Control Optim 16(5):790–802

    Article  MATH  MathSciNet  Google Scholar 

  49. Sussmann HJ (1983) Lie brackets and local controllability: a sufficient condition for scalar-input systems. SIAM J Control Optim 21(5):686–713

    Article  MATH  MathSciNet  Google Scholar 

  50. Sussmann HJ (1987) A general theorem on local controllability. SIAM J Control Optim 25(1):158–194

    Article  MATH  MathSciNet  Google Scholar 

  51. 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

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  53. Tabuada P, Pappas GJ (2005) Quotients of fully nonlinear control systems. SIAM J Control Optim 43(5):1844–1866

    Article  MATH  MathSciNet  Google Scholar 

  54. Wells RO Jr (2008) Differential analysis on complex manifolds, 3 edn, No. 65 in Graduate Texts in Mathematics. Springer, New York

    Google Scholar 

  55. Zabczyk J (1989) Some comments on stabilizability. Appl Math Optim 19(1):1–9

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Queen’s University, Kingston, Canada

    Andrew D. Lewis

Authors
  1. Andrew D. Lewis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrew D. Lewis .

Rights and permissions

Reprints and permissions

Copyright information

© 2014 The Author(s)

About this chapter

Cite this chapter

Lewis, A.D. (2014). Ongoing and Future Work. In: Tautological Control Systems. SpringerBriefs in Electrical and Computer Engineering(). Springer, Cham. https://doi.org/10.1007/978-3-319-08638-5_7

Download citation

Publish with us

Policies and ethics

Search

Navigation