Skip to main content

Part of the book series: Lecture Notes in Biomathematics ((LNBM,volume 30))

  • 385 Accesses

Abstract

Control theory arose in two fields, biology and engineering. At the time when biologists were first beginning to formulate some of the concepts of control theory in a descriptive and qualitative manner, engineers already had made considerable progress in developing and applying quantitative methods. However, these two lines of development did not mingle until the 1940’s. It now appears that control theory will be one of the major conceptual frameworks for biological research in the near future.

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

Access this chapter

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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adolph, E.F., Origins of Physiological Regulations, Academic Press, New York, 1968.

    Google Scholar 

  2. Athans, M. and Falb, P.L., Optimal Control Theory: An Introduction to the Theory and Its Applications, McGraw-Hill, New York, 1966.

    Google Scholar 

  3. Banks, H.T., Modeling and Control in the Biomedical Sciences, Vol. 6 of Lecture Notes in Biomathematics, Springer-Verlag, New York, 197.

    Google Scholar 

  4. Bayliss, L.E., Living Control Systems, English Univ. Press, London, 1966.

    Google Scholar 

  5. Bellman, R., Adaptive Control Processes: A Guided Tour, Princeton University Press, Princeton, 1961.

    MATH  Google Scholar 

  6. Berkovitz, L.D., Variational methods in problems of control and programming, J. Math. Anal. Appl., 3, 145–69, 1961.

    Article  MATH  MathSciNet  Google Scholar 

  7. Berkovitz, L.D., Optimal Control Theory, Am. Math. Monthly, 83, 225–239, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  8. Berkvotiz, L.D., Optimal Control Theory, Springer-Verlag, New York, 1974.

    Google Scholar 

  9. Bernard, C., Les Phenomenes de la Vie, Paris, 1878.

    Google Scholar 

  10. Bode, H.W., Feedback Amplifier Design, Bell System Tech. J., 19, 42, 1940.

    Google Scholar 

  11. Bode, H.W., Network Analysis and Feedback Amplifier Design, Van Nostrand, Princeton, 1945.

    Google Scholar 

  12. Brogan, W.L., Modern Control Theory, Quantum Press, New York, 1974.

    Google Scholar 

  13. Brown, J.H.U. and Grann, D., Engineering Principles in Physiology, Vol. 1–3, Academic Press, New York, 1973.

    Google Scholar 

  14. Brown, J.H.U., Jacobs, J.E. and Stark, L. (editors). Biomedical Engineering, F.A. Davis, Philadelphia, 1971.

    Google Scholar 

  15. Budak, B.M., Difference Approximations in Optimal Control Problems, SIAM J. Control, 7, 18–31, 1969.

    Article  MathSciNet  Google Scholar 

  16. Clark, C.W., Mathematical Bioeconomics. The Optimal Management of Renewable Resources, J. Wiley, New York, 1976.

    MATH  Google Scholar 

  17. Clynes, M. and Milsum, J.H. (editors). Biomedical Engineering Systems, McGraw-Hill, New York, 1970.

    Google Scholar 

  18. Cannon, W.B., Organization for physiological homeostasis, Physiol. Revs., 9, 399, 1929.

    Google Scholar 

  19. Coddington, E.A. and Levinson, N., Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.

    MATH  Google Scholar 

  20. Cullum, J., Discrete approximations to continuous optimal control problems, SIAM J. Control, 7, 32–49, 1969.

    Article  MATH  MathSciNet  Google Scholar 

  21. Cullum, J., An explicit method for discretizing continuous optimal control problems, J. O. T. A., 5, 1970.

    Google Scholar 

  22. Dorf, R.C., Modern Control Systems, Addison-Wesley, Reading, 1967.

    MATH  Google Scholar 

  23. Fredericq, L., Influences du milieu ambiant sur la composition du sang des animaux aquatiques, Arch. de Zool. Exper. et. Gen., 3, 35, 1885.

    Google Scholar 

  24. Funk, J.E. and Gilbert, E.G., Some sufficient conditions for optimality in control problems with state space constraints, SIAM J. Control, 8, 498–504, 1970.

    Article  MATH  MathSciNet  Google Scholar 

  25. Gilbert, E.G., Controllability and observability in multivariable control systems, J. Soc. Ind. Appl. Math. — Control Series, Series A, Vol. 1, No. 2, 128–151, 1963.

    Article  Google Scholar 

  26. Goodwin, B.C., Temporal Organization in Cells, A Dynamic Theory of Cellular Control Processes, Academic Press, New York, 1963.

    Google Scholar 

  27. Grodins, F.S., Control Theory and Biological Systems, Columbia Univ. Press, New York, 1963.

    Google Scholar 

  28. Guyton, A.C. and Crowell, J.W., Cardiac deterioration in shock: I. Its progressive nature, in Hershey, S.G. (editor). Shock, Little, Brown and Co., Boston, 1–12, 1964.

    Google Scholar 

  29. Guyton, A.C., Taylor, A.E. and Granger, H.J., Circulatory Physiology. I Cardiac Output and Its Regulation, Saunders, Philadelphia, 1973. II Dynamics and Control of the Body Fluids, ibid, 1975.

    Google Scholar 

  30. Hautus, M.L.J., Necessary conditions for multiple constraint optimization problems, SIAM J. Control, 11, No. 4, 653–69, 1973.

    Article  MATH  MathSciNet  Google Scholar 

  31. Hazen, H.L., Theory of servo-mechanisms, J. Franklin Instl., 218, 279, 1934.

    Article  Google Scholar 

  32. Hurwitz, A., On the conditions under which an equation has only roots with negative real parts, Math. Annalen, 46, 273–284, 1895. In Selected Papers on Mathematical Trends in Control Theory, Dover, New York, 70–82, 1964.

    Article  MathSciNet  Google Scholar 

  33. Iberall, A.S. and Guyton, A.C. (editors). Regulation and Control in Physiological Systems, Instrument Soc. of America, Pittsburgh, 1973.

    Google Scholar 

  34. Intriligator, M.D., Mathematical Optimization and Economic Theory, Prentice Hall, Englewood Cliffs, 1971.

    Google Scholar 

  35. Kaiman, R.E., On the general theory of control systems, in Automatic and Remote Control (Proc. IFAC Moscow 1960). Vol. I, Butterworth, London, 481–492, 1961.

    Google Scholar 

  36. Kaiman, R.E., Mathematical description of linear dynamical systems, J. Soc. Ind. App. Math. — Control Series, Series A, Vol. 1, No. 2, 152–192, 1963.

    Article  Google Scholar 

  37. Kalmus, H., Regulation and Control in Living Systems, Wiley, New York, 1966.

    Google Scholar 

  38. Kernevez, J.P., Control, optimization and parameter identification in immobilized enzyme systems, in Analysis and Control of Immobilized Enzyme Systems, edited by Thomas, D. and Kernevez, J.P., North-Holland, Amsterdam, 199–225, 1976.

    Google Scholar 

  39. Kernevez, J.P. and Thomas, D., Numerical analysis and control of some biochemical systems, Appl. Math and Optimization, 1, 222–285, 1975.

    Article  MATH  MathSciNet  Google Scholar 

  40. Kline, J., Biological Foundations of Biomedical Engineering, Little, Brown and Co., Boston, 1976.

    Google Scholar 

  41. Leitmann, G., An Introduction to Optimal Control, McGraw-Hill, New York, 1966.

    MATH  Google Scholar 

  42. Leitmann, G., Cooperative and Non-Cooperative Many Player Differential Games, Springer-Verlag, Vienna, 1974.

    Google Scholar 

  43. Longini, Jr., I.M., Ackerman, E. and Elveback, L.R., An optimization model for influenza epidemics, Math. Biosc, 38, 141–157, 1978.

    Article  Google Scholar 

  44. Mayr, O., The Origins of Feedback Control, M.I.T.Press, Cambridge, 1970.

    MATH  Google Scholar 

  45. Maxwell, J.C., On governors, Proc. of the Roy. Soc. of London, 16, 1868, in Selected Papers on Mathematical Trends in Control Theory, Dover, New York, 270–283, 1964.

    Google Scholar 

  46. McClamroch, N.H., A sufficiency condition for discrete optimal processes, Int. J. Control, 12, 157–61, 1970.

    Article  MATH  MathSciNet  Google Scholar 

  47. Milsum, J.H., Biological Control System Analysis, McGraw-Hill, New York, 1966.

    Google Scholar 

  48. Milhorn, H.T., Applications of Control Theory to Physiological Systems, Saunders, Philadelphia, 1966.

    Google Scholar 

  49. Minorsky, N., Directional stability of automatically steered bodies, J. Am. Soc. Naval Eng., 34, 280, 1922.

    Article  Google Scholar 

  50. Nyquist, H., Regeneration theory, Bell System Tech. Journal, 11, 126–147, 1932.

    MATH  Google Scholar 

  51. Papoulis, A., The Fourier Integral and Its Applications, McGraw-Hill, New York, 1962.

    MATH  Google Scholar 

  52. Pareto, V., Cours d Economie Politique, Lausanne, Rouge, 1896.

    Google Scholar 

  53. Pontryagin, L.S., Boltyanskii, v.G.; Gamkrelidge, R.V. and Mischenko, E.F. The Mathematical Theory of Optimal Processes, Interscience (Wiley), New York, 1962.

    MATH  Google Scholar 

  54. Riggs, D.S., Control Theory and Physiological Feedback Mechanisms, William and Wilkins, Baltimore, 1970.

    Google Scholar 

  55. Rosenbleuth, A., Wiener, N. and Biglow, J.H., Behavior, purpose, and teleology, Phil. Sci. 10, 18–24, 1943.

    Article  Google Scholar 

  56. Routh, E.J., Dynamics of a System of Rigid Bodies, Macmillan, New York, 1892.

    MATH  Google Scholar 

  57. Sage, A.P. and White, C.C., Optimum Systems Control, Prentice Hall, Englewood Cliffs, N.J., 1977.

    MATH  Google Scholar 

  58. Saucedo, R. and Schiring, E., Introduction to Continuous and Digital Control Systems, MacMillan, New York, 1968.

    Google Scholar 

  59. Schmitendorf, W.E. and Leitman, G. A simple derivation of necessary conditions for Pareto optimality, IEEE Trans. on Automatic Control, AC — 19, 601, 1974.

    Google Scholar 

  60. Schwan, H.P. (editor). Biological Engineering, McGraw-Hill, New York, 1969.

    Google Scholar 

  61. Stear, E.B., Application of control theory to endocrine regulation and control, Annals of Biomed. Eng., 3, 439–455, 1975.

    Article  Google Scholar 

  62. Swan, G.W., Mathematical Studies of the Harvesting and Control of Biologically Renewable Resources. Unpublished report to the National Science Foundation, Washington, D.C., 1974.

    Google Scholar 

  63. Talbot, S.A. and Gessner, Systems Physiology, Wiley, New York, 1973.

    Google Scholar 

  64. Valentine, F.A., The problem of Lagrange with differential inequalities as added side conditions, in Contributions to the Calculus of Variations, 1933–1937, University of Chicago Press, Chicago, 1937.

    Google Scholar 

  65. Van Valeknburg, M.E., Network Analysis, 2nd ed., Prentice-Hall, Englewood Cliffs, N.J., Chapter 10, 1964.

    Google Scholar 

  66. Vyshnegradskii, I.A., On controllers of direct action, Izv., SPB Tekhnolog. Inst., 1877.

    Google Scholar 

  67. Wiener, N., Cybernetics, MIT Press, Cambridge, Rev. ed., Wiley, New York, 1961.

    Google Scholar 

  68. Wickwire, K., Mathematical models for the control of pests and infectious diseases: a survey, Theor. Pop. Biol. 11, 182–238, 1977.

    Article  MathSciNet  Google Scholar 

  69. Yamamoto, W.S. and Brobeck, J.R. (editors). Physiological Controls and Regulations, Saunders, Philadelphia, 1965.

    Google Scholar 

  70. Yu, P.L. and Leitmann, G., Non dominated decisions and cone convexity dynamic multicriteria decision problems, JOTA, 14, No. 5, 573–84, 1974.

    Article  MATH  MathSciNet  Google Scholar 

  71. Zadeh, L.A. and Desoer, C.A., Linear System Theory, The State Space Approach, McGraw-Hill, New York, 1963.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1979 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Eisen, M. (1979). Control Theory. In: Mathematical Models in Cell Biology and Cancer Chemotherapy. Lecture Notes in Biomathematics, vol 30. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-93126-0_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-93126-0_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09709-9

  • Online ISBN: 978-3-642-93126-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics