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Necessary and Sufficient Conditions for Optimality

  • Kun S. Chang
Part of the International Centre for Mechanical Sciences book series (CISM, volume 135)

Abstract

For the last few years, an enormous number of research publications on periodic processes have appared in the literature (see Fjeld [1], Guardabassi, Locatelli and Rinaldi [2] for the publications up to 1971). As the theory has reached a certain degree of growth and perfection, certain necessary and sufficient conditions have become available. Because of the peculiar behaviour of periodic processes, however, any extension of necessary and sufficient conditions of optimality for non-periodic processes to periodic processes is neither trivial nor straight-forward. The main difficulty is in the matching of the periodicity boundary conditions. This brings us to the problem of necessary and sufficient conditions for the existence and uniqueness of periodic solutions, a problem which was not severe in non-periodic cases. With this in mind, certain necessary and sufficient conditions for the existence and uniqueness of periodic solutions are first presented. Then the necessary conditions for optimality and the sufficient conditions, in particular the Hamilton-Jacobi theory, are covered. Certain conditions of relaxed controls are also briefly treated in the last section.

Keywords

Periodic Solution Riccati Equation Fundamental Matrix Periodic Process Catalyst Selectivity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    M. Fjeld, “ Stability and Control of Periodic Processes’; Rep. 71–52-W, Division of Automatic Control, The Technical University of Norway, Trondheim (April 1971).Google Scholar
  2. [2]
    G. Guardabassi, A. Locatelli, and S. Rïnaldi, “The Status of Periodic Optimization of Dynamical Systems”, Int. Rep. LCA 71–20, Istituto di Elettrotecnica ed Elettronica, Politecnico di Milano (Nov. 1971).Google Scholar
  3. [3]
    P. Hartman, “Ordinary Differential Equations”, John Wiley and Sons, Inc. (1964).MATHGoogle Scholar
  4. [4]
    A. Halanay, “Differential Equations Academic Press (1966).Google Scholar
  5. [5]
    P.L. Falb and J.L. de Jong, “ Some Successive Approximation Methods in Control and Oscillation Theory”? Academic Press, (1969).Google Scholar
  6. [6]
    S. Bittanti, A. Locatelli and C. Maffezzoni, “ Second Variation Methods in Periodic Optimization”, Int. Rep. LCA 71–17, Istituto di Elettrotecnica ed Elettronica, Politecnico di Milano (Oct. 1971).Google Scholar
  7. [7]
    M.A. Krasnoselskii, “ Positive Solutions of Operator Equations” P. Noordhoff Ltd. (1964).Google Scholar
  8. [8]
    M. Kielkiewicz, “ Existence of Positive Periodic Solutions of Certain Ordinary Differential Equations”, Electronics Letters, 6, N°15, 481–482 (1970).Google Scholar
  9. [9]
    I. Lee, “ On the Theory of Linear Dynamic System with Periodic Parameters”, Inf. Control, 6, 265–275 (1963).CrossRefMATHGoogle Scholar
  10. [10]
    F.J.M. Horn and R.C. Lin, “ Periodic Processes: Variational Approach”, Ind. Eng. Chem. Proc. Des. Dev., 6, 21–30 (1967).CrossRefGoogle Scholar
  11. [11]
    J.E. Bailey, “ Necessary Conditions for Optimality in a General Class of Nonlinear Mixed Boundary Value Control Problems”, (to appear in Int. J. Control).Google Scholar
  12. [12]
    M. Fjeld and T. Kristiansen, “ Optimization of a Nonlinear Distributed Parameter System using Periodic Boundary Control”, Int. J. Control, 10, 601–624 (1969).MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    L.I. Rozonoér, “ L.S. Pontryagin Maximum Principle in the Theory of Optimum Systems, I”, Autom. Rem. Control, 20, 1288–1302 (1960).Google Scholar
  14. [14]
    K.S. Chang, “ Second Variation for Periodic Optimization Problems”, In this volume.Google Scholar
  15. [15]
    J.E. Bailey and F.J.M. Horn, “ Comparison between Two Sufficient Conditions for Improvement of an Optimal Steady-State Process by Periodic Operation”, J. Opt. Theory Appl., 7, 378–384 (1971).MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    G. Guardabassi, “ Steady-State versus Periodic Optimization: A Circle Criterion”, Int. Rep. LCA 71–13, Istituto di Elettrotecnica ed Elettronica, Politecnico di Milano (June 1971).Google Scholar
  17. [17]
    C. Maffezzoni, “ Hamilton-Jacobi Theory for a Periodic Control Problem”, Int. Rep. LCA 71–16, Istituto di Elettrotecnica ed Elettronica, Politecnico di Milano (Sept. 1971).Google Scholar
  18. [18]
    L.C. Young, “ Generalized Curves and the Existence of an Attained Absolute Minimum in the Calculus of Variations”, Compt. Rend. Soc. Sci, et Lettres Varsovie, Cl. III, 30, 212–234 (1937).MATHGoogle Scholar
  19. [19]
    E.J. McShane, “ Generalized Curves”, Duke Math. J., 6, 513 (1940).MathSciNetGoogle Scholar
  20. [20]
    E.J. McShane, “ Necessary Conditions for Generalized-Curve Problems of the Calculus of Variations”, Duke Math. J., 7, 1 (1940).MathSciNetGoogle Scholar
  21. [21]
    E.J. McShane, “ Existence Theorem for Bolza Problems in the Calculus of Variations, Duke Math. J., 7, 28 (1940).Google Scholar
  22. [22]
    E.J. McShane, “ Some Existence Theorems in the Calculus of Variations. III Existence Theorems for Non-Regular Problems”, Trans. Amer. Math. Soc., 45, 151 (1939).MathSciNetGoogle Scholar
  23. [23]
    J. Warga, “ Relaxed Variational Problems”, J. Math. Anal. Appt., 4, 111–128 (1962).MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    J. Warga, “ Necessary Conditions for Minimum in Relaxed Variational Problems”, J. Math. Anal. Apple, 4, 129–145 (1962).MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    A.F. Filippov, “ On Certain Questions in the Theory of Optimal Control ”, SIAM J. Control, 1, 76–84 (1962).MATHGoogle Scholar
  26. [26]
    E.B. Lee and L. Markus, “ Foundations of Optimal Control Theory”, John Wiley and Sons, Inc. (1967).MATHGoogle Scholar
  27. [27]
    E.J. McShane, “ Relaxed Controls and Variational Problems”, SIAM J. Control, 5, 438–485 (1967).MathSciNetMATHGoogle Scholar
  28. [28]
    J. Warga, “Functions of Relaxed Controls”, SIAM J., 5, 628–641 (1967).MathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    F.J.M. Horn and J.E. Bailey, “ An Application of the Theorem of Relaxed Control to the Problem of Increasing Catalyst Selectivity”, J. Opt. Theory Appl., 2, 441–449 (1968).MathSciNetCrossRefGoogle Scholar
  30. [30]
    F.J.M. Horn and J.E. Bailey, “Catalyst Selectivity under Steady-State and Dynamic Operation”, Ber. Bunsenges. Phys. Chemie, 73, 274–279 (1969).Google Scholar
  31. [31]
    F.J.M. Horn and J.E. Bailey, “ Catalyst Selectivity under Steady-State and Dynamic Operation: An Investigation of Several Kinetic Mechanisms”, Ber, Bunsenges. Phys. Chemie, 74, 611 (1970).Google Scholar
  32. [32]
    J.E. Bailey and F.J.M. Horn, “ Improvement of the Performance of a Fixed Bed Catalytic Reactor by Relaxed Steady-state Operation”, AIChE J., 550–553 (1971).Google Scholar
  33. [33]
    J.E. Bailey and F.J.M. Horn, “Cyclic Operation of Reaction Systems: The Influence of Diffusion on Catalyst Selectivity”, Chem. Eng. Sci., 22, 109–119 (1972).CrossRefGoogle Scholar
  34. [34]
    NaJ.E. Bailey, F.J.M. Horn, and R.C. Lin, “ Cyclic Operation of Reaction Systems: Effect of Heat and Mass Transfer Resistance”, AIChE J., 17, 818–825 (1971).CrossRefGoogle Scholar
  35. [35]
    J.E. Bailey, “Tangent Cone Analysis of Periodic Precesses”, AIChE J., 18, 459–461 (1972).CrossRefGoogle Scholar
  36. [36]
    M. Fjeld, “ On Periodic Processes. A Preliminary Study and Propositions for Further Investigations’; Rep. 67–54-D, Division of Automatic Control, The Technical University of Norway, Trondheim (April 1967).Google Scholar
  37. [37]
    S.G. Bankoff and K.S. Chang, “Oscillatory Operation of Jacketed Tubular Reactors”, Ind. Eng. Chem. Fundamentals, 9, 301 (1970).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 1972

Authors and Affiliations

  • Kun S. Chang
    • 1
  1. 1.Department of Chemical EngineeringUniversity of WaterlooWaterlooCanada

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