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Differential games, dynamic optimization, and generalized control theory

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Abstract

In this paper, we examine the problem of control in a generalized framework in which there are more than one criterion, more than one intelligent controller, each of which has access to different information. It is argued that optimal control, differential games, dynamic team theory, etc., can all be viewed as special cases of this generalized control theory. Useful concepts and potential difficulties are discussed.

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References

  1. Collected Bibliography on Differential Games, Proceedings of the First International Conference on the Theory and Applications of Differential Games, Edited by Y. C. Ho and G. Leitmann, Amherst, Massachusetts, 1969.

  2. Ho, Y. C., andLeitmann, G., Editors,Proceedings of the First International Conference on the Theory and Application of Differential Games, Amherst, Massachussets, 1969.

  3. Isaacs, R.,Differential Games: Their Scope, Nature, and Future, Journal of Optimization Theory and Applications, Vol. 3, No. 5, 1969.

  4. Behn, R. D., andHo, Y. C.,On a Class of Linear Stochastic Differential Games, IEEE Transactions on Automatic Control, Vol. AC-13, No. 3, 1968.

  5. Rhodes, I. B., andLuenberger, D. G.,Differential Games with Imperfect State Information, IEEE Transactions on Automatic Control, Vol. AC-14, No. 1, 1969.

  6. Willman, W. W.,Formal Solutions for a Class of Stochastic Pursuit-Evasion Games, IEEE Transactions on Automatic Control, Vol. AC-14, No. 5, 1969.

  7. Zadeh, L. A.,Optimality and Non-Scalar Valued Performance Criteria, IEEE Transactions on Automatic Control, Vol. AC-8, No. 1, 1963.

  8. Da Cunha, N. O., andPolak, E.,Constrained Minimization under Vector-Valued Criteria in Finite-Dimensional Spaces, University of California at Berkeley, Memorandum No. ERL-M188, 1966.

  9. Case, J. H.,Towards a Theory of Many Players Differential Games, SIAM Journal on Control, Vol. 7, No. 2, 1969.

  10. Starr, A. W., andHo, Y. C.,Nonzero-Sum Differential Games, Journal of Optimization Theory and Applications, Vol. 3, No. 3, 1969.

  11. Starr, A. W., andHo, Y. C.,Further Properties of Nonzero-Sum Differential Games, Journal of Optimization Theory and Applications, Vol. 3, No. 4, 1969.

  12. Radner, R.,Team Decision Problems, Annals of Mathematical Statistics, Vol. 33, 1962.

  13. Radner, R.,The Evaluation of Information in Organizations, Proceedings of the Fourth Berkeley Symposium on Probability and Mathematical Statistics, University of California Press, Berkeley, California, 1961.

    Google Scholar 

  14. Witsenhausen, H.,A Counter Example in Stochastic Optimum Control, SIAM Journal on Control, Vol. 6, No. 1, 1968.

  15. Harsanyi, J.,Games with Incomplete Information Played by Bayesian Players, Parts I, II, III, Management Science, Vol. 14, Nos, 3, 5, 7, 1967, 1968.

  16. Shapley, L. S., N-Person Game Theory, The RAND Corporation, Report No. P-3752, 1968.

  17. Schelling, T. C.,The Strategy of Conflict, Oxford University Press, Cambridge, England, 1963.

    Google Scholar 

  18. Owens, G.,Game Theory, W. B. Saunders Company, Philadelphia, Pennsylvania, 1968.

    Google Scholar 

  19. MacCrimmon, K. R.,Decision Making Among Multiple-Attribute Alternatives: A Survey and Consolidated Approach, The RAND Corporation, Research Memorandum No. RM-4823-ARPA, 1969.

  20. Raiffa, H.,Preference for Multi-Attributed Alternatives, The RAND Corporation, Research Memorandum No. RM-5868-DOT/RC, 1968.

  21. Bryson, A. E., andHo, Y. C.,Applied Optimal Control, Blaisdell Publishing Company, Waltham, Massachusetts, 1969.

    Google Scholar 

  22. Behn, R.,Linear Stochastic Differential Games, Harvard University, Ph.D. Thesis, 1968.

  23. Rhodes, I. B., andLuenberger, D. G.,Nondeterministic Differential Games with Constrained State Estimators, IEEE Transactions on Automatic Control, Vol. AC-14, No. 5, 1969.

  24. Bley, K., andStear, E.,Game Optimal Control Strategies for Multistage Linear Stochastic Differential Games, Paper Presented at the First International Conference on the Theory and Application of Differential Games, Amherst, Massachusetts, 1969.

  25. Willman, W. W.,Diffrentiale Games with Measurement Uncertainties, Harvard University, Ph.D. Thesis, 1969.

  26. Beckman, M. T.,Decision and Team Problems in Airline Reservations, Econometrica, Vol. 26, No. 2, 1958.

  27. Kriebel, C. H.,Quadratic Team Information Economics and Aggregate Planning Decisions, Econometrica, Vol. 36, Nos. 3–4, 1968.

  28. LaValle, I.,Strategic Situation Theory: A Bayesian Approach to an Individual Player's Choice of Strategies in Noncooperative Games, Harvard University, Business School, Ph.D. Thesis, 1966.

  29. Berkovitz, L. D.,A Survey of Differential Games, Mathematical Theory of Control, Edited by A. V. Balakrishnan and L. W. Neustadt, Academic Press, New York, 1967.

    Google Scholar 

  30. Simakova, E. N.,Differential Games, Automation and Remote Control, Vol. 27, No. 11, 1966.

  31. Breakwell, J. V., andMerz, A. W.,Towards A Solution of the Homicidal Chauffeur Games, Paper Presented at the First International Conference of the Theory and Application of Differential Games, Amherst, Massachusetts, 1969.

  32. Varaiya, P., andLin, J.,Existence of Saddle Point in Differential Games, SIAM Journal on Control, Vol. 7, No. 1, 1969.

  33. Friedman, A.,On the Definition of Differential Games and the Existence of Value and Saddle Point, Journal of Differential Equations, Vol. 7, No. 1, 1970.

  34. Blaquiere, A., Gerard, F., andLeitmann, G.,Quantitative and Qualitative Games, Academic Press, New York, 1969.

    Google Scholar 

  35. Nykolskii, M., and Gosjatinkov, P. B.,Optimality of Pursuit Time, Soviet Mathematics Doklady, Vol. 10, No. 1, 1969.

  36. Varaiya, P.,Linear Pursuit Games, Paper Presented at the First International Conference on the Theory and Application of Differential Games, Amherst, Massachusetts, 1969.

  37. Rogers, P.,Sum Stochastic Games, University of California at Berkeley, Operations Research Center, Report No. ORC69-8, 1969.

  38. Kushner, R. J., andChamberlain, S. G.,Finite State Stochastic Games: Existence Theorems and Computational Procedures, IEEE Transactions on Automatic Control, Vol. AC-14, No. 3, 1969.

  39. Eaton, J. H., andZadeh, L. A.,Optimal Pursuit Strategies in Discrete-State Probabilistic Systems, ASME Transactions, Series D, Journal of Basic Engineering, Vol. 84, 1962.

  40. Greene, T. E., andHuntzicker, J. H.,A Survey of Methods for Studying Air-to-Air Combat: World War II to the Present, The RAND Corporation, Research Memorandum No. RM-5351/1-PR, 1967.

  41. Rutowski, E. S.,Energy Approach to the General Aircraft Performance Problem, Journal of the Aerospace Sciences, Vol. 21, No. 3, 1954.

  42. Salmon, D.,Policy and Controller and Controller Design for a Pursuing Vehicle, IEEE Transactions on Automatic Control, Vol. AC-14, No. 5, 1969.

  43. Meier, L.,A New Technique for Solving Pursuit-Evasin Differential Games, IEEE Transactions on Automatic Control, Vol. AC-14, No. 4, 1969.

  44. Bird, J.,The Differential Maneuver Simulator, Private Communication, 1969.

  45. Greene, T. E., andHuntzicker, J. H.,Current Techniques at RAND for Study of Air-to-Air Combat, The RAND Corporation, Research Memorandum No. RM-5351/2PR, 1967.

  46. Discussion at the Workshop on the Application of Differential Games to Air Combat, USAF Academy, Colorado, 1968.

  47. Spicer, R., Private Communication, 1969.

  48. Miller, L. E.,Tactics Optimization Study: Variables Speed Case, Paper Presented at the First International Conference on the Theory and Application of Differential Games, Amherst, Massachusetts, 1969.

  49. Baron, S., Chu, K. C., Ho, Y. C., andKleinman, D.,Application of Differential Game Theory to Tactical Maneuvering Aircraft, Bolt, Beranek, and Neuman, Cambridge, Massachusetts, Final Report to NASA, Contract No. NAS1-8296, 1970.

    Google Scholar 

  50. Clemhout, S., Leitmann, G., andWan, H. Y.,A Differential Game Model of Duopoly, Paper Presented at the First International Conference on the Theory and Application of Differential Games, Amherst, Massachusetts, 1969.

  51. Case, J.,A Problem in International Trade, Paper Presented at the First International Conference on the Theory and Application of Differential Games, Amherst, Massachusetts, 1969.

  52. Starr, A. W.,Nonzero-Sum Differential Games: Concepts and Models, Harvard University, Division of Engineering and Applied Physics, Technical Report No. TR 590, 1969.

  53. Kuhn, H.,Extensive Games and the Problem of Information, Annals of Mathematical Studies, No. 28, 1953.

  54. Aumann, R.,Mixed and Behavior Strategies in Infinite Extensive Games, Annals of Mathematical Studies, 1964.

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Authors and Affiliations

  1. Harvard University, Cambridge, Massachusetts

    Y. C. Ho (Gordon McKay Professor of Engineering and Applied Mathematics)

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  1. Y. C. Ho
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The work reported in this paper is a result of research support extended over a period of years by Joint Services Electronic Contract No. N00014-67-A-0298-0006, NASA Grant No. NGR-22-007-068, and AFOSR Grant No. 69-1768 administered through Harvard University. The author has benefited from discussions with numerous colleagues, in particular, K. C. Chu, A. Starr, W. Willman, H. Witsenhausen, and S. Mitter.

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Ho, Y.C. Differential games, dynamic optimization, and generalized control theory. J Optim Theory Appl 6, 179–209 (1970). https://doi.org/10.1007/BF00926600

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