臺灣博碩士論文加值系統

在本論文中，我們探討了一種追逐逃逸對局；其中追逐者以作簡單運動的方式移動，同時逃逸者以定速率但具有轉向限制的方式移動，這樣的對局正是一般所謂「殺人司機對局」的逆型式。我們以對局結束當時追逐者與逃逸者間距離的平方作為代價函數。針對這樣的一個零和對局，我們將應用變分法來解決我們的問題，並將就所考慮的各種情況提出一套足以決定對局鞍點與對局解值的演算法則。

In this thesis, a pursuit-evasion game, in which the pursuer moves with simple motion whereas the evader moves at a fixed speed but with a curvature constraint, is investigated. The game is the inverse of the usual homicidal chauffeur game. Square of the distance between the pursuer and the evader when the game is terminated is selected as the cost function. To solve such a zero-sum game, the variational approach will be employed to solve the problem. An algorithm will be proposed to determine a saddle point and the value of the game under consideration

誌謝 ........................................ i
中文摘要 .................................... ii
ABSTRACT .................................... iii
NOMENCLATURE ................................ iv
CHAPTER 1 . INTRODUCTION ..................... 1
1.1 Motivation ........................ 1
1.2 Brief Sketch of the Contents ...... 4
CHAPTER 2 . PRELIMINARIES .................... 5
2.1 Brief Review of the Differential Game
Theory ............................ 5
2.2 Brief Review of the Variational
Approach to Optimal Control Problems
................................... 24
CHAPTER 3 . A PURSUIT-EVASION GAME WITH ONE
CURVATURE CONSTRAINT ............. 34
3.1 Introduction and Problem Formulation
................................... 34
3.2 Analysis and Main Results ......... 38
3.3 Illustrative Examples ............. 55
CHAPTER 4. DISCUSSIONS AND CONCLUSIONS ...... 67
REFERENCES .................................. 76
APPENDIX
A. The Program for Simulating the Case
when Both Players Play Optimally
.................................. 84
B. The Program for Simulating the Case
when Only the Plays Optimally
.................................. 89
C. The Program for Simulating the Case
when Only the Pursuers Plays
Optimally ........................ 94

[Ard.1]M. Ardema and N. Rajan, An approach to three-dimensional
aircraft pursuit-evasion, Computer Mathematics Applications,
Vol. 13, pp.97-110, 1987.

[Ber.1]L. Berkovitz, A variational approach to differential games,
Annals of Mathematics Study, Vol. 52, pp. 127-174, 1964.

[Bor.1]P. Borowko and W. Rzymowski, On the game of two cars,
Journal of Optimization Theory and Applications, Vol. 44, No. 3,
pp. 381-396, 1984.

[Bre.1]J. V. Breakwell, Pursuit of a faster evader, The Theory and
Application of Differential games, J. D. Grote, ed., Vol. 13,
Dordrecht, Holland, 1974.

[Bry.1]A. E. Bryson and Jr. Y. C. Ho, Applied Optimal Control:
Optimization, Estimation, and Control, Blaisdell, M.A, 1969.

[Chu.1]H. J. Chu, J. G. Hsieh, K. H. Hsia and L. W. Chen, Fuzzy
differential game of guarding a movable territory, Information
Sciences, Vol. 91, pp. 113-131, 1996.

[Chu.2]H. J. Chu, J. G. Hsieh, Y. S. Lee and K. H. Hsia, A pursuit-
evasion game with one curvature constraint, Optimal Control
Applications & Methods, Vol. 21, pp. 47-61, 2000.

[Coc.1]E.J. Cockayne, Plane pursuit with curvature constraints, SIAM
Journal on Applied Mathematics, Vol. 15, pp. 1511-1516, 1967.

[Coc.2]E.J. Cockayne and G.W.C. Hall, Plane motion of a particle
subject to curvature constraints, SIAM Journal of Control, Vol.
13, pp. 197-220, 1975.

[Fri.1]A. Friedman, Differential Games, John Wiley & Sons, New
York, 1971.

[Get.1]W. M. Getz and M. Pachter, Capturability in a two-target game
of two cars, Journal of Guidance and Control, Vol. 4, No. 1, pp.
15-21, 1981.

[Get.2]W. M. Getz and M. Pachter, Two-target pursuit-evasion
differential games in the plane, Journal of Optimization Theory
and Applications, Vol. 34, No. 3, pp. 383-403, 1981.

[Gup.1]N. K. Gupta, An overview of differential games, Control and
Dynamic Systems: Advanced in Theory and Applications, C. T.
Leondes, ed., Vol. 17, Academic Press, New York, 1981.

[Haj.1]O. Hajek, Pursuit Games, Academic Press, New York, 1975.

[Hsi.1]K. H. Hsia and J. G. Hsieh, A first approach to fuzzy differential
game problem: guarding a territory, Fuzzy Sets and Systems, Vol.
55, No. 2, pp. 157-167, 1993.

[Hsi.2]K. H. Hsia and J. G. Hsieh, Fuzzy differential game of guarding
a territory: Different speed case, First International Conference
on Fuzzy Theory and Technology, Control and Decision,
Durham, North Carolina, 1992.

[Hsi.3]K. H. Hsia, J. G. Hsieh and H. J. Chu, Fuzzy differential game of
guarding a territory in the presence of an obstacle, Proceeding of
the First European Congress on Fuzzy and Intelligent
Technologies, Aachen, Germany, pp. 1250-1254, 1993.

[Hsi.4]K. H. Hsia, J. G. Hsieh, H. J. Chu and L. W. Chen, Some results
of fuzzy differential game of guarding a movable territory,
Proceeding of the First National Symposium on Fuzzy Set
Theory and Applications, Hsing-Chu, Taiwan, pp. 149-156, 1993.

[Hsi.5]K. H. Hsia, Research on the fuzzy differential game problems,
Ph.D. Dissertation, National Sun Yat-sen University, Kaohsiung,
Taiwan, 1994.

[Hsi.6]謝哲光、夏郭賢、朱弘仁，模糊微分對局簡介，全華科技圖
書股份有限公司，台北市，1993。

[Hsi.7]K. H. Hsia, Y. S. Lee and J. G. Hsieh, Differential game with
approximate reasoning: reasoning differential game,
International Journal of Approximate Reasoning, submitted,
1998.

[Isa.1]R. Isaacs, Differential Games, John Wiley and Sons, New York,
1965.

[Isa.2]R. Isaacs, The past and some bits of the future, The Theory and
Application of Differential games, J. D. Grote, ed., Vol. 13,
Dordrecht, Holland, 1974.

[Kal.1]N. J. Kalton, Differential games of survival, The Theory and
Application of Differential games, J. D. Grote, ed., Vol. 13,
Dordrecht, Holland, 1974.

[Kas.1]B. Kaskosz, On a nonlinear evasion problem, SIAM Journal on
Control and Optimization, Vol. 15, No. 4, pp. 661-673, 1977.

[Kir.1]D.E. Kirk, Optimal Control Theory: An Introduction, Prentice-
Hall, New Jersey, 1970.

[Lee.1]Y. S. Lee, K. H. Hsia and J. G. Hsieh, A reasonable strategy
based on similarity measure for a reasoning differential game,
Proceeding of International Conference on Automation
Technology, Taipei, Taiwan, accepted, 2000.

[Lee.2]Y. S. Lee, K. H. Hsia and J. G. Hsieh, A fuzzy reasoning
differential game, Proceeding of the International Fuzzy
Systems Association World Congress, Taipei, Taiwan, Vol. 2, pp.
772-776, 1999.

[Lei.1]G. Leitmann, The Calculus of Variations and Optimal Control:
An Introduction, Plenum, New York, 1983.

[Leo.1]C. T. Leondes, Control and Dynamic Systems: Advanced in
Theory and Applications, Vol. 17, Academic Press, New York,
1981.

[Mer.1]A. W. Merz, The homicidal chauffeur, a differential game, Ph.D.
Dissertation, Stanford University, Stanford, U.S.A., 1971.

[Mer.2]A. W. Merz, The game of two identical cars, Journal of
Optimization Theory and Applications, Vol. 9, No. 5, pp. 324-
343, 1972.

[Mil.1]T. Miloh, A note on three-dimensional pursuit-evasion game
with bounded curvature, IEEE Transactions on Automatic
Control, Vol. 27, No. 3, pp. 739-741, 1982.

[Pac.1]M. Pachter, Simple-motion pursuit-evasion in the half plane,
Computers and Mathematics with Applications, Vol. 13, pp. 69-
82, 1987.

[Pac.2]M. Pachter and Y. Yavin, Simple-motion pursuit-evasion games-
part I: stroboscopic strategies in collision course ..., Journal of
Optimization Theory and Applications, Vol. 51, pp. 95-127,
1986.

[Pac.3]M. Pachter and Y. Yavin, Simple-motion pursuit-evasion
differential games-part II: optimal evasion from ..., Journal of
Optimization Theory and Applications, Vol. 51, pp. 129-159,
1986.

[Pet.1]L. A. Petrosjan, Differential Games of Pursuit, World Scientific,
Singapore, New Jersey, London, HongKong, 1993.

[Pra.1]U. R. Prasad, N. Rajan and N. J. Rao, Planar pursuit-evasion
with variable speeds: extremal trajectory maps, Journal of
Optimization Theory and Applications, Vol. 33, pp. 401-418,
1981.

[Pra.2]U. R. Prasad and N. Rajan, Aircraft pursuit-evasion problems
with variable speeds, Computers and Mathematics with
Applications, Vol. 13, pp. 111-122, 1987.

[Rad.1]J. E. Rader, Use of parameter optimization methods to determine
the existence of game theoretic saddle points, Control and
Dynamic Systems: Advanced in Theory and Applications, C. T.
Leondes, ed., Vol. 17, Academic Press, New York, 1981.

[Rag.1]Raghavan, T. E. S., T. S. Ferguson and O. J. Vrieze, eds.
Stochastic games and related topics, Kluwer Academic Press,
Massachusetts, 1989.

[Raj.1]N. Rajan, U. R. Prasad and N. J. Rao, Pursuit-evasion of two
aircraft in a horizontal plane, Journal of Guidance and Control,
Vol. 3, No. 3, pp. 261-267, 1980.

[Raj.2]N. Rajan, U. R. Prasad and N. J. Rao, Planar pursuit-evasion
with variable speeds: barrier sections, Journal of Optimization
Theory and Applications, Vol. 33, pp. 419-432, 1981.

[Rod.1]E. Y. Rodin, A pursuit-evasion bibliography - version 2,
Computers and Mathematics with Applications, Vol. 18, No. 1-3,
pp. 245-320, 1989.

[Rub.1]G. T. Rublein, On pursuit with curvature constraints, SIAM
Journal of Control, Vol. 10, No. 1, pp. 37-39, 1972.

[Shi.1]J. Shinar, Solution techniques for realistic pursuit-evasion games,
Control and Dynamic Systems: Advanced in Theory and
Applications, C. T. Leondes, ed., Vol. 17, Academic Press, New
York, 1981.

[Sta.1]H. Stalford, Sufficient conditions for optimal control with state
and control constraints, Journal of Optimization Theory and
Applications, Vol. 7, No. 2, pp. 118-135, 1971.

[Sta.2]H. Stalford, Sufficient conditions for optimality in two-person
zero-sum differential games with state and strategy constraints,
Journal of Mathematical Analysis and Applications, Vol. 33, pp.
650-654, 1971.

[Yav.1]Y. Yavin and R. de Villiers, The game of two cars with a
containment probability as a cost function: the case of variable
speed, Computer Mathematics Applications, Vol. 18, pp. 61-67,
1989.

[Yav.2]Y. Yavin and R. de Villiers, Proportional navigation and the
game of two cars: the case of a pursuer with variable speed,
Computer and Mathematics Applications, Vol. 18, pp. 69-75,
1989.