Nonlinear optimal control theory

Nonlinear Optimal Control Theory

Volumn , Issue , 2012, Pages 1-379

  Berkovitz, Leonard D ^a   Medhin, Negash G ^b  

^a PURDUE UNIVERSITY   (United States)

^b NORTH CAROLINA STATE UNIVERSITY   (United States)

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EID: 85055378544     PISSN: None     EISSN: None     Source Type: Book    
DOI: None     Document Type: Book

References (94)

1. N. U. Ahmed, Properties of relaxed trajectories for a class of nonlinear evolution equations on Banach space, SIAM J. Control and Optimization, 21 (6), (1983), 953-967.
2. K. J. Arrow, S. Karlin, and H. Scarf, Studies in the Mathematical Theory of Inventory and Production, Stanford University Press, Stanford, CA, 1958.
3. M. Athans and P. Falb, Optimal Control, McGraw-Hill, New York, 1966.
4. Jean-Pierre Aubin, Optima and Equilibria, 2nd ed., Springer, Berlin, Heidelberg, New York, 1998, Corrected second printing 2003.
5. M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhauser, Boston, 1997.
6. R. Bellman, I. Glicksberg, and O. Gross, On the “bang-bang” control problem, Quart. Appl. Math., 14 (1956), 11-18.
7. L. D. Berkovitz, Variational methods in problems of control and programming, J. Math. Anal. Appl., 3 (1961), 145-169.
8. L. D. Berkovitz, Necessary conditions for optimal strategies in a class of differential games and control problems, SIAM J. Control 5 (1967), 1-24.
9. L. D. Berkovitz, An existence theorem for optimal controls, J. Optimization Theory Appl., 6 (1969), 77-86.
10. L. D. Berkovitz, Existence theorems in problems of optimal control, Studia Math., 44 (1972), 275-285.
11. L. D. Berkovitz, Existence theorems in problems of optimal control without property (Q), in Techniques of Optimization, A. V. Balakrishnan Ed., Academic Press, New York and London, 1972, 197-209.
12. L. D. Berkovitz, Optimal Control Theory, Springer-Verlag, New York, 1974.
13. L. D. Berkovitz, A penalty function proof of the maximum principle, Applied Mathematics and Optimization, Vol. 2, (1976), 291-303.
14. L. D. Berkovitz, Optimal feedback controls, SIAM J. Control and Optimization, 27 (5), (1989), 991-1006.
15. G. A. Bliss, The problem of Lagrange in the calculus of variations. Am. J. Math., 52 (1930), 673-741.
16. G. A. Bliss, Lectures on the Calculus of Variations, The University of Chicago Press, Chicago, 1946.
17. V. G. Boltyanskii, R. V. Gamkrelidze, and L. S. Pontryagin, The theory of optimal processes I, the maximum principle. Izv. Akad. Nauk SSSR. Ser Mat., 24 (1960), 3-42. English translation in Am. Math. Soc. Transl. Ser., 2, 18 (1961), 341-382.
18. O. Bolza, Vorlesungen uber Variationsrechnung, Reprint of 1909 edition, Chelsea Publishing Co., New York.
19. A. E. Bryson and Y. C. Ho, Applied Optimal Control, Waltham, Toronto, London, 1969.
20. Z. Brzezniak and R. Serrano, Optimal relaxed control of dissipative stochastic partial differential equations in Banach spaces, arXiv: 1001.3165v2 (2010).
21. D. Bushaw, Optimal discontinuous forcing terms, Contributions to the Theory of Nonlinear Oscillations IV, Annals of Math Study 41, S. Lefschetz, Ed., Princeton University Press, Princeton, 1958, 29-52.
22. C. Casting, Sur les multi-applications measurables, Rev. Francaise Automat. Informat. Recherche Operationnelle 1 (1967), 91-126.
23. L. Cesari, Existence theorems for optimal solutions in Pontryagin and Lagrange problems, SIAM J. Control, 3 (1966), 475-498
24. L. Cesari, Existence theorems for weak and usual optimal solutions in Lagrange problems with unilateral constraints I, Trans. Am. Math. Soc., 124 (1966), 369-412.
25. L. Cesari, Existence theorems for optimal controls of the Mayer type, SIAM J. Control, 9(1968), 517-552.
26. L. Cesari, Closure, lower closure, and semicontinuity theorems in optimal control, SIAM J. Control, 9(1971), 287-315.
27. L. Cesari, Optimization—Theory and Applications, Vol. 17 of Applications of Mathematics, Springer-Verlag, New York, 1983.
28. M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Ams. 277 (1983), 1-42.
29. M. G. Crandall, H. Ishii, and P.-L. Lions, Users guide to viscosity solutions of second order partial differential equations, Bull. Am. Math. Soc., 279, 1-67.
30. B. Dacorogna, Direct Methods in the Calculus of Variations, Springer-Verlag, Berlin, New York, 1989.
31. N. Dunford and J. T. Schwartz, Linear Operators Part I: General Theory, Interscience, New York, 1958.
32. H. G. Eggleston, Convexity, Cambridge University Press, Cambridge, 1958.
33. A. F. Filippov, On certain questions in the theory of optimal control, SIAM J. Control, 1 (1962), 76-89. Orig. Russ. Article in Vestnik Moskov. Univ. Ser. Mat. Mech. Astr., 2 (1959), 25-32.
34. W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal, Springer-Verlag, New York, 1975.
35. W. H. Fleming and H. Mete Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993, Second Edition 2006.
36. R. V. Gamkrelidze, Theory of time-optimal process for linear systems, Izv. Akad. Nauk. SSSR. Ser Mat., 22 (1958), 449-474 (Russian).
37. R. V. Gamkrelidze, On sliding optimal regimes, Dokl. Akad. Nauk SSSR. 143 (1962), 1243-1245. Translated as Soviet Math. Dokl., 3 (1962), 1243-1245. 3 (1962), 390-395.
38. R. V. Gamkrelidze, On some extremal problems in the theory of differential equations with applications to the theory of optimal control, SIAM J. Control, 3 (1965), 106-128.
39. R. V. Gamkrelidze, Principles of Optimal Control Theory, Plenum, New York, 1978.
40. M. E. Gurtin and L. F. Murphy, On the optimal harvesting of persistent age-structured populations, J. Math. Biol., 13 (2) (1981), 131-148.
41. J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.
42. H. Hermes and J. P. LaSalle, Functional Analysis and Time Optimal Control, Academic Press, New York, 1969.
43. M. R. Hestenes, Calculus of Variations and Optimal Control Theory, John Wiley, New York, 1966.
44. E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Revised Ed., American Mathematical Society, Providence, RI, 1957.
45. J. G. Hocking and G. S. Young, Topology, Addison-Wesley, Reading, MA, 1961.
46. M. Q. Jacobs, Attainable sets in systems with unbounded controls, J. Differential Equations, 4 (1968), 408-423.
47. G. S. Jones and A. Strauss, An example of optimal control, SIAM Review, 10 (1) (1968), 25-55.
48. J. P. LaSalle, Study of the Basic Principle Underlying the Bang-Bang Servo, Goodyear Aircrafts Corp. Report GER-5518 (July 1953). Abstract 247t. Bull. Am. Math. Soc., 60 (1954), 154.
49. J. P. LaSalle, The time optimal control problem, Contributions to the Theory of Nonlinear Oscillations, Vol. 5, Annals of Math Study No. 45 Princeton University Press, Princeton, 1960, 1-24.
50. E. B. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley, New York, 1967.
51. G. Leitmann, On a class of variational problems in rocket flight, J. Aero/Space Sci., 26 (1959), 586-591.
52. G. Leitman, An Introduction to Optimal Control, McGraw-Hill, New York, 1966.
53. J. Lindenstrauss, A short proof of Liapounoff’s convexity theorem, J. Math. Mech., 15 (1966), 971-972.
54. H. Lou, Analysis of the optimal relaxed control to an optimal control problem, Appl. Math. Optim., 59 (2009), 75-97.
55. H. Lou, Existence and non-existence results of an optimal control problem by using relaxed control, SIAM J. Control Optim., 46 (2007) 923-1941.
56. H. Maurer and M. D. Mittelmann, The non-linear beam via optimal control with bounded state vriables, Optimal Control Applications and Methods, 12 (1991), 19-31.
57. D. McDonald, Nonlinear techniques for improving servo performance, National Electronics Conference, 6 (1950), 400-421.
58. E. J. McShane, Integration, Princeton University Press, Princeton, 1944.
59. E. J. McShane, On multipliers for Lagrange problems, Am. J. Math., 61 (1939), 809-819.
60. E. J. McShane, Necessary conditions in generalized-curve problems in the calculus of variations, Duke Math. J., 6 (1940), 1-27.
61. E. J. McShane, Existence theorems for Bolza problems in the calculus of variations, Duke Math. J., 7 (1940), 28-61.
62. E. J. McShane, Generalized curves, Duke Math. J., 6 (1940), 513-536.
63. E. J. McShane, Relaxed controls and variational problems, SIAM J. Control, 5 (1967), 438-485.
64. E. J. McShane and R. B. Warfield Jr., On Filippov’s implicit functions lemma, Proc. Am. Math. Soc., 18 (1967), 41-47.
65. N. G. Medhin, Optimal processes governed by integral equations, J. Math. Anal. App., 120 (1) (1986), 1-12.
66. N. G. Medhin, Necessary conditions for optimal control problems with bounded state by a penalty method, JOTA, 52 (1) (1987), 97-110.
67. N. G. Medhin, Optimal processes governed by integral equations with unilateral constraint, J. Math. Anal. App., 129 (1) (1988), 269-283.
68. N. G. Medhin, Optimal harvesting in age-structured populations, JOTA, 74 (3) (1992), 413-423.
69. N. G. Medhin, Characterization of optimal pairs for hereditary control problems, JOTA, 75 (2) (1992), 355-367.
70. N. G. Medhin, On optimal control of functional differential systems, JOTA, 85 (2) (1995), 363-376.
71. N. G. Medhin, Bounded state problem for hereditary control problems, JOTA, 79 (1) (1993), 87-103.
72. S. Mirica, On the admissible synthesis in optimal control theory and differential games, SIAM J. Control, 7 (1969), 292-316.
73. B. Mordukhovich, Variational analysis of evolution inclusions, SIAM J. Optim., 18 (3) (2007), 752-777.
74. I. P. Natanson, Theory of Functions of a Real Variable, Eng. trans. by Leo Boron, revised ed., F. Ungar, New York, 1961.
75. L. W. Neustadt, The existence of optimal controls in the absence of convexity conditions, J. Math. Anal. Appl., 7 (1963), 110-117.
76. L. W. Neustadt, Optimization: A Theory of Necessary Conditions, Princeton University Press, Princeton, NJ, 1977.
77. C. Olech, Extremal solutions of a control system, J. Differential Equations, 2 (1966), 74-101.
78. L. A. Pontryagin, V. G. Boltyanskii, R.V. Gamkrelidze, and E. F. Mischenko, The Mathematical Theory of Optimal Processes (Translated by K. N. Trirogoff, L. W. Neustadt), John Wiley, Ed., New York, 1962.
79. L. S. Pontryagin, Optimal regulation processes, Uspehi Mat. Nauk (N.S.) 14 (1959), 3-20. English translation in Am. Math. Soc. Transl. Ser., 2 18(1961), 321-339.
80. R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
81. E. Roxin, The existence of optimal controls, Mich. Math. J., 9 (1962), 109-119.
82. W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966.
83. P. Sattayatham, Relaxed control for a class of strongly nonlinear impulsive evolution equations, Comp. Math. Appl., 52 (2006), 779-790.
84. G. Scorza-Dragoni, Un teorema sulle funzioni continue rispetto ad una a misurabile rispetto ad un’altra variable, Rend. Sem. Mat. Univ. Padova, 17 (1948), 102-106.
85. L. M. Sonneborn and F. S. Van Vleck, The bang-bang principle for linear control systems, SIAM J. Control, 2 (1964), 151-159.
86. L. Tonelli, Fondamenti del calcolo delle variazioni I, II, Zanichelli, Bologna, 1921, 1923.
87. John L. Troutman, Variational Calculus and Optimal Control, Springer, New York, 1996.
88. J. Warga, Relaxed variational problems, J. Math. Anal. Appl., 4 (1962), 111-128.
89. J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.
90. X. Xiang, W. Wei, and H. Liu, Relaxed trajectories of integrodifferential equations and optimal control on Banach Space, Comp. Math. Appl., 52 (2006), 735-748.
91. X. Xiang, P. Sattayatham, and W. Wei, Relaxed controls for a class of strongly nonlinear delay evolution equations, Nonlinear Analysis, 52 (2003), 703-723.
92. L. C. Young, Generalize curves and the existence of an attained absolute minimum in the calculus of variations, Compt. Rend. Soc. Sci. et Lettres., Varsovie, Cl III 30 (1937), 212-234.
93. L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, W. B. Saunders Co., Philadelphia, 1969.
94. R. Yue, Lipschitz Continuity of the Value Function in a Class of Optimal Control Problems Governed by Orindary Differential Equations, Control Theory, Stochastic Analysis, and Applications, World Scientific Publishing, New Jersey, 1991, 125-136.