Chapter 3 Differential Equations on Closed Sets

Handbook of Differential Equations: Ordinary Differential Equations

Volumn 2, Issue , 2006, Pages 147-238

  Crjǎ, Ovidiu ^a  

^a Faculty of Mathematics   (Romania)

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[No Author keywords available]

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EID: 66149134013     PISSN: 18745725     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1874-5725(05)80005-7     Document Type: Article

References (112)

1. Aubin J.P. Viability Theory (1991), Birkhäuser, Boston
2. Aubin J.P. Boundary-value problems for systems of Hamilton-Jacobi-Bellman inclusions with constraints. SIAM J. Control Optim. 41 (2002) 56-425
3. Aubin J.P., and Cellina A. Differential Inclusions (1984), Springer-Verlag, Boston
4. Aubin J.P., and Frankowska H. Set-Valued Analysis (1990), Birkhäuser, Berlin
5. Barbu V., and Pavel N.H. Flow-invariant closed sets with respect to nonlinear semigroup flows. NoDEA Nonlinear Differential Equations Appl. 10 (2003) 57-72
6. Bebernes W., and Schuur I.D. The Wazewski topological method for contingent equations. Ann. Mat. Pura Appl. 87 (1970) 271-278
7. Bellman R. Bottleneck problems and dynamic programming. Proc. Natl. Acad. Sci. USA 39 (1953) 947-951
8. Bellman R. Bottleneck problems, functional equations and dynamic programming. Econometrica 23 (1955) 73-87
9. Bellman R. Dynamic Programming (1957), Princeton Univ. Press
10. Berliocchi H., and Lasry J.M. Intégrandes normales et mesures paramétrées en calcul des variations. Bull. Soc. Math. France 101 (1973) 129-184
11. in Russian. Boltyanskii V.G. Sufficient optimality conditions. Dokl. Akad. Nauk SSSR 140 (1961) 994-997
12. in Russian. Boltyanskii V.G. Sufficient conditions for optimality and the justification of the programming method. Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964) 481-514
13. Bony J.M. Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier (Grenoble) 19 (1969) 277-304
14. Bothe D. Flow invariance for perturbed nonlinear evolution equations. Abstr. Appl. Anal. 1 (1996) 417-433
15. Bothe D. Reaction diffusion systems with discontinuities. A viability approach. Nonlinear Anal. 30 (1997) 677-686
16. Bouligand H. Sur les surfaces dépourvues de points hyperlimités. Ann. Soc. Polon. Math. 9 (1930) 32-41
17. Bressan A., and Staicu V. On nonconvex perturbations of maximal monotone differential inclusions. Set-Valued Anal. 2 (1994) 415-437
18. Brezis H. On a characterization of flow-invariant sets. Comm. Pure Appl. Math. 23 (1970) 261-263
19. Brezis H., and Browder F. A general principle on ordered sets in nonlinear functional analysis. Adv. Math. 21 (1976) 355-364
20. Cârjǎ O. Unele metode de analizǎ funcţnalǎ neliniarǎ (2003), MATRIXROM, Princeton, NJ
21. Cârjǎ O., and Monteiro Marques M.D.P. Viability for nonautonomous semilinear differential equations. J. Differential Equations 166 (2000) 328-346
22. Cârjǎ O., and Monteiro Marques M.D.P. Viability results for nonautonomous differential inclusions. J. Convex Anal. 7 (2000) 437-443
23. Cârjǎ O., and Monteiro Marques M.D.P. Weak tangency, weak invariance and Carathédory mappings. J. Dynam. Control Systems 8 (2002) 445-461
24. Cârjǎ O., Necula M., and Vrabie I.I. Local invariance via comparison functions. Electron. J. Differential Equations 2004 50 (2004) 1-14
25. Cârjǎ O., and Ursescu C. The characteristics method for a first order partial differential equation. An. Ştiin. Univ. Al. I. Cuza Iaşi, Secţ. I a Mat. 39 (1993) 367-396
26. Cârjǎ O., and Ursescu C. Viscosity solutions and partial differential inequalities. Evolution Equations, Control Theory, and Biomathematics. Lecture Notes in Pure and Appl. Math. Vol. 155 (1994), Marcel Dekker, Bucureşti 39-44
27. Cârjǎ O., and Vrabie I.I. Some new viability results for semilinear differential inclusions. NoDEA Nonlinear Differential Equations Appl. 4 (1997) 24-401
28. Cârjǎ O., and Vrabie I.I. Viability results for nonlinear perturbed differential inclusions. Panamer. Math. J. 9 (1999) 63-74
29. Carathéodory C. Variationsrechnung und partielle Differentialgleichungen erster Ordnung (1935), Teubner, New York
30. Cesari L. Optimization-Theory and Applications (1983), Springer, Leipzig-Berlin
31. Chiş-Şter I. Existence of monotone solutions for semilinear differential inclusions. NoDEA Nonlinear Differential Equations Appl. 6 (1999) 63-78
32. Clarke F.H. Generalized gradients and applications. Trans. Amer. Math. Soc. 205 (1975) 247-262
33. Clarke F.H., Ledyaev Yu.S., and Stern R.J. Invariance, monotonicity and applications. In: Clarke F.H., and Stern R.J. (Eds). Nonlinear Analysis, Differential Equations and Control (1999), Kluwer Academic Publishers, New York 207-305
34. Clarke F.H., Ledyaev Yu.S., Stern R.J., and Wolenski P.R. Nonsmooth Analysis and Control Theory (1998), Springer-Verlag, Dordrecht
35. Clarke F.H., and Vinter R.B. Local optimality conditions and lipschitzean solutions to the Hamilton-Jacobi equation. SI AM J. Control Optim. 21 (1983) 856-870
36. Cornet B. Regularity properties of open tangent cones. Nonlinear Analysis and Optimization, Louvain-la-Neuve, 1983, Math. Programming Stud. 30 (1987) 17-33
37. Courant R., and Hilbert D. Methoden der Mathematischen Physik (1937), Akademische Verlagsgesellschaft, Berlin
38. Crandall M.G. A generalization of Peano's existence theorem and flow-invariance. Proc. Amer. Math. Soc. 36 (1972) 151-155
39. Crandall M.G., and Lions PL. Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983) 1-42
40. Part I. Dunford N., and Schwartz J.T. Linear Operators. General Theory (1958), Interscience Publishers, Berlin
41. Federer H. Curvature measures. Trans. Amer. Math. Soc. 93 (1959) 418-491
42. Feferman S. Independence of the axiom of choice from the axiom of independent choices. J. Symbolic Logic 29 (1967) 226
43. Filippov A.F. Classical solutions of differential equations with multi-valued right-hand side. SIAM J. Control 5 (1967) 609-621
44. Frankowska H. Equations d'Hamilton-Jacobi contingentes. C. R. Acad. Sci. Paris Sér. I Math. 304 (1987) 295-298
45. Frankowska H., Plaskacz S., and Rzezuchowski T. Measurable viability theorems and Hamilton-Jacobi-Bellman equations. J. Differential Equations 116 (1995) 265-305
46. Fryszkowski A. Continuous selections for a class of non-convex multivalued maps. Studia Math. 76 (1983) 163-174
47. Fukuhara M. Sur les systèmes des équations différentielles ordinaires II. Japan. J. Math. 6 (1929) 269-299
48. Gautier S. Equations differentielles multivoques sur unfermé (1973), Publications de l'Université de Pau, New York
49. Girsanov I.V. Lekciipo Matematičeskoi Teorii Èkstremal 'nyh Zada (1970), Moscow University
50. English translation. Lecture on Mathematical Theory of Extremum Problems. Lectures Notes in Econom. and Math. Systems Vol. 67 (1972), Springer-Verlag, Moscow
51. Gonzales R. Sur I'existence d'une solution maximale de Véquation de Hamilton-Jacobi. C. R. Acad. Sci. Paris Sér. I Math. 282 (1976) 1287-1290
52. Goursat E. Lecons sur l'Integration des Équation Partielles du Premier Ordre (1891), Hermann, New York
53. Haddad G. Monotone trajectories of differential inclusions and functional differential inclusions with memory. Israel J. Math. 39 (1981) 83-100
54. Hájek O. On differentiability of the minimal time function. Funkcial.Ekvac. 20 (1977) 97-114
55. Hartman P. Ordinary Differential Equations (1964), Wiley, Paris
56. Hartman P. On invariant sets and on a theorem of Ważewski. Proc. Amer. Math. Soc. 32 (1972) 511-520
57. Hobson E.W. The Theory of Functions of a Real Variable and the Theory of Fourier's Series. third (1927), Cambridge Univ. Press, New York
58. Howard P., and Rubin J.E. The Boolean prime ideal theorem plus countable choice do [does] not imply dependent choice. Math. Logic Quart. 42 (1996) 20-410
59. Ishii H. A simple, direct proof of uniqueness for solutions of the Hamilton-Jacobi equations of eikonal type. Proc. Amer. Math. Soc. 100 (1987) 247-251
60. Kamke E. Dijferentialgleichungen reeler Funktionen (1930), Akademische Verlagsgesellschaft, Cambridge
61. Kenmochi N., and Takahashi T. Nonautonomous differential equations in Banach spaces. Nonlinear Anal. 4 (1980) 1109-1121
62. Kneser H. Siz.-Ber. d. Preuss Akademie d. Wissensch. Phys.-Math. Klasse (1923) 171-174
63. Kobayashi J. Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups. J. Math. Soc. Japan 27 (1975) 640-665
64. Kucia A. Scorza Dragoni type theorems. Fund. Math. 138 (1991) 197-203
65. Ledyaev Yu. Criteria for viability of trajectories of nonautonomous inclusions and their applications. J. Math. Anal. Appl. 182 (1994) 165-188
66. Lupulescu V., and Necula M. Viability and local invariance for non-convex semilinear differential inclusions. Nonlinear Funct. Anal. Appl. 9 (2004) 495-512
67. Martin Jr. R.H. Differential equations on closed subsets of a Banach space. Trans. Amer. Math. Soc. 179 (1973) 399-414
68. Michael E. Continuous selectors I. Ann. of Math. 63 (1956) 361-382
69. Michael E. Continuous selectors II. Ann. of Math. 64 (1956) 62-580
70. Mordukhovich B.S. Maximum principle in the problem of time optimal control with nonsmooth constraints. J. Appl. Math. Mech. 40 (1976) 960-969
71. Nagumo M. Über die Lage der Integralkurven gewönlicher Differentialgleichungen. Proc. Phys. Math. Soc. Japan 24 (1942) 551-559
72. Necula M. Viability of variable domains for nonautonomous semilinear differential equations. Panamer. Math. J. 11 (2001) 65-80
73. Necula M. Viability of variable domains for differential equations governed by Caratheodory perturbations of nonlinear m-accretive operators. An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Secţ. I a Mat. 48 (2002) 41-60
74. M. Necula, Personal communication
75. M. Necula and I.I. Vrabie, Paper in preparation
76. Pavel N.H. Invariant sets for a class of semi-linear equations of evolution. Nonlinear Anal. 1 (1977) 187-196
77. Pavel N.H. Differential Equations, Flow Invariance and Applications. Res. Notes Math. Vol. 113 (1984), Pitman, Leipzig
78. Pavel N.H., and Ursescu C. Flow-invariant sets for second order differential equations and applications in Mechanics. Nonlinear Anal. 6 (1982) 35-74
79. Pavel N.H., and Vrabie I.I. Equations dévolution multivoques dans des espaces de Banach. C. R. Acad. Sci. Paris Sér. I Math. 287 (1978) 315-317
80. Pavel N.H., and Vrabie I.I. Semilinear evolution equations with multivalued right-hand side in Banach spaces. An. Ştiinţ. Univ. Al. I. Cuza Iaşi Secţ. I a Mat. 25 (1979) 137-157
81. Peano G. Démonstration de Vintégrabilité des équations différentielles ordinaires. Math. Ann. 37 (1890) 182-228
82. Perron O. Partieller Differentialgleichungssysteme im reellen Gebiet. Math. Z. 27 (1928) 549-564
83. Plaskacz S. On the solution set for differential inclusions. Boll. Unione Mat. Ital. Sez. A 6 7 (1992) 387-394
84. in Russian. Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., and Mishchenko E.F. The Mathematical Theory of Optimal Processes (1961), GIF-ML, Boston
85. Quincampoix M. Differential inclusions and target problems. SIAM J. Control Optim. 30 (1992) 324-335
86. Redheffer R. The theorem of Bony and Brezis on flow-invariant sets. Amer. Math. Monthly 79 (1972) 740-747
87. Roxin E. Stability in general control systems. J. Differential Equations 1 (1965) 115-150
88. Scorza Dragoni G. Un teorema sulle funzioni continue rispetto ad una e misurabili rispetto ad un'altra variabile. Rend. Sem. Mat. Univ. Padova 17 (1948) 102-106
89. Scorza Dragoni G. Una applicazione della quasicontinuita semiregolare delle funzioni misurabili rispetto ad una e continue rispetto ad un'altra variabile. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 12 8 (1952) 55-61
90. Severi F. Su alcune questioni di topologia infinitesimale. Ann. Polon. Soc. Math. 9 (1930) 97-108
91. Severi F. Sulla differenziabilità totale delle funzioni di più variabili reali. Ann. Mat. Pura Appl. 13 (1936) 1-35
92. Shi S.Z. Viability theorems for a class of differential operator inclusions. J. Differential Equations 79 (1989) 232-257
93. Tallos P. Viability problems for nonautonomous differential inclusions. SI AM J. Control Optim. 29 (1991) 253-263
94. Treiman J.S. Characterization of Clarke's tangent and normal cones in finite and infinite dimensions. Nonlinear Anal. T.M.A. 7 (1983) 771-783
95. Ursescu C. Sur une généralisation de la notion de différentiabilité. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 54 7 (1973) 199-204
96. 1976. Ursescu C. Tangent sets and differentiate functions. Math. Control Theory Proc. Conf. Zakopane (1974), Banach Center Publications, Moscow 151-155
97. Grant 11996/25.09.1980, Iaşi. Ursescu C. Probleme de tangenţǎ cu aplicaţii n teoria optimizǎrii. Internal Report (1981)
98. Ursescu C. Tangent sets' calculus and necessary conditions for extremality. SIAM J. Control Optim. 20 (1982) 563-574
99. Ursescu C. Carathéodory solutions of ordinary differential equations on locally closed sets infinite dimensional spaces. Math. Japon. 31 (1986) 483-491
100. 0-semigroups, characteristics method, and generic differential equations. An. Univ. Timişoara Ser. Mat-Inform. 34 (1996) 283-290
101. Ursescu C. Evolutors and invariance. Set-Valued Anal. 11 (2003) 69-89
102. C. Ursescu, Personal communication
103. Vrabie I.I. Compactness Methods for Nonlinear Evolutions, second edition. Pitman Monographs Surveys Pure Appl. Math. Vol. 75 (1995), Longman
104. Vrabie I.I. Compactness methods and flow-invariance for perturbed nonlinear semigroups. An. Ştiinţ. Univ. Al. I. Cuza Iaşi Secţ. I a Mat. 27 (1981) 117-125
105. Vrabie I.I. Periodic solutions for nonlinear evolution equations in a Banach space. Proc. Amer. Math. Soc. 109 (1990) 653-661
106. 0-Semigroups and Applications. North-Holland Math. Stud. Vol. 191 (2003)
107. Vrabie I.I. Differential Equations. An Introduction to Basic Results, Concepts and Applications (2004), World Scientific, Harlow
108. Ważewski T. Sur un condition équivalent à Véquation au contingent. Bull. Polon. Acad. Sci. Ser. Sci. Math. Astr. Phys. 9 (1961) 865-867
109. Yorke J.A. Invariance for ordinary differential equations. Math. Systems Theory 1 (1967) 353-372
110. Yorke J.A. Invariance for ordinary differential equations: Correction. Math. Systems Theory 2 (1968) 381
111. Yoshizawa T. Stability theory by Liapunov's second method (1966), The Mathematical Society of Japan, New Jersey
112. Zaremba S.C. Sur les équations auparatingent. Bull. Sci. Math. 60 2 (1936) 139-160