A viability theorem for morphological inclusions

SIAM Journal on Control and Optimization

Volumn 47, Issue 3, 2008, Pages 1591-1614

  Lorenz, Thomas ^a  

^a UNIVERSITY OF HEIDELBERG   (Germany)

Author keywords

Clarke's generalized shape derivative; Morphological equations; Nagumo's theorem; Shape evolutions with state constraints; Velocity method (speed method); Viability condition

Indexed keywords

DIFFERENTIAL EQUATIONS; INCLUSIONS; OPTIMAL CONTROL SYSTEMS; POWDERS; SET THEORY; THEOREM PROVING; VECTORS;

CLARKE'S GENERALIZED SHAPE DERIVATIVE; MORPHOLOGICAL EQUATIONS; NAGUMO'S THEOREM; SHAPE EVOLUTIONS WITH STATE CONSTRAINTS; VIABILITY CONDITION;

SHAPE OPTIMIZATION;

EID: 55549094817     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/060670778     Document Type: Article

References (38)

1. Z. ARTSTEIN, A calculus for set-valued maps and set-valued evolution equations, Set-Valued Anal., 3 (1995), pp. 213-261.
2. J.-P. AUBIN, Viability Theory, Systems Control Found. Appl., Birkhäuser, Boston, 1991.
3. J.-P. AITBIN, A note on differential calculus in metric spaces and its applications to the evolution of tubes, Bull. Pol. Acad. Sci. Math., 40 (1992), pp. 151-162.
4. J.-P. AIJBIN, Mutational equations in metric spaces, Set-Valued Anal., 1 (1993), pp. 3-46.
5. J.-P. AUBIN, Mutational and Morphological Analysis: Tools for Shape Evolution and Morphogenesis, Systems Control Found. Appl., Birkhäuser, Boston, 1999.
6. J.-P. AUBIN AND A. CELLINA, Differential Inclusions, Grundlehren Math. Wiss. 264, Springer-Verlag, Berlin, 1984.
7. J.-P. AUBIN AND H. FRANKOWSKA, Set-Valued Analysis, Systems Control Found. Appl., Birkhäuser, Boston, 1990.
8. D. BÁRCENAS, Weak compactness criteria for set-valued inteqrals and Radon-Nikodym theorem for vector-valued multimeasures, Czech. Math. J., 51 (2001), pp. 493-504.
9. J. CÉA, Une méthode numérique pour la recherche d'un domaine optimal, in Computing Methods in Applied Sciences and Engineering. Part 1, R. Glowinski and J. L. Lions, eds., Lecture Notes in Econom. and Math. Systems 134, Springer-Verlag, Berlin, 1976, pp. 245-257.
10. F. H. CLARKE, Optimization and Nonsmooth Analysis, Canad. Math. Soc. Ser. Monogr. Adv. Texts, Wiley-Interscience, New York, 1983.
11. M. C. DELFOUR, AND J.-P. ZOLÉSIO, Velocity method and Lagrangian formulation for the computation of the shape Hessian, SIAM J. Control Optim., 29 (1991), pp. 1414-1442.
12. M. C. DELFOUR AND J.-P. ZOLÉSIO, Shapes and Geometries: Analysis, Differential Calculus, and Optimization, Adv. Des. Control 4, SIAM, Philadelphia, 2001.
13. 1(μ, X), Glasgow Math. J., 18 (1977), pp. 87-91.
14. 1(μ, X), Proc. Amer. Math. Soc., 118 (1993), pp. 443-453.
15. J. DIESTEL AND J. UHL, Vector Measures, Math. Surveys 15, AMS, Providence, Rl, 1977.
16. L. DOYEN, Filippov and invariance theorems for mutational inclusions of tubes, Set-Valued Anal., 1 (1993), pp. 289-303.
17. L. DOYEN, Shape Lyapunov functions and stabilization of reachable tubes of control problems, J. Math. Anal. Appl., 184 (1994), pp. 222-228.
18. L. DOYEN, Inverse function theorems and shape optimization, SIAM J. Control Optim., 32 (1994), pp. 1621-1642.
19. L. DOYEN, Mutational equations for shapes and vision-based control, J. Math. Imaging Vis., 5 (1995), pp. 99-109.
20. H. FRANKOWSKA, S. PLASKACZ, AND T. RZEŻUCHOWSKI, Measurable viability theorems and the Hamilton-Jacobi-Bellman equation, J. Differential Equations, 116 (1995), pp. 265-305.
21. A. GORRE, Evolutions of tubes under operability constraints, J. Math. Anal. Appl., 216 (1997), pp. 1-22.
22. J. W. GREEN AND F. A. VALENTINE, On the Arzela-Ascoli theorem, Math. Mag., 34 (1960/61), pp. 199-202.
23. M. KISIELEWICZ, Weak compactness in spaces C(S, X), in Transactions of the 11th Prague Conference on Information Theory, Statistical Decision Functions, and Random Processes, Prague, Czechoslovakia, Vol. B, 1992, pp. 101-106.
24. TH. LORENZ, Set-valued maps for image segmentation, Comput. Vis. Sci., 4 (2001), pp. 41-57.
25. TH. LORENZ, Reynold's transport theorem for differential inclusions, Set-Valued Anal., 14 (2006), pp. 209-247.
26. M. NAGUMO, Über die Lage der Integralkurven gewöhnlicher Differentialgleichungen, Proc. Phys.-Math. Soc. Japan (3), 24 (1942), pp. 551-559.
27. A. I. PANASYUK, Quasidifferential equations in metric spaces, Differential Equations, 21 (1985), pp. 914-921.
28. A. I. PANASYUK, Properties of solutions of a quasidifferential approximation equation and the equation of an integral funnel, Differential Equations, 28 (1992), pp. 1259-1266.
29. A. I. PANASYUK, Quasidifferential equations in a complete metric space under conditions of the Carathéodory type. I, Differential Equations, 31 (1995), pp. 901-910.
30. M. QUINCAMPOIX AND V. VELIOV, Open-loop viable control under uncertain initial state information, Set-Valued Anal., 7 (1999), pp. 55-87.
31. B. RICCERI AND A. VILLANI, Separability and Scorza-Dragoni's property, Matematiche (Catania), 37 (1983), pp. 156-161.
32. G. SCORZA-DRAGONI, Un teorema sulle funzioni continue rispetto ad una e misurabili rispetto ad un'altra variabile, Rend. Sem. Mat. Univ. Padova, 17 (1948), pp. 102-108.
33. J. SOKOLOWSKI AND J.-P. ZOLÉSIO, Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer Ser. Comput. Math. 16, Springer-Verlag, Berlin, 1992.
34. 1(μ, X), Proc. Amer. Math. Soc., 113 (1991), pp. 143-149.
35. R. VINTER, Optimal Control, Systems Control Found. Appl., Birkhäuser, Boston, 2000.
36. K. YOSIDA, Functional Analysis, 5th ed., Grundlehren Math. Wiss. 123, Springer-Verlag, Berlin, 1978.
37. W. ZIEMER, Weakly Differentiable Functions, Grad. Texts in Math. 120, Springer-Verlag, New York, 1989.
38. J.-P. ZOLÉSIO, Identification de domaine par déformations, Thèse de doctorat d'état, Université de Nice, Nice, France, 1979.