Set-Valued Analysis

, Volume 3, Issue 3, pp 213–261 | Cite as

A calculus for set-valued maps and set-valued evolution equations

  • Zvi Artstein


A definition of differentiability of a set-valued map is offered. As derivatives, which are called directives in the set-valued setting, unions of affine maps are used; these are called multiaffines. A multiaffine is a directive if it is a first-order approximation of the set-valued map. One application is a necessary condition for maximin optimality of constrained decisions. A distance among multiaffines permits the development of set-valued evolution equations along the lines of ordinary differential equations in a vector space. The theory is displayed along with some comments on applications.

Mathematics Subject Classifications (1991)

26E25 34G20 54C60 58C06 

Key words

set-valued maps set-valued calculus set-valued evolution equations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arahovitis, J.: Multivalued linear mappings and matrices,Math. Balkanica 3 (1973), 3–8.Google Scholar
  2. 2.
    Artstein, Z.: On the calculus of set-valued functions,Indiana Univ. Math. J. 24 (1974), 433–441.Google Scholar
  3. 3.
    Artstein, Z.: Piecewise linear approximations of set-valued maps,J. Approx. Theory 56 (1989), 41–45.Google Scholar
  4. 4.
    Artstein, Z.: First order approximations for differential exclusions,Set-Valued Anal. 2 (1994), 7–17.Google Scholar
  5. 5.
    Aubin, J.-P.: Contingent derivatives of set-valued maps and existence of solutions to non-linear inclusions and differential inclusions, in L. Nachbin (ed.),Mathematical Analysis and Applications, Academic Press, New York, 1981, pp. 159–229.Google Scholar
  6. 6.
    Aubin, J.-P.: Mutational equations in metric spaces,Set-Valued Anal. 1 (1993), 3–46.Google Scholar
  7. 7.
    Aubin, J.-P. and Cellina, A.:Differential Inclusions, Springer-Verlag, Berlin, 1984.Google Scholar
  8. 8.
    Aubin, J.-P. and Frankowska, H.:Set-Valued Analysis, Birkhäuser, Basel, 1990.Google Scholar
  9. 9.
    Aumann, R. J.: Integrals of set-valued functions,J. Math. Anal. Appl. 12 (1965), 1–12.Google Scholar
  10. 10.
    Banks, H. T. and Jacobs, M. Q.: A differential calculus for multifunctions,J. Math. Anal. Appl. 29 (1970), 246–272.Google Scholar
  11. 11.
    Bhatia, N. P. and Szegö, G. P.:Stability Theory of Dynamical Systems, Springer-Verlag, Berlin, 1970.Google Scholar
  12. 12.
    Brandao Lopes Pinto, A. J., De Blasi, F. S. and Iervolino, F.: Uniqueness and existence theorems for differential equations with compact convex valued solutions,Boll. Un. Mat. Ital. 4 (1971), 47–54.Google Scholar
  13. 13.
    Bridgland, T. F.: Trajectory integrals of set-valued functions,Pacific J. Math. 33 (1970), 43–67.Google Scholar
  14. 14.
    Burden, R. L., Faires, J. D. and Reynolds, A. C.:Numerical Analysis, Pindle, Weber & Schmidt, Boston, 1978.Google Scholar
  15. 15.
    Bushaw, D.: Dynamical polysystems and optimization,Contrib. Differential Equations 2 (1963), 351–365.Google Scholar
  16. 16.
    De Blasi, F. S.: On the differentiability of multifunctions,Pacific J. Math. 66 (1976), 67–81.Google Scholar
  17. 17.
    De Blasi, F. S. and Iervolino, F.: Euler method for differential equations with set-valued solutions,Boll. Un. Mat. Ital. 4 (1971), 941–949.Google Scholar
  18. 18.
    Debreu, G.: Integration of correspondences, inProc. 5th Berkeley Symp. Math. Stat. and Prob. Vol. II, University of California Press, 1967, pp. 351–372.Google Scholar
  19. 19.
    Deimling, K.:Multivalued Differential Equations, De Gruyter, Berlin, 1992.Google Scholar
  20. 20.
    Delfour, M. C. and Zolesio, J.-P.: Shape sensitivity analysis via min max differentiability,SIAM J. Control Optim. 26 (1988), 834–862.Google Scholar
  21. 21.
    Delfour, M. C. and Zolesio, J.-P.: Velocity method and Lagrangian formulation for the computation of the shape Hessian,SIAM J. Control Optim. 29 (1991), 1414–1442.Google Scholar
  22. 22.
    Deutsch, F. and Singer, I.: On single-boundedness of convex set-valued maps,Set-Valued Anal. 1 (1993), 97–103.Google Scholar
  23. 23.
    Donchev, A. and Lempio, F.: Difference methods for differential inclusions: A survey,SIAM Rev. 34 (1992), 263–294.Google Scholar
  24. 24.
    Doyen, L.: Filippov and invariance theorems for mutational inclusions of tubes,Set-Valued Anal. 1 (1993), 289–303.Google Scholar
  25. 25.
    Doyen, L.: Shape Lyapunov functions and stabilization of reachable tubes of control problems,J. Math. Anal. Appl. 184 (1994), 222–228.Google Scholar
  26. 26.
    Doyen, L.: Inverse function theorems and shape optimization,SIAM J. Control Optim. 32 (1994), 1621–1642.Google Scholar
  27. 27.
    Fleming, W. H.:Functions of Several Variables, Addison-Wesley, Reading, MA, 1965.Google Scholar
  28. 28.
    Frankowska, H.: Local controllability and infinitesimal generators of semigroups of set-valued maps,SIAM J. Control Optim. 25 (1987), 412–432.Google Scholar
  29. 29.
    Frankowska, H.: Some inverse mapping theorems,Ann. Inst. Henri Poincaré, Analyse non linéaire 7 (1990), 183–234.Google Scholar
  30. 30.
    Hermes, H.: Calculus of set-valued functions and control,J. Math. Mech. 18 (1968), 47–60.Google Scholar
  31. 31.
    Hukuhara, M.: Integration des applications mesurable dont la valeur est compact convexe,Funk. Eku. 10 (1967), 205–223.Google Scholar
  32. 32.
    Klein, E. and Thompson, A. C.:Theory of Correspondences, Wiley, New York, 1984.Google Scholar
  33. 33.
    Kurzhanski, A. B. and Filippova, T. F.: On the set-valued calculus in problems of viability and control for dynamic processes: The evolution equation, in Attouchet. al. (eds),Analyse non linéaire, Gauthier-Villars, Paris, 1989, pp. 339–363.Google Scholar
  34. 34.
    Kurzhanski, A. B. and Nikonov, O. I.: On the problem of synthesizing control strategies. Evolution equations and multivalued integration,Soviet Math. Dokl. 41 (1990), 300–305.Google Scholar
  35. 35.
    Lemaréchal, C. and Zowe, J.: The eclipsing concept to approximate a multi-valued mapping,Optimization 22 (1991), 3–37.Google Scholar
  36. 36.
    Lin, Y., Sontag, E. D. and Wang, Y.: A smooth converse Lyapunov theorem for robust stability,SIAM J. Control Optim,, to appear (preliminary version entitled ‘Recent results on Lyapunov-theoretic techniques for nonlinear stability’, Proc. Amer. Automatic Control Conf., Baltimore, 1994, pp. 1771–1775).Google Scholar
  37. 37.
    Lintz, R. G. and Buonomano, V.: The concept of differential equations in topological spaces and generalized mechanics,J. Reine Angew. Math. 265 (1974), 31–70.Google Scholar
  38. 38.
    Martelli, M. and Vignoli, A.: On differentiability of multi-valued maps,Boll. Un. Mat. Ital. 10 (1974), 701–712.Google Scholar
  39. 39.
    Matheron, G.:Random Sets and Integral Geometry, Wiley, New York, 1975.Google Scholar
  40. 40.
    Mordukhovich, B.: Sensitivity analysis for constraint and variational systems by means of set-valued differentiation,Optimization, to appear.Google Scholar
  41. 41.
    Nikolski, M. S.: Local approximation of first order to set-valued mappings, in: A. Kurzhanski and V. Veliov (eds),Set-Valued Analysis and Differential Inclusions, Birkhäuser, Boston, to appear.Google Scholar
  42. 42.
    Panasyuk, A. I.: Differential equations in metric spaces,Differential Equations 21 (1985), 914–921.Google Scholar
  43. 43.
    Panasyuk, A. I.: Equations of attainable set dynamics, Part I: Integral funnel equations,J. Optim. Theory Appl. 64 (1990), 349–366.Google Scholar
  44. 44.
    Panasyuk, A. I.: Quasidifferential approximation equations in metric spaces under Caratheodory-type conditions,Differential Equations 28 (1992), 1073–1083.Google Scholar
  45. 45.
    Penot, J.-P.: Differentiability of relations and differential stability of perturbed optimization problems,SIAM J. Control Optim. 22 (1984), 529–551; erratum:26 (1988), 997–998.Google Scholar
  46. 46.
    Polovinkin, E. S. and Smirnov, G. V.: Differentiation of multivalued mappings and properties of solutions of differential inclusions,Soviet Math. Dokl. 33 (1986), 662–666.Google Scholar
  47. 47.
    Rockafellar, R. T.: Proto-differentiability of set-valued mappings and its applications in optimization, in Attouch,et. al. (eds),Analyse non linéaire, Gauthier-Villars, Montreal, 1989, pp. 449–482.Google Scholar
  48. 48.
    Roxin, E.: Stability in general control systems,J. Differential. Equations 1 (1965), 115–150.Google Scholar
  49. 49.
    Roxin, E.: On generalized dynamical systems defined by contingent equations,J. Differential Equations 2 (1965), 188–205.Google Scholar
  50. 50.
    Rubinov, A. M.: The contingent derivative of a multivalued mapping and differentiability of the maximum under connected constraints,Sib. Math. Zh. 26 (1985), 147–155.Google Scholar
  51. 51.
    Wolenski, P.: The exponential formula for the reachable set of Lipschitz differential inclusions,SIAM J. Control Optim. 28 (1990), 1148–1161.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Zvi Artstein
    • 1
  1. 1.Department of Theoretical MathematicsThe Weizmann Institute of ScienceRehovotIsrael

Personalised recommendations