Existence and density results for retarded subdifferential evolution inclusions

Tiziana Cardinali; Simona Pieri

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (1996)

  • Volume: 16, Issue: 1, page 53-74
  • ISSN: 1509-9407

Abstract

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In this paper we study Cauchy problems for retarded evolution inclusions, where the Fréchet subdifferential of a function F:Ω→R∪{+∞} (Ω is an open subset of a real separable Hilbert space) having a φ-monotone subdifferential of oder two is present. First we establish the existence of extremal trajectories and we show that the set of these trajectories is dense in the solution set of the original convex problem for the norm topology of the Banach space C([-r, T₀], Ω) ("strong relaxation theorem"). Then we prove that this density result allows one to establish a nonlinear "bang-bang principle" for a class of control systems with dealy on a nonconvex constraint. We want to observe that the results here extend those of [11] and [13].

How to cite

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Tiziana Cardinali, and Simona Pieri. "Existence and density results for retarded subdifferential evolution inclusions." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 16.1 (1996): 53-74. <http://eudml.org/doc/275937>.

@article{TizianaCardinali1996,
abstract = {In this paper we study Cauchy problems for retarded evolution inclusions, where the Fréchet subdifferential of a function F:Ω→R∪\{+∞\} (Ω is an open subset of a real separable Hilbert space) having a φ-monotone subdifferential of oder two is present. First we establish the existence of extremal trajectories and we show that the set of these trajectories is dense in the solution set of the original convex problem for the norm topology of the Banach space C([-r, T₀], Ω) ("strong relaxation theorem"). Then we prove that this density result allows one to establish a nonlinear "bang-bang principle" for a class of control systems with dealy on a nonconvex constraint. We want to observe that the results here extend those of [11] and [13].},
author = {Tiziana Cardinali, Simona Pieri},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {Fréchet subdifferential; subdifferential evolution inclusion; dealy; strong solution; extremal trajectory; solution set; relaxation theorem; bang-bang principle; control system; retarded subdifferential evolution inclusions},
language = {eng},
number = {1},
pages = {53-74},
title = {Existence and density results for retarded subdifferential evolution inclusions},
url = {http://eudml.org/doc/275937},
volume = {16},
year = {1996},
}

TY - JOUR
AU - Tiziana Cardinali
AU - Simona Pieri
TI - Existence and density results for retarded subdifferential evolution inclusions
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 1996
VL - 16
IS - 1
SP - 53
EP - 74
AB - In this paper we study Cauchy problems for retarded evolution inclusions, where the Fréchet subdifferential of a function F:Ω→R∪{+∞} (Ω is an open subset of a real separable Hilbert space) having a φ-monotone subdifferential of oder two is present. First we establish the existence of extremal trajectories and we show that the set of these trajectories is dense in the solution set of the original convex problem for the norm topology of the Banach space C([-r, T₀], Ω) ("strong relaxation theorem"). Then we prove that this density result allows one to establish a nonlinear "bang-bang principle" for a class of control systems with dealy on a nonconvex constraint. We want to observe that the results here extend those of [11] and [13].
LA - eng
KW - Fréchet subdifferential; subdifferential evolution inclusion; dealy; strong solution; extremal trajectory; solution set; relaxation theorem; bang-bang principle; control system; retarded subdifferential evolution inclusions
UR - http://eudml.org/doc/275937
ER -

References

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  11. [11] N.S. P, F. P, On retarded evolution inclusion of the subdifferetial type, to appear. 
  12. [12] N.S. P, On the solution set of nonconvex subdifferential evolution inclusions, to appear. Zbl0868.34010
  13. [13] F. P, Properties of the solution set of evolution inclusions, to appear. 
  14. [14] J. S, C. W, Convexity and Optimization in Finite Dimension I, Springer-Verlag Berlin, Heidelberg, New York 1970. 
  15. [15] A.A. T, Extremal selections of multivalued mappings and the 'bang - bang' principle for evolution inclusions, Soviet. Math. Dokl 317 (1991), 481-485. Zbl0784.54024
  16. [16] M. T, Quasi-autonomous parabolic evolution equations associated with a class of nonlinear operators, Ricerche di Matematica 38 (1989), 63-92. Zbl0736.47026
  17. [17] D.H. W, Survey of measurable selection theorems, SIAM J. Control Optim., 15 (1977), 859-903. Zbl0407.28006

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