# 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

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topTiziana 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

top- [1] V. B, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, The Netherlands 1976.
- [2] M. B, Points extremaux multi-applications et fonctionelles integrales, These du 3éme cycle, Université de Grenoble, France.
- [3] H. B, Analisi funzionale, Teoria e applicazioni, Liguori 1986.
- [4] T. C, G. C, F. P, M. T, On a class of evolution equations without convexity, to appear.
- [5] G. C, M. T, Multivalued perturbations for a class of nonlinear evolution equations, Ann. di Mat. Pura Appl., 160 (1991), 147-162. Zbl0752.34013
- [6] M. D, A. M, M. T, Evolution equations with lack of convexity, J. Nonlinear Anal. Theory Meth. Appl., 9 (1985), 1401-1443. Zbl0545.46029
- [7] J. D, J. Uhl, Vector Measures, Mat. Surveys 15, A.M.S., Providence, R.I. 1977.
- [8] A. F, Continuos selections for a class of non-convex multivalued maps, Studia Math. 76 (1983), 163-174. Zbl0534.28003
- [9] J. G, Convex Analysis with Application in Differentiation of Convex Functions, Pitman, Boston 1982. Zbl0486.46001
- [10] C.J. H, Measurable relations, Fund. Math., LXXXVII (1975) 53-72.
- [11] N.S. P, F. P, On retarded evolution inclusion of the subdifferetial type, to appear.
- [12] N.S. P, On the solution set of nonconvex subdifferential evolution inclusions, to appear. Zbl0868.34010
- [13] F. P, Properties of the solution set of evolution inclusions, to appear.
- [14] J. S, C. W, Convexity and Optimization in Finite Dimension I, Springer-Verlag Berlin, Heidelberg, New York 1970.
- [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] M. T, Quasi-autonomous parabolic evolution equations associated with a class of nonlinear operators, Ricerche di Matematica 38 (1989), 63-92. Zbl0736.47026
- [17] D.H. W, Survey of measurable selection theorems, SIAM J. Control Optim., 15 (1977), 859-903. Zbl0407.28006

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