Near viability for fully nonlinear differential inclusions

Irina Căpraru; Alina Lazu

Open Mathematics (2014)

  • Volume: 12, Issue: 10, page 1447-1459
  • ISSN: 2391-5455

Abstract

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We consider the nonlinear differential inclusion x′(t) ∈ Ax(t) + F(x(t)), where A is an m-dissipative operator on a separable Banach space X and F is a multi-function. We establish a viability result under Lipschitz hypothesis on F, that consists in proving the existence of solutions of the differential inclusion above, starting from a given set, which remain arbitrarily close to that set, if a tangency condition holds. To this end, we establish a kind of set-valued Gronwall’s lemma and a compactness theorem, which are extensions to the nonlinear case of similar results for semilinear differential inclusions. As an application, we give an approximate null controllability result.

How to cite

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Irina Căpraru, and Alina Lazu. "Near viability for fully nonlinear differential inclusions." Open Mathematics 12.10 (2014): 1447-1459. <http://eudml.org/doc/269131>.

@article{IrinaCăpraru2014,
abstract = {We consider the nonlinear differential inclusion x′(t) ∈ Ax(t) + F(x(t)), where A is an m-dissipative operator on a separable Banach space X and F is a multi-function. We establish a viability result under Lipschitz hypothesis on F, that consists in proving the existence of solutions of the differential inclusion above, starting from a given set, which remain arbitrarily close to that set, if a tangency condition holds. To this end, we establish a kind of set-valued Gronwall’s lemma and a compactness theorem, which are extensions to the nonlinear case of similar results for semilinear differential inclusions. As an application, we give an approximate null controllability result.},
author = {Irina Căpraru, Alina Lazu},
journal = {Open Mathematics},
keywords = {Nonlinear differential inclusions; A priori estimates; Near viability; viability; differential inclusion; tangency; evolution inclusion; m-dissipative; Lipschitz; compactness; control system},
language = {eng},
number = {10},
pages = {1447-1459},
title = {Near viability for fully nonlinear differential inclusions},
url = {http://eudml.org/doc/269131},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Irina Căpraru
AU - Alina Lazu
TI - Near viability for fully nonlinear differential inclusions
JO - Open Mathematics
PY - 2014
VL - 12
IS - 10
SP - 1447
EP - 1459
AB - We consider the nonlinear differential inclusion x′(t) ∈ Ax(t) + F(x(t)), where A is an m-dissipative operator on a separable Banach space X and F is a multi-function. We establish a viability result under Lipschitz hypothesis on F, that consists in proving the existence of solutions of the differential inclusion above, starting from a given set, which remain arbitrarily close to that set, if a tangency condition holds. To this end, we establish a kind of set-valued Gronwall’s lemma and a compactness theorem, which are extensions to the nonlinear case of similar results for semilinear differential inclusions. As an application, we give an approximate null controllability result.
LA - eng
KW - Nonlinear differential inclusions; A priori estimates; Near viability; viability; differential inclusion; tangency; evolution inclusion; m-dissipative; Lipschitz; compactness; control system
UR - http://eudml.org/doc/269131
ER -

References

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  1. [1] Barbu V., Analysis and Control of Nonlinear Infinite Dimensional Systems, Math. Sci. Engrg., 190, Academic Press, Boston, 1993 
  2. [2] Bothe D., Flow invariance for perturbed nonlinear evolution equations, Abstr. Appl. Anal., 1996, 1(4), 417–433 http://dx.doi.org/10.1155/S1085337596000231 Zbl0954.34054
  3. [3] Cârjă O., On the minimal time function for distributed control systems in Banach spaces, J. Optim. Theory Appl., 1984, 44(3), 397–406 http://dx.doi.org/10.1007/BF00935459 Zbl0536.49023
  4. [4] Cârjă O., On constraint controllability of linear systems in Banach spaces, J. Optim. Theory Appl., 1988, 56(2), 215–225 http://dx.doi.org/10.1007/BF00939408 Zbl0625.93011
  5. [5] Cârjă O., On the minimum time function and the minimum energy problem; a nonlinear case, Systems Control Lett., 2006, 55(7), 543–548 http://dx.doi.org/10.1016/j.sysconle.2005.11.005 Zbl1129.49304
  6. [6] Cârjă O., Donchev T., Postolache V., Nonlinear evolution inclusions with one-sided Perron right-hand side, J. Dyn. Control Syst., 2013, 19(3), 439–456 http://dx.doi.org/10.1007/s10883-013-9187-2 Zbl1300.34144
  7. [7] Cârjă O., Lazu A. I., Approximate weak invariance for differential inclusions in Banach spaces, J. Dyn. Control Syst., 2012, 18(2), 215–227 http://dx.doi.org/10.1007/s10883-012-9140-9 Zbl1267.34116
  8. [8] Cârjă O., Monteiro Marques M. D. P., Weak tangency, weak invariance, and Carathéodory mappings., J. Dynam. Control Systems, 2002, 8(4), 445–461 http://dx.doi.org/10.1023/A:1020765401015 Zbl1025.34057
  9. [9] Cârjă O., Necula M., Vrabie I. I., Viability, Invariance and Applications, North-Holland Math. Stud., 207, Elsevier Science B.V., Amsterdam, 2007 
  10. [10] Cârjă O., Necula M., Vrabie I. I., Necessary and sufficient conditions for viability for nonlinear evolution inclusions, Set-Valued Anal., 2008, 16(5–6), 701–731 http://dx.doi.org/10.1007/s11228-007-0063-7 Zbl1179.34068
  11. [11] Cârjă O., Postolache V., Necessary and sufficient conditions for local invariance for semilinear differential inclusions, Set-Valued Var. Anal., 2011, 19(4), 537–554 http://dx.doi.org/10.1007/s11228-011-0173-0 Zbl1257.34045
  12. [12] Clarke F. H., Ledyaev Yu. S., Radulescu M. L., Approximate invariance and differential inclusions in Hilbert spaces, J. Dynam. Control Systems, 1997, 3(4), 493–518 Zbl0951.49007
  13. [13] Din Q., Donchev T., Kolev D., Filippov-Pliss lemma and m-dissipative differential inclusions, Journal of Global Optimization, 2013, 56(4), 1707–1717 http://dx.doi.org/10.1007/s10898-012-9963-7 Zbl1286.34088
  14. [14] Donchev T., Multi-valued perturbations of m-dissipative differential inclusions in uniformly convex spaces, New Zealand J. Math., 2002, 31(1), 19–32 
  15. [15] Filippov A. F., Classical solutions of differential equations with multivalued right hand side, SIAM J. Control Optim., 1967, 5, 609–621 http://dx.doi.org/10.1137/0305040 
  16. [16] Frankowska H., A priori estimates for operational differential inclusions, J. Differential Equations, 1990, 84(1), 100–128 http://dx.doi.org/10.1016/0022-0396(90)90129-D 
  17. [17] Goreac D., Serea O.-S., Discontinuous control problems for non-convex dynamics and near viability for singularly perturbed control systems, Nonlinear Anal., 2010, 73(8), 2699–2713 http://dx.doi.org/10.1016/j.na.2010.06.050 Zbl1195.49007
  18. [18] Lazu A. I., Postolache V., Approximate weak invariance for semilinear differential inclusions in Banach spaces, Cent. Eur. J. Math., 2011, 9(5), 1143–1155 http://dx.doi.org/10.2478/s11533-011-0051-x Zbl1247.34110
  19. [19] Tolstonogov A. A., Properties of integral solutions of differential inclusions with m-accretive operators, Mat. Zametki, 1991, 49(6), 119–131, 159 Zbl0734.34020
  20. [20] Vrabie I. I., Compactness Methods for Nonlinear Evolutions, 2nd ed., Pitman Monogr. Surveys Pure Appl. Math., 75, Longman, New York, 1995 Zbl0842.47040

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