Derived cones to reachable sets of a nonlinear differential inclusion

Aurelian Cernea

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 4, page 567-575
  • ISSN: 0862-7959

Abstract

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We consider a nonlinear differential inclusion defined by a set-valued map with nonconvex values and we prove that the reachable set of a certain variational inclusion is a derived cone in the sense of Hestenes to the reachable set of the initial differential inclusion. In order to obtain the continuity property in the definition of a derived cone we use a continuous version of Filippov's theorem for solutions of our differential inclusion. As an application, in finite dimensional spaces, we obtain a sufficient condition for local controllability along a reference trajectory.

How to cite

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Cernea, Aurelian. "Derived cones to reachable sets of a nonlinear differential inclusion." Mathematica Bohemica 139.4 (2014): 567-575. <http://eudml.org/doc/269856>.

@article{Cernea2014,
abstract = {We consider a nonlinear differential inclusion defined by a set-valued map with nonconvex values and we prove that the reachable set of a certain variational inclusion is a derived cone in the sense of Hestenes to the reachable set of the initial differential inclusion. In order to obtain the continuity property in the definition of a derived cone we use a continuous version of Filippov's theorem for solutions of our differential inclusion. As an application, in finite dimensional spaces, we obtain a sufficient condition for local controllability along a reference trajectory.},
author = {Cernea, Aurelian},
journal = {Mathematica Bohemica},
keywords = {derived cone; $m$-dissipative operator; local controllability; derived cone; -dissipative operator; local controllability},
language = {eng},
number = {4},
pages = {567-575},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Derived cones to reachable sets of a nonlinear differential inclusion},
url = {http://eudml.org/doc/269856},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Cernea, Aurelian
TI - Derived cones to reachable sets of a nonlinear differential inclusion
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 4
SP - 567
EP - 575
AB - We consider a nonlinear differential inclusion defined by a set-valued map with nonconvex values and we prove that the reachable set of a certain variational inclusion is a derived cone in the sense of Hestenes to the reachable set of the initial differential inclusion. In order to obtain the continuity property in the definition of a derived cone we use a continuous version of Filippov's theorem for solutions of our differential inclusion. As an application, in finite dimensional spaces, we obtain a sufficient condition for local controllability along a reference trajectory.
LA - eng
KW - derived cone; $m$-dissipative operator; local controllability; derived cone; -dissipative operator; local controllability
UR - http://eudml.org/doc/269856
ER -

References

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  2. Barbu, V., Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden (1976). (1976) Zbl0328.47035MR0390843
  3. Căpraru, I., Cernea, A., On the existence of solutions for nonlinear differential inclusions, 14 pages, DOI:102478/aicu-2014-0016 (to appear) in Anal. Univ. "Al. I. Cuza", Iaşi. MR3300732
  4. Cernea, A., Local controllability of hyperbolic differential inclusions via derived cones, Rev. Roum. Math. Pures Appl. 47 (2002), 21-31. (2002) Zbl1055.49002MR1978185
  5. Cernea, A., Derived cones to reachable sets of differential-difference inclusions, Nonlinear Anal. Forum 11 (2006), 1-13. (2006) Zbl1131.34047MR2251460
  6. Cernea, A., Derived cones to reachable sets of discrete inclusions, Nonlinear Stud. 14 (2007), 177-187. (2007) Zbl1213.93012MR2327830
  7. Cernea, A., Mirică, Ş., Derived cones to reachable sets of differential inclusions, Mathematica 40 (1998), 35-62. (1998) Zbl1281.34020MR1701249
  8. Hestenes, M. R., Calculus of Variations and Optimal Control Theory, Wiley, New York (1966). (1966) Zbl0173.35703MR0203540
  9. Lakshmikantham, V., Leela, S., Nonlinear Differential Equations in Abstract Spaces, International Series in Nonlinear Mathematics: Theory, Methods and Applications 2 Pergamon Press, Oxford (1981). (1981) Zbl0456.34002MR0616449
  10. Mirică, Ş., 10.1007/BF00940323, J. Optimization Theory Appl. 74 (1992), 487-508. (1992) Zbl0795.49013MR1181848DOI10.1007/BF00940323
  11. Mirică, Ş., 10.1080/02331939908844449, Optimization 46 (1999), 135-163. (1999) Zbl0959.90048MR1729825DOI10.1080/02331939908844449

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