Skip to main content Skip to main navigation menu Skip to site footer
  • Login
  • Language
    • English
    • Język Polski
  • Menu
  • Home
  • Current
  • Online First
  • Archives
  • About
    • About the Journal
    • Submissions
    • Editorial Team
    • Privacy Statement
    • Contact
  • Login
  • Language:
  • English
  • Język Polski

Topological Methods in Nonlinear Analysis

Trajectory approximately controllability and optimal control for noninstantaneous impulsive inclusions without compactness
  • Home
  • /
  • Trajectory approximately controllability and optimal control for noninstantaneous impulsive inclusions without compactness
  1. Home /
  2. Archives /
  3. Vol 58, No 1 (September 2021) /
  4. Articles

Trajectory approximately controllability and optimal control for noninstantaneous impulsive inclusions without compactness

Authors

  • Shengda Liu
  • JinRong Wang
  • Donal O'Regan

DOI:

https://doi.org/10.12775/TMNA.2020.069

Keywords

Noninstantaneous impulsive inclusions, controllability, upper and almost lower semicontinuous, optimal control, stability

Abstract

In this paper, a noninstantaneous impulsive differential inclusion model is established based on the heating phenomenon of the rod. The controllability problem for this system governed by a semilinear differential inclusion with noninstantaneous impulses is studied in a Banach space and in this differential inclusion system we assume that the semigroup generated by the linear part of the inclusion is not compact. We suppose that the set-valued nonlinearity satisfies a regularity condition expressed in terms of the Hausdorff measure of noncompactness and some sufficient conditions for approximately controllability for both upper and almost lower semicontinuous types of nonlinearity are presented. Also we discuss existence and the stability of optimal control. As an application, the controllability for a differential inclusion system governed by a heat equation is considered.

References

R.R. Akhmerov, M.I. Kamenskiı̆, A.S. Potapov, A.E. Rodkina and B.N. Sadovskiı̆, Measures of Noncompactness and Condensing Operators, Birkhauser, Boston, 1992.

J.P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, 1990.

K. Balachandran, J.P. Dauer, Controllability of nonlinear systems in Banach spaces: a survey, J. Optim. Theory Appl. 115 (2002), 7–28.

M. Benchohra and S.K. Ntouyas, Controllability for functional differential and integrodifferential inclusions in Banach spaces, J. Optim. Theory Appl. 113 (2002), 449–472.

M. Benchohra and A. Ouahab, Controllability results for functional semilinear differential inclusions in Frechet spaces, Nonlinear Anal. 61 (2005), 405–423.

I. Benedetti, V. Obukhovskiı̆ and V. Taddei, Controllability for systems governed by semilinear evolution inclusions without compactness, Nonlinear Differential Equations Appl. 21 (2017), 795- -812.

I. Benedetti, V. Obukhovskiı̆ and P. Zecca, Controllability for impulsive semilinear functional differential inclusions with a non-compact evolution operator, Discuss. Math. Differ. Incl. Control Optim. 31 (2011), 39–69.

Y. Chang and W. Li, Controllability of functional integro-differential inclusions with an unbounded delay, J. Optim. Theory Appl. 132 (2007), 125–142.

Y. Chang, W. Li and J. Nieto, Controllability of evolution differential inclusions in Banach spaces, Nonlinear Anal. 67 (2007), 623–632.

V.I. Chilin and B.A. Rakhimov, Criteria of compactness in Lp -spaces, Internat. J. Modern Phys. Conference Series, International Conference Mathematical and Computational Biology 2011, 9 (2012), 520–528.

J. Diblik, Relative and trajectory controllability of linear discrete systems with constant coefficients and a single delay, IEEE Trans. Automat. Control 64 (2019), 2158–2165.

H. Deng and W. Wei, Existence and stability analysis for nonlinear optimal control problem with 1-mean equicontinuous controls, J. Ind. Manag. Optim. 11 (2015), 1409–1422.

M.K. Fort, Points of continuity of semicontinuous functions, Publ. Math. Debrecen 2 (1951), 100–102.

L. Górniewicz, S.K. Ntouyas and D. O’Regan, Controllability of semilinear differential equations and inclusions via semigroup theory in Banach spaces, Rep. Math. Phys. 56 (2005), 437–470.

L. Górniewicz, S.K. Ntouyas, D. O’Regan, Controllability results for first and second order evolution inclusions with nonlocal conditions, Ann. Polon. Math. 89 (2006), 65–101.

L. Górniewicz, S.K. Ntouyas and D. O’Regan, Existence and controllability results for first- and second-order functional semilinear differential inclusions with nonlocal conditions, Numer. Funct. Anal. Optim. 28 (2007), 53–82.

L. Górniewicz, S.K. Ntouyas and D. O’Regan, Controllability of evolution inclusions in Banach spaces with nonlocal conditions, Nonlinear Anal. Forum 12 (2007), 103–117.

M. Kamenskiı̆, V. Obukhovskiı̆ and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter, Berlin, 2001.

S. Liu, J. Wang and Y. Zhou, Optimal control of non instantaneous impulsive differential equations, J. Franklin Inst. 354 (2017), 7668–7698.

S. Liu, A. Debbouche, J. Wang, ILC method for solving approximate controllability of fractional differential equations with non instantaneous impulses, J. Comput. Appl. Math. 339 (2018), 343–355.

N.I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim. 42 (2003), 1604–1622.

V. Obukhovskiı̆ and P. Zecca, Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup, Nonlinear Anal. 70 (2009), 3424–3436.

M. Riesz, Sur les ensembles compacts de fonctions sommables, Acta Szeged Sect. Math. 6 (1933), 136–142. (in French)

K. Rykaczewski, Approximate controllability of differential inclusions in Hilbert spaces, Nonlinear Anal. 75 (2012), 2701–2712.

R. Triggiani, A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optim. 15 (1977), 407–411.

R. Triggiani, Addendum: A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optim 18 (1980), 98–99.

J. Wang, A.G. Ibrahim, M. Feckan and Y. Zhou, Controllability of fractional noninstantaneous impulsive differential inclusions without compactness, IMA J. Math. Control Inform. 36 (2019), 443–460.

J. Wang, A.G. Ibrahim, D. O’Regan and Y. Zhou, Controllability for noninstantaneous impulsive semilinear functional differential inclusions without compactness, Indag. Math. 29 (2018), 1362–1392.

J. Wang, A.G. Ibrahim, D. O’Regan and Y. Zhou, Controllability for non instantaneous impulsive semilinear functional differential inclusions without compactness, Indag. Math. 29 (2018), 1362–1392.

J. Yu, Z. Liu, D. Peng, D. Xu and Y. Zhou, Existence and stability analysis of optimal control, Optim. Control Appl. Methods 35 (2014), 721–729.

Downloads

  • PREVIEW
  • FULL TEXT

Published

2021-10-11

How to Cite

1.
LIU, Shengda, WANG, JinRong and O’REGAN, Donal. Trajectory approximately controllability and optimal control for noninstantaneous impulsive inclusions without compactness. Topological Methods in Nonlinear Analysis. Online. 11 October 2021. Vol. 58, no. 1, pp. 19 - 49. [Accessed 7 July 2025]. DOI 10.12775/TMNA.2020.069.
  • ISO 690
  • ACM
  • ACS
  • APA
  • ABNT
  • Chicago
  • Harvard
  • IEEE
  • MLA
  • Turabian
  • Vancouver
Download Citation
  • Endnote/Zotero/Mendeley (RIS)
  • BibTeX

Issue

Vol 58, No 1 (September 2021)

Section

Articles

License

Copyright (c) 2021 Topological Methods in Nonlinear Analysis

Creative Commons License

This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.

Stats

Number of views and downloads: 0
Number of citations: 0

Search

Search

Browse

  • Browse Author Index
  • Issue archive

User

User

Current Issue

  • Atom logo
  • RSS2 logo
  • RSS1 logo

Newsletter

Subscribe Unsubscribe
Up

Akademicka Platforma Czasopism

Najlepsze czasopisma naukowe i akademickie w jednym miejscu

apcz.umk.pl

Partners

  • Akademia Ignatianum w Krakowie
  • Akademickie Towarzystwo Andragogiczne
  • Fundacja Copernicus na rzecz Rozwoju Badań Naukowych
  • Instytut Historii im. Tadeusza Manteuffla Polskiej Akademii Nauk
  • Instytut Kultur Śródziemnomorskich i Orientalnych PAN
  • Instytut Tomistyczny
  • Karmelitański Instytut Duchowości w Krakowie
  • Ministerstwo Kultury i Dziedzictwa Narodowego
  • Państwowa Akademia Nauk Stosowanych w Krośnie
  • Państwowa Akademia Nauk Stosowanych we Włocławku
  • Państwowa Wyższa Szkoła Zawodowa im. Stanisława Pigonia w Krośnie
  • Polska Fundacja Przemysłu Kosmicznego
  • Polskie Towarzystwo Ekonomiczne
  • Polskie Towarzystwo Ludoznawcze
  • Towarzystwo Miłośników Torunia
  • Towarzystwo Naukowe w Toruniu
  • Uniwersytet im. Adama Mickiewicza w Poznaniu
  • Uniwersytet Komisji Edukacji Narodowej w Krakowie
  • Uniwersytet Mikołaja Kopernika
  • Uniwersytet w Białymstoku
  • Uniwersytet Warszawski
  • Wojewódzka Biblioteka Publiczna - Książnica Kopernikańska
  • Wyższe Seminarium Duchowne w Pelplinie / Wydawnictwo Diecezjalne „Bernardinum" w Pelplinie

© 2021- Nicolaus Copernicus University Accessibility statement Shop