Araştırma Makalesi
BibTex RIS Kaynak Göster

Approximate controllability result for backward stochastic evolution inclusions in Hilbert spaces

Yıl 2025, , 834 - 844, 24.06.2025

Cesim Temel , Müberra Selah

https://doi.org/10.15672/hujms.1487942

Öz

In this paper, we study semilinear backward stochastic evolution inclusion systems in Hilbert spaces. First, we prove the existence of mild solution of the semilinear backward stochastic evolution inclusion systems using a multivalued fixed point theorem. Then, we obtain the approximate controllability result for semilinear backward stochastic evolution inclusion systems through the linear systems corresponding to these semilinear backward stochastic evolution inclusion systems under appropriate conditions. In particular, our study extends the results of the concept of approximate controllability to backward stochastic evolution inclusion systems.

Anahtar Kelimeler

backward stochastic evolution inclusions approximate controllability mild solutions multivalued maps Hausdorff measure of noncompactness

Kaynakça

  • [1] S. Arora and J. Dabas, Existence and approximate controllability of non-autonomous functional impulsive evolution inclusions in Banach spaces, J. Differ. Equ. 307, 83–113, 2022.
  • [2] P. Balasubramaniam and P. Tamilalagan, Approximate controllability of a class of fractional neutral stochastic integro-differential inclusions with infinite delay by using Mainardi’s function, Appl. Math. Comput. 256, 232–246, 2015.
  • [3] F. Confortola and M. Fuhrman, Backward stochastic differential equations and optimal control of marked point processes, SIAM J. Control Optim. 51, 3592–3623, 2013.
  • [4] J.P. Dauer, N.I. Mahmudov and M.M. Matar, Approximate controllability of backward stochastic evolution equations in Hilbert spaces, J. Math. Anal. Appl. 323(1), 42-56, 2006.
  • [5] J. P. Dauer and N. I.,Mahmudov, Controllability of stochastic semilinear functional differential equations in Hilbert spaces J. Math. Anal. Appl. 290, 373-394,2004.
  • [6] K. Deimling, Multivalued differential equation, de Gruyter Series in Nonlinear Analysis and Applications,1, de Gruyter, Berlin, 1992.
  • [7] E. H. Essaky, M. Hassani and C. E. Rhazlane, Backward stochastic evolution inclusions in UMD Banach spaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 26 (04):2350013, 2023.
  • [8] I. Exarchos and E.A. Theodorou, Stochastic optimal control via forward and backward stochastic differential equations and importance sampling, Automatica. 87, 159–165, 2018.
  • [9] T.E. Govindan, Stability of mild solutions of stochastic evolution equations with variable delay Stochastic Anal. Appl. 21 (5), 1059–1077, 2003.
  • [10] K. Kavitha, V. Vijayakumar, A. Shukla, K. S. Nisar and R. Udhayakumar, Results on approximate controllability of Sobolev-type fractional neutral differential inclusions of Clarke subdifferential type, Chaos Soliton Fract. 151, 111264, 2021).
  • [11] K.S. Nisar and V. Vijayakumar, An analysis concerning approximate controllability results for second-order Sobolev-type delay differential systems with impulses, J. Inequal Appl. 2022, 53, 2022.
  • [12] W. A. Kirk, A remark on condensing mappings, J.Math.Anal.Appl. 51, 629–632, 1975.
  • [13] M. Kisielewicz, Backward stochastic differential inclusions, Dynamic Syst. Appl. 16, 121–140, 2007.
  • [14] H. Leiva and J. Uzcategui, Controllability of linear difference equations in Hilbert Spaces and applications, IMA Journal of Math. Control and Information. 25, 323–340, 2008.
  • [15] Z. Liu, J. Lv and R. Sakthivel, Approximate controllability of fractional functional evolution inclusions with delay in Hilbert spaces IMA Journal of Math. Control and Information. 31, 363–383, 2013.
  • [16] Q. Lü and J. Neerven, Backward Stochastic Evolution Equations in UMD Banach Spaces, Positivity and Noncommutative Analysis, 381, 2019.
  • [17] N. I. Mahmudova and M. A. McKibben, On backward stochastic evolution equations in Hilbert spaces and optimal control, Nonlinear Anal. 67, 1260–1274, 2007.
  • [18] N.I. Mahmudov, Controllability of linear stochastic systems, IEEE Trans. Automatic Control AC-46, 724–731, 2001.
  • [19] M. Mohan Raja, V. Vijayakumar, R. Udhayakumar and Y. Zhou, A new approach on the approximate controllability of fractional differential evolution equations of order $1 < r < 2$ in Hilbert spaces, Chaos Soliton Fract. 141, 1–11, 2020.
  • [20] M. Mohan Raja, V. Vijayakumar, L.N. Huynh, R. Udhayakumar and K.S. Nisar, Results on the approximate controllability of fractional hemivariational inequalities of order $1 < r < 2$, Adv. Differ. Equ. 2021 237, 2021.
  • [21] M. Mohan Raja, V. Vijayakumar, A. Shukla, K.S. Nisar and H.M. Baskonus, On the approximate controllability results for fractional integrodifferential systems of order $1 < r < 2$ with sectorial operators, J. Comput. Appl. Math. 415, 114492, 2022.
  • [22] M. Mohan Raja and V. Vijayakumar, Approximate controllability results for the Sobolev type fractional delay impulsive integrodifferential inclusions of order $r\in (1,2)$ via sectorial operator, Fract. Calc. Appl. Anal. 26, 1740–1769, 2023.
  • [23] E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14, 55–61, 1990.
  • [24] R. Sakthivel, Y. Ren, A. Debbouche and N.I. Mahmudov, Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions, Appl. Anal. 95, 2361–2382, 2016.
  • [25] R. Subalakshmi and K. Balachandran, Approximate controllability of nonlinear stochastic impulsive integrodifferential systems in Hilbert spaces, Chaos Soliton Fract. 42, 2035–2046, 2009.
  • [26] C.Temel, Multivalued types of Krasnosel’skii’s fixed point theorem for weak topology U.P.B. Sci. Bull., Series A, 81 (2), 139-148, 2019.
  • [27] C.Temel, On some results of Krasnosel’skii’s theorem for weak topology in Banach Space. Fixed Point Theory, 21 (1), 309-318, 2020.
  • [28] V. Vijayakumar, K.S. Nisar, D. Chalishajar, A. Shukla, M. Malik, A. Alsaadi and S.F. Aldosary, A note on approximate controllability of fractional semilinear integrodifferential control systems via resolvent operators, Fractal Fract. 6, 73, 2022.
  • [29] W. Ye and Z. Yu, Exact controllability of linear mean-field stochastic systems and observability inequality for mean-field backward stochastic differential equations, Asian J. Control. 1–12, 2020.
  • [30] E. Zeidler, Nonlinear functional analysis and its applications, I: Fixed point theorems, Springer-Verlag, New York, 1986.

Yıl 2025, , 834 - 844, 24.06.2025

Cesim Temel , Müberra Selah

https://doi.org/10.15672/hujms.1487942

Öz

Kaynakça

  • [1] S. Arora and J. Dabas, Existence and approximate controllability of non-autonomous functional impulsive evolution inclusions in Banach spaces, J. Differ. Equ. 307, 83–113, 2022.
  • [2] P. Balasubramaniam and P. Tamilalagan, Approximate controllability of a class of fractional neutral stochastic integro-differential inclusions with infinite delay by using Mainardi’s function, Appl. Math. Comput. 256, 232–246, 2015.
  • [3] F. Confortola and M. Fuhrman, Backward stochastic differential equations and optimal control of marked point processes, SIAM J. Control Optim. 51, 3592–3623, 2013.
  • [4] J.P. Dauer, N.I. Mahmudov and M.M. Matar, Approximate controllability of backward stochastic evolution equations in Hilbert spaces, J. Math. Anal. Appl. 323(1), 42-56, 2006.
  • [5] J. P. Dauer and N. I.,Mahmudov, Controllability of stochastic semilinear functional differential equations in Hilbert spaces J. Math. Anal. Appl. 290, 373-394,2004.
  • [6] K. Deimling, Multivalued differential equation, de Gruyter Series in Nonlinear Analysis and Applications,1, de Gruyter, Berlin, 1992.
  • [7] E. H. Essaky, M. Hassani and C. E. Rhazlane, Backward stochastic evolution inclusions in UMD Banach spaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 26 (04):2350013, 2023.
  • [8] I. Exarchos and E.A. Theodorou, Stochastic optimal control via forward and backward stochastic differential equations and importance sampling, Automatica. 87, 159–165, 2018.
  • [9] T.E. Govindan, Stability of mild solutions of stochastic evolution equations with variable delay Stochastic Anal. Appl. 21 (5), 1059–1077, 2003.
  • [10] K. Kavitha, V. Vijayakumar, A. Shukla, K. S. Nisar and R. Udhayakumar, Results on approximate controllability of Sobolev-type fractional neutral differential inclusions of Clarke subdifferential type, Chaos Soliton Fract. 151, 111264, 2021).
  • [11] K.S. Nisar and V. Vijayakumar, An analysis concerning approximate controllability results for second-order Sobolev-type delay differential systems with impulses, J. Inequal Appl. 2022, 53, 2022.
  • [12] W. A. Kirk, A remark on condensing mappings, J.Math.Anal.Appl. 51, 629–632, 1975.
  • [13] M. Kisielewicz, Backward stochastic differential inclusions, Dynamic Syst. Appl. 16, 121–140, 2007.
  • [14] H. Leiva and J. Uzcategui, Controllability of linear difference equations in Hilbert Spaces and applications, IMA Journal of Math. Control and Information. 25, 323–340, 2008.
  • [15] Z. Liu, J. Lv and R. Sakthivel, Approximate controllability of fractional functional evolution inclusions with delay in Hilbert spaces IMA Journal of Math. Control and Information. 31, 363–383, 2013.
  • [16] Q. Lü and J. Neerven, Backward Stochastic Evolution Equations in UMD Banach Spaces, Positivity and Noncommutative Analysis, 381, 2019.
  • [17] N. I. Mahmudova and M. A. McKibben, On backward stochastic evolution equations in Hilbert spaces and optimal control, Nonlinear Anal. 67, 1260–1274, 2007.
  • [18] N.I. Mahmudov, Controllability of linear stochastic systems, IEEE Trans. Automatic Control AC-46, 724–731, 2001.
  • [19] M. Mohan Raja, V. Vijayakumar, R. Udhayakumar and Y. Zhou, A new approach on the approximate controllability of fractional differential evolution equations of order $1 < r < 2$ in Hilbert spaces, Chaos Soliton Fract. 141, 1–11, 2020.
  • [20] M. Mohan Raja, V. Vijayakumar, L.N. Huynh, R. Udhayakumar and K.S. Nisar, Results on the approximate controllability of fractional hemivariational inequalities of order $1 < r < 2$, Adv. Differ. Equ. 2021 237, 2021.
  • [21] M. Mohan Raja, V. Vijayakumar, A. Shukla, K.S. Nisar and H.M. Baskonus, On the approximate controllability results for fractional integrodifferential systems of order $1 < r < 2$ with sectorial operators, J. Comput. Appl. Math. 415, 114492, 2022.
  • [22] M. Mohan Raja and V. Vijayakumar, Approximate controllability results for the Sobolev type fractional delay impulsive integrodifferential inclusions of order $r\in (1,2)$ via sectorial operator, Fract. Calc. Appl. Anal. 26, 1740–1769, 2023.
  • [23] E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14, 55–61, 1990.
  • [24] R. Sakthivel, Y. Ren, A. Debbouche and N.I. Mahmudov, Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions, Appl. Anal. 95, 2361–2382, 2016.
  • [25] R. Subalakshmi and K. Balachandran, Approximate controllability of nonlinear stochastic impulsive integrodifferential systems in Hilbert spaces, Chaos Soliton Fract. 42, 2035–2046, 2009.
  • [26] C.Temel, Multivalued types of Krasnosel’skii’s fixed point theorem for weak topology U.P.B. Sci. Bull., Series A, 81 (2), 139-148, 2019.
  • [27] C.Temel, On some results of Krasnosel’skii’s theorem for weak topology in Banach Space. Fixed Point Theory, 21 (1), 309-318, 2020.
  • [28] V. Vijayakumar, K.S. Nisar, D. Chalishajar, A. Shukla, M. Malik, A. Alsaadi and S.F. Aldosary, A note on approximate controllability of fractional semilinear integrodifferential control systems via resolvent operators, Fractal Fract. 6, 73, 2022.
  • [29] W. Ye and Z. Yu, Exact controllability of linear mean-field stochastic systems and observability inequality for mean-field backward stochastic differential equations, Asian J. Control. 1–12, 2020.
  • [30] E. Zeidler, Nonlinear functional analysis and its applications, I: Fixed point theorems, Springer-Verlag, New York, 1986.
Toplam 30 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Adi Diferansiyel Denklemler, Fark Denklemleri ve Dinamik Sistemler, Operatör Cebirleri ve Fonksiyonel Analiz, Varyasyon Hesabı, Sistem Teorisinin Matematiksel Yönleri ve Kontrol Teorisi
Bölüm Matematik
Yazarlar

Cesim Temel 0000-0002-9015-4155

Müberra Selah 0000-0001-6218-398X

Erken Görünüm Tarihi 27 Ağustos 2024
Yayımlanma Tarihi 24 Haziran 2025
Gönderilme Tarihi 23 Mayıs 2024
Kabul Tarihi 5 Temmuz 2024
Yayımlandığı Sayı Yıl 2025

Kaynak Göster

APA Temel, C., & Selah, M. (2025). Approximate controllability result for backward stochastic evolution inclusions in Hilbert spaces. Hacettepe Journal of Mathematics and Statistics, 54(3), 834-844. https://doi.org/10.15672/hujms.1487942
AMA Temel C, Selah M. Approximate controllability result for backward stochastic evolution inclusions in Hilbert spaces. Hacettepe Journal of Mathematics and Statistics. Haziran 2025;54(3):834-844. doi:10.15672/hujms.1487942
Chicago Temel, Cesim, ve Müberra Selah. “Approximate Controllability Result for Backward Stochastic Evolution Inclusions in Hilbert Spaces”. Hacettepe Journal of Mathematics and Statistics 54, sy. 3 (Haziran 2025): 834-44. https://doi.org/10.15672/hujms.1487942.
EndNote Temel C, Selah M (01 Haziran 2025) Approximate controllability result for backward stochastic evolution inclusions in Hilbert spaces. Hacettepe Journal of Mathematics and Statistics 54 3 834–844.
IEEE C. Temel ve M. Selah, “Approximate controllability result for backward stochastic evolution inclusions in Hilbert spaces”, Hacettepe Journal of Mathematics and Statistics, c. 54, sy. 3, ss. 834–844, 2025, doi: 10.15672/hujms.1487942.
ISNAD Temel, Cesim - Selah, Müberra. “Approximate Controllability Result for Backward Stochastic Evolution Inclusions in Hilbert Spaces”. Hacettepe Journal of Mathematics and Statistics 54/3 (Haziran 2025), 834-844. https://doi.org/10.15672/hujms.1487942.
JAMA Temel C, Selah M. Approximate controllability result for backward stochastic evolution inclusions in Hilbert spaces. Hacettepe Journal of Mathematics and Statistics. 2025;54:834–844.
MLA Temel, Cesim ve Müberra Selah. “Approximate Controllability Result for Backward Stochastic Evolution Inclusions in Hilbert Spaces”. Hacettepe Journal of Mathematics and Statistics, c. 54, sy. 3, 2025, ss. 834-4, doi:10.15672/hujms.1487942.
Vancouver Temel C, Selah M. Approximate controllability result for backward stochastic evolution inclusions in Hilbert spaces. Hacettepe Journal of Mathematics and Statistics. 2025;54(3):834-4.

For more information about the journal, please visit: https://dergipark.org.tr/en/pub/hujms