Araştırma Makalesi
Yıl 2025,
Cilt: 54 Sayı: 3, 834 - 844, 24.06.2025
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,
Cilt: 54 Sayı: 3, 834 - 844, 24.06.2025
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 | |
| 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 Cilt: 54 Sayı: 3 |