Controllability of Semilinear Integrodifferential Degenerate Sobolev Equations

GE Zhaoqiang

系统科学与复杂性(英文) ›› 2024, Vol. 37 ›› Issue (5) : 1923-1936.

PDF(218 KB)
PDF(218 KB)
系统科学与复杂性(英文) ›› 2024, Vol. 37 ›› Issue (5) : 1923-1936. DOI: 10.1007/s11424-024-3181-5

Controllability of Semilinear Integrodifferential Degenerate Sobolev Equations

    GE Zhaoqiang
作者信息 +

Controllability of Semilinear Integrodifferential Degenerate Sobolev Equations

    GE Zhaoqiang
Author information +
文章历史 +

摘要

In this paper, the approximate controllability of semilinear integrodifferential degenerate Sobolev equations with nonlocal conditions is investigated in the sense of integral solution in Hilbert spaces. Some sufficient and necessary conditions are obtained. Firstly, the existence and uniqueness of integral solutions of semilinear integrodifferential degenerate Sobolev equations with nonlocal conditions are considered by GE-evolution operator theory and Sadovskii's fixed point theorem, the existence and uniqueness theorem of solutions is given. Secondly, the approximate controllability of semilinear integrodifferential degenerate Sobolev equations with nonlocal conditions is studied in the sense of integral solution. The criterion for approximate controllability is provided. The obtained results have important theoretical and practical value for the study of controllability of semilinear integrodifferential degenerate Sobolev equations with nonlocal conditions.

Abstract

In this paper, the approximate controllability of semilinear integrodifferential degenerate Sobolev equations with nonlocal conditions is investigated in the sense of integral solution in Hilbert spaces. Some sufficient and necessary conditions are obtained. Firstly, the existence and uniqueness of integral solutions of semilinear integrodifferential degenerate Sobolev equations with nonlocal conditions are considered by GE-evolution operator theory and Sadovskii's fixed point theorem, the existence and uniqueness theorem of solutions is given. Secondly, the approximate controllability of semilinear integrodifferential degenerate Sobolev equations with nonlocal conditions is studied in the sense of integral solution. The criterion for approximate controllability is provided. The obtained results have important theoretical and practical value for the study of controllability of semilinear integrodifferential degenerate Sobolev equations with nonlocal conditions.

关键词

Approximate controllability / GE-evolution operator / integral solution / Sadovskii's fixed point theorem / semilinear integrodifferential degenerate Sobolev equations

Key words

Approximate controllability / GE-evolution operator / integral solution / Sadovskii's fixed point theorem / semilinear integrodifferential degenerate Sobolev equations

引用本文

导出引用
GE Zhaoqiang. Controllability of Semilinear Integrodifferential Degenerate Sobolev Equations. 系统科学与复杂性(英文), 2024, 37(5): 1923-1936 https://doi.org/10.1007/s11424-024-3181-5
GE Zhaoqiang. Controllability of Semilinear Integrodifferential Degenerate Sobolev Equations. Journal of Systems Science and Complexity, 2024, 37(5): 1923-1936 https://doi.org/10.1007/s11424-024-3181-5

参考文献

[1] Kalman R E, Controllability of linear systems, Contrib. Differ., 1963, 1: 190-213.
[2] Barnett S, Introduction to Mathematical Control Theory, Clarendom Press, Oxford, UK, 1975.
[3] Curtain R and Zwart H J, An Introduction to Infinite Dimensional Linear Systems Theory, Springer-Verlag, New York, 1995.
[4] Guo Y, Shu X B, and Yang C, HJB equation for optimal control system with random impulses, Optimization, 2022, 7: 1-25.
[5] Bian W, Approximate controllability of semilinear systems, Acta Math. Hungary, 1998, 81: 41-57.
[6] Dauer J P and Mahmudov N I, Approximate controllability of semilinear functional equations in Hilbert spaces, J. Math. Anal. Appl., 2002, 273: 310-327.
[7] Klamka J, Contained exact controllability of semilinear systems, Systems and Control Letters, 2002, 47: 139-147.
[8] Wang L W, Approximate controllability and approximate null controllability of semilinear systems, Communications on Pure and Applied Analysis, 2006, 5: 953-962.
[9] Fu X L, Controllability of non-densely defined functional differential systems in abstract space, Applied Mathematical Letters, 2006, 19: 369-377.
[10] Jeong J M and Roh H H, Approximate controllability for semilinear retarded systems, J. Math. Anal. Appl., 2006, 321: 961-975.
[11] Fu X L and Liu X B, Controllability of non-densely defined neutral functional differential systems in abstract space, Chin. Ann. Math., 2007, 28: 243-252.
[12] Jeong J M and Kim H G, Controllability for semilinear functional systems integrodifferential equation, Bull. Korean Math. Soc., 2009, 46: 463-475.
[13] Kavitha V and Mallika A M, Controllability of non-densely defined impulsive neutral functional differential systems with infinite delay in Banach spaces, Nonlinear Analysis: Hybrid Systems, 2010, 4: 441-450.
[14] Tomar N K and Sukavanam N, Approximate controllability of non-densely defined semilinear delayed control systems, Nonlinear Studies, 2011, 18: 229-234.
[15] Sakthivel P, Mahmudov N I, and Nieto J J, Controllability for a class of fractional order neutral evolution control systems, Applied Mathematics and Computation, 2012, 218: 10334-10340.
[16] Bashirov A E and Jneid M, On partial complete controllability of semilinear systems, Abstract and Applied Analysis, 2013, 2013: 1-8.
[17] Bashirov A E and Ghahramanlou N, On partial approximate controllability of semilinear systems, Cogent Engineering, 2014, 1: 1-13.
[18] Shukla A S, Sukavanam N, and Pandey D N, Approximate controllability of semilinear systems with state delay using sequence method, Journal of the Franklin Institute, 2015, 352: 5380-5392.
[19] Ahluwalia D, Sukavanam N, and Shukla A S, On the approximate controllability of semilinear control systems in Hilbert spaces, Cogent Mathematics, 2016, 3: 1-10.
[20] Arora and Sukavanam N, Approximate controllability of non-densely defined semilinear control system with nonlocal conditions, Nonlinear Dynamics and Systems Theory, 2017, 17: 5-18.
[21] Kumar S, Kumar M, and Sukavanam N, Approximate controllability of non-densely defined semilinear control systems for two classes of nonlinearity, International Journal of Dynamics and Control, 2018, 6: 1807-1815.
[22] Klamka J, Controllability of semilinear systems with multiple variable delays in control, Mathematics, 2020, 8: 1-9.
[23] Bashirow A E, On exact controllability of semilinear systems, Math. Meth. Appl. Sci., 2021, 44: 7455-7462.
[24] Zhu B and Han B, Existence and uniqueness of mild solutions for fractional partial integrodifferential equations, Mediterr. J. Math., 2020, 17(113): 1-11.
[25] Zhu B, Han B, and Yu W, Existence of mild solutions for a class of fractional non-autonomous evolution equations with delay, Acta Math. Appl. Sin. Engl. Ser., 2020, 36: 870-878.
[26] Shu L X, Shu X B, and Mao J Z, Approxomate controllability and existence of mild solution for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1< α < 2, Fractional Calculus and Applied Analysis, 2019, 22: 1086-1112.
[27] Zhu B and Han B, Approximate controllability for mixed type non-autonomous fractional differential equations, Qual. Theory Dyn. Syst., 2022, 21: 111.
[28] Kobayasi K, The equivalence of weak solution and entropy solution of nonlinear degenerate seconder-order equation, J. Diff. Equation, 2003, 189: 383-395.
[29] Amara M, Obeid G, and Vallet G, Existence results for a degenerate nonlinear elliptic partial differential equation, J. Math. Anal. Appl., 2005, 310: 641-656.
[30] Andreianov B, Bendahmane M, Karlsen K H, et al, Well-posedness results for triply nonlinear degenerate parabolic equations, J. Diff. Equation, 2009, 247: 277-302.
[31] Showalter R E, Nonlinear degenerate evolution equations in mixed formulation, SIAM J. Math. Anal., 2010, 42: 2114-2131.
[32] Su N, Extinction in finite time of solution to degenerate parabolic equations with nonlinear boundary conditions, J. Math. Anal. Appl., 2000, 246: 503-519.
[33] Ge Z Q and Feng D X, Well-posed problem of nonlinear singular distributed parameter systems and nonlinear GE-semigroup, Sci. China Ser. F Inf., 2013, 56: 1-14.
[34] Ge Z Q and Feng D X, Well-posed problem of nonlinear time varying singular distributed parameter systems, Sci. Sin. Math., 2014, 44: 1277-1298.
[35] Ge Z Q, Impulse observability and impulse controllability of regular degenerate evolution systems, Journal of Systems Science & Complexity, 2016, 29(4): 933-945.
[36] Ge Z Q and Ge X C, Controllability of singular distributed parameter systems in the sense of mild solution, Journal of Systems Science & Complexity, 2020, 33(5): 1485-1496.
[37] Ge Z Q, Impulse controllability and impulse observability of stochastic singular systems, Journal of Systems Science & Complexity, 2021, 34(3): 899-911.
[38] Ge Z Q, Controllability and observability of stochastic singular systems in Banach spaces, Journal of Systems Science & Complexity, 2022, 35(1): 194-204.
[39] Ge Z Q, Review of the latest progress in controllability of stochastic linear systems and stochastic GE-evolution operators, Mathematics, 2021, 9: 1-42.
[40] Ge Z Q, Approximate controllability of semilinear stochastic generalized systems in Hilbert spaces, Mathematics, 2022, 11(17): 1-30.
[41] Anguraj A and Ramkumar K, Approximate controllability of semilinear stochastic integrodifferential equations with nonlocal conditions, Fractal and Fractional, 2018, 2(4): 29-42.

基金

This research was supported by the National Natural Science Foundation of China under Grant Nos. 12126401 and 11926402.
PDF(218 KB)

62

Accesses

0

Citation

Detail

段落导航
相关文章

/