JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2024, Vol. 59 ›› Issue (2): 110-119.doi: 10.6040/j.issn.1671-9352.0.2022.592

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Existence and approximate controllability of mild solutions for a class of Riemann-Liouville fractional evolution equations

Yuxin FENG(),He YANG*()   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Received:2022-11-09 Online:2024-02-20 Published:2024-02-20
  • Contact: He YANG E-mail:fengyuxin202211@163.com;yanghe@nwnu.edu.cn

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Abstract:

The existence of mild solutions for a class of Riemann-Liouville fractional semilinear evolution equations in Banach space is proved by utilizing the cosine family theory of linear operators and Schauder′s fixed point theorem. The approximate controllability result is also established for the related control systems. An example is given to illustrate the application of abstract conclusions in the end.

Key words: fractional evolution equation, mild solution, approximate controllability, cosine family

CLC Number: 

  • O175.15
1 FECKAN M , WANG Jinrong , ZHOU Yong . Controllability of fractional functional evolution equations of Sobolev type via characteristic solution operators[J]. Journal of Optimization Theory and Applications, 2013, 156 (1): 79- 95.
doi: 10.1007/s10957-012-0174-7
2 CHANG Y K , PEI Y T , PONCE R . Existence and optimal controls for fractional stochastic evolution equations of Sobolev type via fractional resolvent operators[J]. Journal of Optimization Theory and Applications, 2019, 182 (2): 558- 572.
doi: 10.1007/s10957-018-1314-5
3 CHANG Y K , PEREIRA A , PONCE R . Approximate controllability for fractional differential equations of Sobolev type via properties on resolvent operators[J]. Fractional Calculus and Applied Analysis, 2017, 20 (4): 963- 987.
doi: 10.1515/fca-2017-0050
4 ZHOU Yong , HE Jiawei . New results on controllability of fractional evolution systems with order α ∈ (1, 2)[J]. Evolution Equations and Control Theory, 2021, 10 (3): 491- 509.
doi: 10.3934/eect.2020077
5 DU Maolin , WANG Zaihua . Initialized fractional differential equations with Riemann-Liouville fractional-order derivative[J]. The European Physical Journal Special Topics, 2011, 193 (1): 49- 60.
doi: 10.1140/epjst/e2011-01380-8
6 HEYMANS N , PODLUBNY I . Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives[J]. Rheologica Acta, 2006, 45 (5): 765- 771.
doi: 10.1007/s00397-005-0043-5
7 KILBAS A A , SRIVASTAVA H M , TRUJILLO J J . Theory and applications of fractional differential equations[M]. Amsterdam: Elsevier, 2006: 69- 70.
8 LIU Zhenhai , LI Xiuwen . Approximate controllability of fractional evolution systems with Riemann-Liouville fractional derivatines[J]. SIAM Journal on Control and Optimization, 2015, 53 (4): 1920- 1933.
doi: 10.1137/120903853
9 TRAVIS C C , WEBB G F . Cosine families and abstract nonlinear second order differential equations[J]. Acta Mathematica Hungarica, 1978, 32 (1): 75- 96.
10 HE Jiawei , PENG Li . Approximate controllability for a class of fractional stochastic wave equations[J]. Computers Mathematics with Applications, 2019, 78 (5): 1463- 1476.
doi: 10.1016/j.camwa.2019.01.012
11 FATTORINI H O . Ordinary differential equations in linear topological spaces[J]. Journal of Difference Equations and Applications, 1968, 5, 72- 105.
12 GOU Haide , LI Yongxiang . Existence and approximate controllability of Hilfer fractional evolution equations in Banach spaces[J]. Journal of Applied Analysis Computation, 2021, 11 (6): 2895- 2920.
doi: 10.11948/20210053
13 TRAVIS C C , WEBB G F . Compactness, regularity, and uniform continuity properties of strongly continuous consine families[J]. Houston Journal of Mathematics, 1977, 3 (4): 555- 567.
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