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A Study on the Approximate Controllability of Damped Elastic Systems Using Sequence Method

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Abstract

In this paper, we study the approximate controllability of damped elastic systems with initial conditions without the assumptions that the corresponding linear system is approximately controllable. Firstly, the existence of mild solution is obtained by means of contraction mapping principle and operator semigroup theory. Secondly, using the sequence method, a new set of sufficient conditions for approximate controllability of damped elastic systems are formulated and proved. The results in this paper are generalization and continuation of the recent results on this issue. Finally, as the application of abstract results, an example is given to illustrate our main results.

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References

  1. Abada, N., Benchohra, M., Hammouche, H.: Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions. J. Differ. Equ. 246, 3834–3863 (2009)

    MathSciNet  Google Scholar 

  2. Arthi, G., Park, J.: On controllability of second-order impulsive neutral integro-differential systems with infinite delay. IMA J. Math. Control Inf. 32, 1–19 (2014)

    Google Scholar 

  3. Arthi, G., Balachandran, K.: Controllability results for damped second-order impulsive neutral integro-differential systems with nonlocal conditions. J. Control Theory Appl. 11, 186–192 (2013)

    MathSciNet  Google Scholar 

  4. Arthi, G., Balachandran, K.: Controllability of damped second-order neutral functional differential systems with impulses. Taiwan. J. Math. 16, 89–106 (2012)

    MathSciNet  Google Scholar 

  5. Amann, H.: Periodic solutions of semilinear parabolic equations. In: Cesari, L., Kannan, R., Weinberger, R. (eds.) Nonlinear Analysis, A Collection of Papers in Honor of Erich H. Rothe, pp. 1–29. Academic Press, New York (1978)

    Google Scholar 

  6. Acquistapace, P.: Evolution operators and strong solution of abstract parabolic equations. Differ. Integr. Equ. 1, 433–457 (1988)

    MathSciNet  Google Scholar 

  7. Acquistapace, P., Terreni, B.: A unified approach to abstract linear parabolic equations. Rend. Semin. Mat. Univ. Padova 78, 47–107 (1987)

    MathSciNet  Google Scholar 

  8. Alkhazzan, A., Jiang, P., Baleanu, D., Khan, H., Khan, A.: Stability and existence results for a class of nonlinear fractional differential equations with singularity. Math. Methods Appl. Sci. 41, 1–14 (2018)

    MathSciNet  Google Scholar 

  9. Ansari, K.J., Ilyas, A.F., Shah, K., Khan, A., Abdeljawad, T.: On New Updated Concept for Delay Differential Equations with Piecewise Caputo Fractional-Order Derivative, pp. 1–20. Waves in Random and Complex Media (2023)

    Google Scholar 

  10. Bedi, P., Kumar, A., Khan, A.: Controllability of neutral impulsive fractional differential equations with Atangana–Baleanu–Caputo derivatives. Chaos Solitons Fractals 150, 111153 (2021)

    MathSciNet  Google Scholar 

  11. Benchohra, M., Gorniewicz, L., Ntouyas, S.K., Ouahab, A.: Controllability results for impulsive functional differential inclusions. Rep. Math. Phys. 54, 211–228 (2004)

    MathSciNet  Google Scholar 

  12. Balachandran, K., Sakthivel, R.: Controllability of integro-differential systems in Banach spaces. Appl. Math. Comput. 118, 63–71 (2001)

    MathSciNet  Google Scholar 

  13. Chen, P., Zhang, X., Li, Y.: Approximate controllability of non-autonomous evolution system with nonlocal conditions. J. Dyn. Control Syst. 26, 1–16 (2020)

    MathSciNet  Google Scholar 

  14. Cao, Y., Sun, J.: Existence of solutions for semilinear measure driven equations. J. Math. Anal. Appl. 425, 621–631 (2015)

    MathSciNet  Google Scholar 

  15. Cao, Y., Sun, J.: Controllability of measure driven evolution systems with nonlocal conditions. Appl. Math. Comput. 299, 119–126 (2017)

    MathSciNet  Google Scholar 

  16. Chen, G., Russell, D.L.: A mathematical model for linear elastic systems with structural damping. Quart. Appl. Math. 39, 433–454 (1982)

    MathSciNet  Google Scholar 

  17. Cao, Y., Sun, J.: Approximate controllability of semilinear measure driven systems. Mathematische Nachrichten 291, 1979–1988 (2018)

    MathSciNet  Google Scholar 

  18. Chen, S., Triggiani, R.: Gevrey class semigroups arising from elastic systems with gentle dissipation: the case \(0 <\alpha < \frac{1}{2}\). Proc. Am. Math. Soc. 110(2), 401–415 (1990). https://doi.org/10.2307/2048084

    Article  MathSciNet  Google Scholar 

  19. Chen, G., Russell, D.L.: A mathematical model for linear elastic systems with structural damping. Quart. Appl. Math. 39(1982), 433–454 (1982). https://doi.org/10.1093/qjmam/35.4.548

    Article  MathSciNet  Google Scholar 

  20. Diagana, T.: Semilinear Evolution Eqautions and Their Applications. Springer, Switzerland (2018)

    Google Scholar 

  21. Djaout, A., Benbachir, M., Lakrib, M., Matar, M.M., Khan, A., Abdeljawad, T.: Solvability and stability analysis of a coupled system involving generalized fractional derivatives. AIMS Math. 8(4), 7817–7839 (2022)

    MathSciNet  Google Scholar 

  22. Dineshkumar, C., Nisar, K.S., Udhayakumar, R., Vijayakumar, V.: A discussion on approximate controllability of Sobolev-type Hilfer neutral fractional stochastic differential inclusions. Asian J. Control 24, 2378–2378 (2022)

    MathSciNet  Google Scholar 

  23. Dineshkumar, C., Udhayakumar, R.: Results on approximate controllability of nondensely defined fractional neutral stochastic differential systems,. Numer. Methods Part. Differ. Equ. 38, 1–27 (2020)

    MathSciNet  Google Scholar 

  24. Dineshkumar, C., Udhayakumar, R., Vijayakumar, K.S., Nisar, V., Shukla, A., Aty, A.H.A., Mahmoud, M.E., Mahmoud, M.: A note on existence and approximate controllability outcomes of Atangana–Baleanu neutral fractional stochastic hemivariational inequality. Results Phys. 38, 105647 (2022)

    Google Scholar 

  25. Diagana, T.: Well-posedness for some damped elastic systems in Banach spaces. Appl. Math. Lett. 71(2017), 74–80 (2017)

    MathSciNet  Google Scholar 

  26. Diagana, T.: Well-posedness for some damped elastic systems in Banach spaces. Appl. Math. Lett. 71, 74–80 (2017)

    MathSciNet  Google Scholar 

  27. Fan, H., Li, Y.: Monotone iterative technique for the elastic systems with structural damping in Banach spaces. Comput. Math. Appl. 68, 384–391 (2014)

    MathSciNet  Google Scholar 

  28. Fan, H., Li, Y.: Analyticity and exponential stability of semigroups for the elastic systems with structural damping in Banach spaces. J. Math. Anal. Appl. 410, 316–322 (2014)

    MathSciNet  Google Scholar 

  29. Fan, H., Gao, F.: Asymptotic stability of solutions to elastis systems with structural damping. Electron. J. Differ. Equ. 245, 1–9 (2014)

    Google Scholar 

  30. Fan, H., Li, Y., Chen, P.: Existence of mild solutions for the elastic systems with structural damping in Banach spaces. Abstract Appl. Anal. 746893, 1–6 (2013). https://doi.org/10.1155/2013/746893

    Article  MathSciNet  Google Scholar 

  31. Fu, X.: Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay. Evol. Equ. Control Theory 6, 517–534 (2017)

    MathSciNet  Google Scholar 

  32. Fu, X., Huang, R.: Approximate controllability of semilinear non-autonomous evolutionary systems with nonlocal conditions. Autom. Remote Control 77, 428–442 (2016)

    MathSciNet  Google Scholar 

  33. Fu, X.T., Zhang, Y.: Exact null controllability of non-autonomous functional evolution systems with nonlocal conditions. Acta Math. Sci. Ser. B Engl. Ed. 33(840), 747–757 (2013)

    MathSciNet  Google Scholar 

  34. Graber, P.J., Lasiecka, I.: Analyticity and Gevrey class regularity for a strongly damped wave equation with hyperbolic dynamic boundary conditions. Semigroup Forum 88(2), 333–365 (2014). https://doi.org/10.1007/s00233-013-9534-3

    Article  MathSciNet  Google Scholar 

  35. George, R.K.: Approximate controllability of non-autonomous semilinear systems. Nonlinear Anal. 1995(24), 1377–1393 (1995)

    Google Scholar 

  36. Ge, F.D., Zhou, H.C., Kou, C.H.: Approximate controllability of semilinear evolution equations of fractional order with nonlocal and impulsive conditions via an approximating technique. Appl. Math. Comput. 275, 107–120 (2016)

    MathSciNet  Google Scholar 

  37. Gou, H., Li, Y.: Mixed monotone iterative technique for damped elastic systems in Banach spaces. J. Pseudo-Differ. Oper. Appl. 11, 917–933 (2020)

    MathSciNet  Google Scholar 

  38. Gou, H., Li, Y.: A study on damped elastic systems in Banach spaces. Numer. Func. Anal. Opt. 41, 542–570 (2020)

    MathSciNet  Google Scholar 

  39. Huang, F.: On the holomorphic property of the semigroup associated with linear elastic systems with structural damping,. Acta Math. Sci. 5, 271–277 (1985). (Chinese)

    MathSciNet  Google Scholar 

  40. Huang, F., Liu, K.: Holomiphic property and exponential stability of the semigroup associated with linear elastic systems with damping. Ann. Differ. Equ. 4(4), 411–424 (1988)

    Google Scholar 

  41. Hernández, E., Regan, D.O.: Controllability of Volterra–Fredholm type systems in Banach spaces. J. Franklin Inst. 346, 95–101 (2009)

    MathSciNet  Google Scholar 

  42. Ji, S., Li, G., Wang, M.: Controllability of impulsive differential systems with nonlocal conditions. Appl. Math. Comput. 217, 6981–6989 (2011)

    MathSciNet  Google Scholar 

  43. Jeong, J.M., Ju, E.Y., Cho, S.H.: Control problems for semilinear second order equations with cosine families. Adv. Differ. Equ. 2016, 125 (2016)

    MathSciNet  Google Scholar 

  44. Ji, S.: Approximate controllability of semilinear nonlocal fractional differential systems via an approximating method. Appl. Math. Comput. 236, 43–53 (2014)

    MathSciNet  Google Scholar 

  45. Kumar, S., Sukavanam, N.: Approximate controllability of fractional order semilinear systems with bounded delay. J. Differ. Equ. 252, 6163–6174 (2012)

    MathSciNet  Google Scholar 

  46. Kalman, R.E.: Controllablity of linear dynamical systems. Contrib. Differ. Equ. 1, 190–213 (1963)

    Google Scholar 

  47. Khan, A., Khan, H., Gómez-Aguilar, J.F., Abdeljawa, T.: Existence and Hyers–Ulam stability for a nonlinear singular fractional differential equations with Mittag–Leffler kernel. Chaos Solitons Fractals 127, 422–427 (2019)

    MathSciNet  Google Scholar 

  48. Khan, A., Alshehri, H.M., Gómez-Aguilar, J.F., Khan, Z.A., Anaya, G.F.: A predator-prey model involving variable-order fractional differential equations with Mittag–Leffler kernel. Adv. Differ. Equ. 2021, 183 (2021)

    MathSciNet  Google Scholar 

  49. Khan, H., Tunc, C., Khan, A.: Stability results and existence theorems for nonlinear delay fractional differential equations with \(\varphi ^*_P\)-operator. J. Appl. Anal. Comput. 10(2), 584–597 (2020)

    MathSciNet  Google Scholar 

  50. Khan, A., Syam, M.I., Zada, A., Khan, H.: Stability analysis of nonlinear fractional differential equations with Caputo and Riemann–Liouville derivatives. Eur. Phys. J. Plus 133, 264 (2018)

    Google Scholar 

  51. Khan, A., Khan, Z.A., Abdeljawad, T., Khan, H.: Analytical analysis of fractional-order sequential hybrid system with numerical application. Adv. Contin. Discrete Models. 2022, 12 (2022)

    MathSciNet  Google Scholar 

  52. Luong, V.T., Tung, N.T.: Decay mild solutions for elastic systems with structural damping involving nonlocal conditions. Mathematics 50, 55–67 (2017)

    Google Scholar 

  53. Luong, V.T., Tung, N.T.: Exponential decay for elastic systems with structural damping and infinite delay. Appl. Anal. 99, 13–28 (2020)

    MathSciNet  Google Scholar 

  54. Liu, Z., Motreanu, D., Zeng, S.: Generalized penalty and regularization method for differential variational hemivariationak inequalities. SIAM J. Optim. 31(2), 1158–1183 (2021)

    MathSciNet  Google Scholar 

  55. Li, X., Liu, Z., Papageorgiou, N.S.: Solvability and pullback attractor for a class of differential hemivariational inequalities with its applications. Nonlinearity 36, 1323–1348 (2023)

    MathSciNet  Google Scholar 

  56. Liu, Y., Liu, Z., Papageorgiou, N.S.: Sensitivity analysis of optimal control problems driven by dynamic history-dependent variational-hemivariational inequalities. J. Differ. Equ. 342, 559–595 (2023)

    MathSciNet  Google Scholar 

  57. Liu, K., Liu, Z.: Analyticity and differentiability of semigroups associated with elastic systems with damping and gyroscopic forces. J. Differ. Equ. 141, 340–355 (1997)

    MathSciNet  Google Scholar 

  58. Liu, Z., Lv, J., Sakthivel, R.: Approximate controllability of fractional functional evolution inclusions with delay in Hilbert spaces. IMA J. Math. Control Inf. 31(3), 363–383 (2014)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  60. Mahmudov, N.I.: Approximate controllability of evolution systems with nonlocal conditions. Nonlinear Anal. 68, 536–546 (2008)

    MathSciNet  Google Scholar 

  61. Mahmudov, N.I., Zorlu, S.: On the approximate controllability of fractional evolution equations with compact analytic semigroup. J. Comput. Appl. Math. 259, 194–204 (2014)

    MathSciNet  Google Scholar 

  62. Mokkedem, F.Z., Fu, X.: Approximate controllability for a semilinear evolution system with infinite delay. J. Dyn. Control Syst. 22, 71–89 (2016)

    MathSciNet  Google Scholar 

  63. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

    Google Scholar 

  64. Pang, X., Li, X., Liu, Z.: Decay mild solutions of Hilfer fractional differential variational-hemivariational inequalities. Nonlinear Anal. Real World Appl. 71, 103834 (2023)

    MathSciNet  Google Scholar 

  65. Singh, V., Pandey, D.N.: Controllability of second-order sobolev type impulsive delay differential systems. Math. Methods Appl. Sci. 42, 1–12 (2009)

    MathSciNet  Google Scholar 

  66. Subalakshmi, R., Balachandran, K.: Approximate controllability of nonlinear stochastic impulsive integrodifferential systems in Hilbert spaces. Chaos Solitons Fractals 42, 2035–2046 (2016)

    MathSciNet  Google Scholar 

  67. Sakthivel, R., Choi, Q.H., Anthoni, S.M.: Controllability result for nonlinear evolution integrodifferential systems. Appl. Math. Lett. 17, 1015–1023 (2004)

    MathSciNet  Google Scholar 

  68. Sakthivel, R., Anthoni, S.M., Kim, J.H.: Existence and controllability result for semilinear evolution integro differential Systems. Math. Comput. Model. 41, 1005–1011 (2005)

    Google Scholar 

  69. Shukla, A., Sukavanam, N., Pandey, D.N.: Approximate controllability of semilinear system with state delay using sequence method. J. Franklin Inst. 352, 5380–5392 (2015)

    MathSciNet  Google Scholar 

  70. Sakthivel, R., Anandhi, E.: Approximate controllability of impulsive differential equations with state-dependent delay. Int. J. Control 83, 387–493 (2010)

    MathSciNet  Google Scholar 

  71. Sakthivel, R., Anandhi, E.R.: Approximate controllability of impulsive differential equations with state-dependent delay. Int. J. Control 83(2), 387–393 (2009)

    MathSciNet  Google Scholar 

  72. Shah, K., Abdalla, B., Abdeljawad, T., Gul, R.: Analysis of multipoint impulsive problem of fractional-order differential equations. Bound. Value Probl. 1, 1–17 (2023)

    MathSciNet  Google Scholar 

  73. Sher, M., Shah, K., Feckan, M., Khan, R.A.: Qualitative analysis of multi-terms fractional order delay differential equations via the topological degree theory. Mathematics 8(2), 218 (2020)

    Google Scholar 

  74. Shah, K., Sher, M., Abdeljawad, T.: Study of evolution problem under Mittag–Leffler type fractional order derivative. Alex. Eng. J. 59(5), 3945–3951 (2020)

    Google Scholar 

  75. Sakthivel, R.: Approximate controllability of nonlinear fractional dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 18, 3498–3508 (2013)

    MathSciNet  Google Scholar 

  76. Sakthivel, R., Ren, Y.: Approximate controllability of fractional differential equations with state-dependent delay. Results Math. 63, 949–963 (2013)

    MathSciNet  Google Scholar 

  77. Shen, L., Sun, J.: Approximate controllability of abstract stochastic impulsive systems with multiple time-varying delays. Int. J. Robust Nonlinear Control 63, 827–838 (2013)

    MathSciNet  Google Scholar 

  78. Tajadodi, H., Khan, A., Aguilar, J.F.G., Khan, H.: Optimal control problems with Atangana–Baleanu fractional derivative. Optim. Control Appl. Methods 42(42), 96–109 (2021)

    MathSciNet  Google Scholar 

  79. Vijayakumar, V., Udhayakumar, R., Zhou, Y., Sakthivel, N.: Approximate controllability results for Sobolev-type delay differential system of fractional order without uniqueness. Numer. Methods Part. Differ. Equ. 8, 1–20 (2020)

    Google Scholar 

  80. Vijayakumar, V.: Approximate controllability results for analytic resolvent integro-differential inclusions in Hilbert spaces. Int. J. Control 91(1), 204–214 (2018)

    MathSciNet  Google Scholar 

  81. Vijayakumar, V., Ravichandran, C., Murugesu, R., Trujillo, J.J.: Controllability results for a class of fractional semilinear integro-differential inclusions via resolvent operators. Appl. Math. Comput. 247, 152–161 (2014)

    MathSciNet  Google Scholar 

  82. Wei, S.: Global existence of mild solutions for the elastic system with structural damping. Ann. Appl. Math. 35, 180–188 (2019)

    MathSciNet  Google Scholar 

  83. Wei, M., Li, Y.: Existence and global asymptotic behavior of mild solutions for damped elastic systems with delay and nonlocal conditions. J. Anal. Appl. Comput. 13(2), 874–892 (2023)

    MathSciNet  Google Scholar 

  84. Wei, M., Li, Y., Li, Q.: Positive mild solutions for damped elastic systems with delay and nonlocal conditions in ordered Banach space. Qual. Theory Dyn. Syst. 21, 128 (2022)

    MathSciNet  Google Scholar 

  85. Yan, Z., Lu, F.: On approximate controllability of fractional stochastic neutral integro-differential inclusions with infinite delay. Appl. Anal. 94, 1235–1258 (2015)

    MathSciNet  Google Scholar 

  86. Zhou, H.X.: Approximate controllability for a class of semilinear abstract equations. SIAM J. Control Optim. 21(4), 551–565 (1983)

    MathSciNet  Google Scholar 

  87. Zhou, Y., Vijayakumar, V., Ravichandran, C., Murugesu, R.: Controllability results for Fractioanl order neutral functional differential inclusions with infinite delay. Fixed Point Theory 18(2), 773–798 (2017)

    MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos.11661071, 12061062). Science Research Project for Colleges and Universities of Gansu Province (No.2022A-010) and Project of NWNU-LKQN2023-02.

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  1. Department of Mathematics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China

    Haide Gou & Yongxiang Li

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  1. Haide Gou
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Supported by the National Natural Science Foundation of China (Grant No. 12061062, 11661071). Science Research Project for Colleges and Universities of Gansu Province (No. 2022A-010) Project of NWNU-LKQN2023-02.

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Gou, H., Li, Y. A Study on the Approximate Controllability of Damped Elastic Systems Using Sequence Method. Qual. Theory Dyn. Syst. 23, 37 (2024). https://doi.org/10.1007/s12346-023-00895-9

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