Existence and approximate controllability of Hilfer fractional evolution equations in Banach spaces. (English) Zbl 07907309
Summary: This paper is concerned with the existence of mild solutions as well as approximate controllability for Hilfer fractional evolution equations in Banach spaces. Firstly, we give an appropriate definition of mild solutions for this type of fractional equations. The definition of mild solutions for studied problem was given based on a cosine family generated by the operator \(A\) and probability density function. Secondly, we discuss the existence results of the mild solutions for our concerned problem under the case sine family is compact. Moreover, we establish the approximate controllability when the corresponding linear system is approximately controllable. At last, as an application, two examples are presented to illustrate the abstract results.
MSC:
| 34K30 | Functional-differential equations in abstract spaces |
| 34K45 | Functional-differential equations with impulses |
| 93B05 | Controllability |
| 26A33 | Fractional derivatives and integrals |
Keywords:
Hilfer fractional evolution equations; mild solution; approximate controllability; cosine familyReferences:
| [1] | R. P. Agarwal, M. Benchohra and S. Hamani, a survey on existence result for boundary value problem of nonlinear fractional differential equations and inclusions, Acta. Appl. Math., 2010, 109, 973-1033. · Zbl 1198.26004 |
| [2] | R. Agarwal, M. Meehan and D. O’Regan, Fixed Point Theory and Applications, Cambridge University Press, 2001. · Zbl 0960.54027 |
| [3] | A. Aghajani, J. Banaś and N. Sabzali, Some generalizations of Darbo fixed point theorem and application, Bull. Belg. Math. Soc. Simon Stevin, 2013, 20(2), 345-358. · Zbl 1290.47053 |
| [4] | B. Ahmad and S. Sivasundaram, Existence results for nonlinear impulsive hy-brid boundary value peoblems involving fractional differential equations, Non-linear Anal. HS, 2009, 3, 251-258. · Zbl 1193.34056 |
| [5] | G. Arthi and J. Park, On controllability of second-order impulsive neutral integro-differential systems with infinite delay, IMA J. Math. Control Inf., 2014, 1-19. |
| [6] | G. Arthi and K. Balachandran, Controllability results for damped second-order impulsive neutral integro-differential systems with nonlocal conditions, J. Con-trol Theory Appl., 2013, 11, 186-192. · Zbl 1299.93027 |
| [7] | G. Arthi and K. Balachandran, Controllability of damped second-order neutral functional differential systems with impulses, Taiwanese Journal of Mathemat-ics, 2012, 16, 89-106 . · Zbl 1235.93042 |
| [8] | M. Benchohra and D. Seba, Impulsive fractional differential equations in Ba-nach spaces, Electron. J. Qual. Theory Differ. Equ., 2009, 8, 1-14. · Zbl 1189.26005 |
| [9] | K. Balachandran and S. Kiruthika, Existence of solutions of abstract frac-tional impulsive semilinear evolution equations, Electron. J. Qual. Theory Dif-fer. Equ., 2010, 4, 1-12. · Zbl 1201.34091 |
| [10] | M. Benchohra, L. Gorniewicz, S. K. Ntouyas and A. Ouahab, Controllability re-sults for impulsive functional differential inclusions, Reports on Mathematical Physics, 2004, 54, 211-228. · Zbl 1130.93310 |
| [11] | Z. Bai, X. Du and C. Yin, Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions, Boundary Value Problem, 2016, 63, 1-11. · Zbl 1407.34006 |
| [12] | P. Chen, X. Zhang and Y. Li, Study on fractiona non-autonomous evolution equations with delay, Computers and Mathematics with Applications, 2017, 73(5), 794-803. · Zbl 1375.34115 |
| [13] | P. Chen, Y. Li, Q. Chen and B. Feng, On the initial value problem of fractional evolution equations with noncompact semigroup, Comput. Math. Appl., 2014, 67, 1108-1115. · Zbl 1381.34011 |
| [14] | P. Chen, X. Zhang and Y. Li, Existence and approximate controllability of frac-tional evolution equations with nonlocal conditions resolvent operators, Frac-tional Calculus & Applied Analysis, 2020, 23, 268-291. · Zbl 1441.34006 |
| [15] | Y. Cao and J. Sun, Approximate controllability of semilinear measure driven systems, Mathematische Nachrichten, 2018, 291, 1979-1988. · Zbl 1401.93034 |
| [16] | Y. Cao and J. Sun, Controllability of measure driven evolution systems with nonlocal conditions, Appl. Math. Comput., 2017, 299, 119-126. · Zbl 1411.93027 |
| [17] | P. Chen, X. Zhang and Y. Li, Approximate controllability of non-autonomous evolution system with nonlocal conditions, J. Dyn. Control. Syst., 2020, 26, 1-16. · Zbl 1439.34065 |
| [18] | M. M. El-Borai, The fundamental solutions for fractional evolution equations of parabolic type, J. Appl. Math. Stoch. Anal., 2004, 3, 197-211. · Zbl 1081.34053 |
| [19] | M. Feckan, Y. Zhou and J. Wang, On the concept and existence of solution for impulsive fractional differential equations, Communn. Nonlinear Sci. Numer. Simul., 2012, 17, 3050-3060. · Zbl 1252.35277 |
| [20] | K. M. Furati, M. D. Kassim and N. e. Tatar, Existence and uniqueness for a problem involving Hilfer factional derivative, Comput. Math. Appl., 2012, 64, 1612-1626. · Zbl 1268.34013 |
| [21] | X. Fu, Approximate controllability of semilinear non-autonomous evolution sys-tems with state-dependent delay, Evol. Equ. Control Theory, 2017, 6, 517-534. · Zbl 1381.34097 |
| [22] | X. Fu and R. Huang, Approximate controllability of semilinear non-autonomous evolutionary systems with nonlocal conditions, Autom. Remote Control, 2016, 77, 428-442. · Zbl 1348.93049 |
| [23] | X. Fu and Y. Zhang, Exact null controllability of non-autonomous functional evolution systems with nonlocal conditions, Acta. Math. Sci. Ser. B. Engl. Ed., 2013, 33, 747-757. · Zbl 1299.34255 |
| [24] | D. Guo and J. Sun, Ordinary Differential Equations in Abstract Spaces, Shan-dong Science and Technology. Ji¡¯nan, 1989. |
| [25] | H. Gu and J. J. Trujillo, Existence of mild solution for evolution equation with Hilfre fractional derivative, Applied Mathematics and Computation, 2015, 257, 344-354. · Zbl 1338.34014 |
| [26] | D. Guo and J. Sun, Ordinary Differential Equations in Abstract Spaces, Shan-dong Science and Technology, Jinan, 1989. |
| [27] | D. Guo, Y. Cho and J. Zhu, Partial Ordering Methods in Nonlinear Problems, NOVA Publishers, 2004. · Zbl 1116.45007 |
| [28] | H. Gou and B. Li, Local and global existence of mild solution to impulsive fractional semilinear integro-differential equation with noncompact semigroup, Commun. Nonlinear Sci. Numer. Simulat., 2017, 42, 204-214. · Zbl 1473.34046 |
| [29] | R. Hilfer, Y. Luchko and Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal., 2009, 12(3), 299-318. · Zbl 1182.26011 |
| [30] | R. Hilfer, Applications of Fractional Caiculus in Physics, World Scientific, Sin-gapore, 2000. · Zbl 0998.26002 |
| [31] | M. Haase, The complex inversion formula revisited, J. Aust. Math. Soc., 2008, 84, 73-83. · Zbl 1149.47031 |
| [32] | J. He and L. Peng, Approximate controllability for a class of fractional stochas-tic wave equations, Computers and Mathematics with Applications, 2019, 78, 1463-1476. · Zbl 1442.93007 |
| [33] | J. M. Jeong, E. Y. Ju and S. H. Cho, Control problems for semilinear second order equations with cosine families, Advances in Difference Equations, 2016, 125. · Zbl 1419.35175 |
| [34] | S. Ji, G. Li and M. Wang, Controllability of impulsive differential systems with nonlocal conditions, Appl. Math. Comput., 2011, 217, 6981-6989. · Zbl 1219.93013 |
| [35] | A. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and application of fractional differential equations,in: North-Holland Mathematics Studies, Else-vier Science B, Amsterdam, 2006, 204. · Zbl 1092.45003 |
| [36] | A. Kumar, M. Muslim and R. Sakthivel, Controllability of the Second-Order Nonlinear Differential Equations with Non-instantaneous Impulses, J. Dyn. Control Syst., 2018, 24, 325-342. · Zbl 1391.34100 |
| [37] | S. Kumar and N. Sukavanam, Approximate controllability of fractional order semilinear systems with bounded delay, J. Differ. Equ., 2012, 252, 6163-6174. · Zbl 1243.93018 |
| [38] | R. Kalman, Controllablity of linear dynamical systems, Contrib. Diff. Equ., 1963, 1, 190-213. |
| [39] | J. Liang and H. Yang, Controllability of fractional integro-differential evolution equations with nonlocal conditions, Appl. Math. Comput., 2015, 254(1), 20-29. · Zbl 1410.93022 |
| [40] | J. Lv and X. Yang, Approximate controllability of Hilfer fractional differential equations, Math. Meth. App. Sci., 2020(43), 242-254. · Zbl 1451.93030 |
| [41] | K. Li, J. Peng and J. Jia, Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives, J. Funct. Anal., 2012, 263, 476-510. · Zbl 1266.47066 |
| [42] | F. Mainardi, Fractional relaxation oscillation and fractional diffusion wave phenomena, Chaos Solitons Fractals, 1996, 7(9), 1461-1477. · Zbl 1080.26505 |
| [43] | N. I. Mahmdov and A. Denker, On controllability of linear stochastic systems, Internat. J. Control, 2000, 73, 144-151. · Zbl 1031.93033 |
| [44] | N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim., 2003, 42, 1604-1622. · Zbl 1084.93006 |
| [45] | N. I. Mahmudov, Approximate controllability of evolution systems with nonlocal conditions, Nonlinear Anal., 2008, 68, 536-546. · Zbl 1129.93004 |
| [46] | I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. · Zbl 0924.34008 |
| [47] | A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-verlag, Berlin, 1983. · Zbl 0516.47023 |
| [48] | T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 1971, 19, 7-15. · Zbl 0221.45003 |
| [49] | M. H. M. Rashid and A. Al-Omari, Local and global existence of mild solutions for impulsive fractional semilinear integro-differential equation, Commun. Non-linear Sci. Numer. Simul., 2011, 16, 3493-503. · Zbl 1223.45009 |
| [50] | X. Shu and Q. Wang, The existence and uniqueness of mild solutions for frac-tional differential equations with nonlocal conditions of order 1 < α < 2, Computers and Mathematics with Applications, 2012, 64, 2100-2110. · Zbl 1268.34155 |
| [51] | R. Sakthivel, Y. Ren, A. Debbouche and N. I. Mahmudov, Approximate control-lability of fractional stochastic differential inclusions with nonlocal conditions, Applicable Analysis, 2016, 95, 2361-2382. · Zbl 1350.93018 |
| [52] | W. Schmaedeke, Optimal control theory for nonlinear vector differential equa-tions containing measures, SIAM J. Control, 1965, 3, 231-280. · Zbl 0161.29203 |
| [53] | R. Sakthivel and E. Anandhi, Approximate controllability of impulsive differ-ential equations with state-dependent delay, International Journal of Control, 2009, 83(2), 387-393. · Zbl 1184.93021 |
| [54] | R. Sakthivel and E. Anandhi, Approximate controllability of impulsive differ-ential equations with state-dependent delay, Int. J. Control., 2010, 83, 387-493. · Zbl 1184.93021 |
| [55] | G. Shen, R. Sakthivel, Y. Ren and M. Li, Controllability and stability of frac-tional stochastic functional systems driven by Rosenblatt process, Collectanea Mathematica, 2020, 71(1), 6382. · Zbl 1450.34058 |
| [56] | A. Shukla, N. Sukavanam and D. N. Pandey, Approximate controllability of semilinear system with state delay using sequence method, Journal of The Franklin Institute, 2015, 352, 5380-5392. · Zbl 1395.93119 |
| [57] | C. C. Travis and G. F. Webb, Cosine families and abstract nonlinear second order differentail equations, Acta. Math. Hungar., 1978, 32, 75-96. · Zbl 0388.34039 |
| [58] | Z. Tomovski, Generalized cauchy type problems for nonlinear fractional differ-ential equations with composite fractional derivative operator, Nonlinear Anal-ysis, 2012, 75, 3364-3384. · Zbl 1241.33020 |
| [59] | G. Wang, L. Zhang and G. Song, Systems of first order impulsive functional differential equations with deviating arguments and nonlinear boundary condi-tions, Nonlinear Anal: TMA, 2011, 74, 974-982. · Zbl 1223.34091 |
| [60] | J. Wang, Y. Zhou and M. Feckan, On recent developments in the theory of boundary value problems for impulsive fractional differentail equations, Com-puters and Mathematics with Applications, 2012, 64, 3008-3020. · Zbl 1268.34032 |
| [61] | J. Wang, X. Li and W. Wei, On the natural solution of an impulsive fractional differential equation of order q ∈ (1, 2), Commun. Nonlinear Sci. Numer. Simul., 2012, 17, 4384-4394. · Zbl 1248.35226 |
| [62] | J. Wang, M. Feckan and Y. Zhou, Relaxed Controls for Nonlinear Frational Impulsive Evolution Equations, J. Optim. Theory. Appl., 2013, 156, 13-32. · Zbl 1263.49038 |
| [63] | J. Wang, M. Feckan and Y. Zhou, Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions, Evol. Equ. Control The-ory, 2017, 6(3), 471-486. · Zbl 06744040 |
| [64] | A. Wehbe and W. Youssef, Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks, J. Math. Phys., 2010, 51(10), Article ID 103523. · Zbl 1314.74035 |
| [65] | Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fractional evolution eqau-tions, Nonlinear analysis: Real World Application, 2010, 5, 4465-4475. · Zbl 1260.34017 |
| [66] | Y. Zhou, L. Zhang and X. Shen, Existence of mild solutions for fractioanl evolution equations, Journal of Integral Equations and Applications, 2013, 25, 557-586. · Zbl 1304.34013 |
| [67] | H. Zhou, Approximate controllability for a class of semilinear abstract equa-tions, SIAM J. Control Optim., 1983, 21(4), 551-565. · Zbl 0516.93009 |
| [68] | Y. Zhou and J. He, New results on controllablity of fractional evolution systems with order α ∈ (1, 2), Evolution Equations and Control Theory, 2019. |
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