Gou, Haide A study on approximate controllability of evolution systems with nonlocal conditions. (English) Zbl 07959964 Topol. Methods Nonlinear Anal. 64, No. 1, 87-106 (2024). Summary: The purpose of this paper is to present the existence of mild solutions and approximate controllability for a class of non-autonomous measure driven evolution systems with nonlocal conditions in Banach spaces. Firstly, we obtain the existence of mild solutions for the concerned problem by the semigroup theory and Schauder’s theorem. Secondly, some sufficient conditions of approximate controllability are proved. At the end, an example is also given to illustrate the feasibility of our theoretical results. MSC: 26A42 Integrals of Riemann, Stieltjes and Lebesgue type 34A38 Hybrid systems of ordinary differential equations 34K30 Functional-differential equations in abstract spaces 34K35 Control problems for functional-differential equations 93B05 Controllability Keywords:approximate controllability; evolution family; Lebesgue-Stieltjes integral; measure driven evolution equation × Cite Format Result Cite Review PDF Full Text: DOI References: [1] S.M. Abdal and S. Kumar, Existence of integro-differential neutral Mmasure driven system using monotone iterative technique and measure of Noncompactness, Differ. Equ. Dynam. Syst. (2022), DOI: 10.1007/s12591-022-00614-x. [2] G. Arthi and K. Balachandran, Controllability results for damped second-order impulsive neutral integro-differential systems with nonlocal conditions, J. Control Theory Appl. 11 (2013), 186-192. · Zbl 1299.93027 [3] G. Arthi and K. Balachandran, Controllability of damped second-order neutral functional differential systems with impulses, Taiwanese J. Math. 16 (2012), 89-106. · Zbl 1235.93042 [4] G. Arthi and J. Park, On controllability of second-order impulsive neutral integrodifferential systems with infinite delay, IMA J. Math. Control Inf. 2014 (2014), 1-19. [5] M. Benchohra, L. Gorniewicz, S.K. Ntouyas and A. Ouahab, Controllability results for impulsive functional differential inclusions, Rep. Math. Phys. 54 (2004), 211-228. · Zbl 1130.93310 [6] Y. Cao and J. Sun, Existence of solutions for semilinear measure driven equations, J. Math. Anal. Appl. 425 (2015), 621-631. · Zbl 1304.34015 [7] Y. Cao and J. Sun, Measures of noncompactness in spaces of regulated functions with application to semilinear measure driven equations, Bound. Value Probl. 2016, 38. · Zbl 1335.34092 [8] Y. Cao and J. Sun, On existence of nonlinear measure driven equations involving nonabsolutely convergent integrals, Nonlinear Anal. Hybrid Syst. 20 (2016b), 72-81. · Zbl 1341.34061 [9] Y. Cao and J. Sun, Approximate controllability of semilinear measure driven systems, Math. Nachr. 2018, 1-10. [10] Y. Cao and J. Sun, Controllability of measure driven evolution systems with nonlocal conditions, Appl. Math. Comput. 299 (2017), 119-126. · Zbl 1411.93027 [11] P. Chen, X. Zhang and Y. Li, Approximate controllability of non-autonomous evolution system with nonlocal conditions, J. Dyn. Control. Syst. 26 (2020), 1-16. · Zbl 1439.34065 [12] M. Cichon, and B.R. Satco, Measure differential inclusions-between continuous and discrete, Adv. Differ. Equ. 2014 (2014), 1-18. · Zbl 1350.49014 [13] P.C. Das and R.R. Sharma, Existence and stability of measure differential equations, Czechoslovak Math. J. 22 (1972), 145-158. · Zbl 0241.34070 [14] K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl. 179 (1993), 630-637. · Zbl 0798.35076 [15] T. Diagana, Semilinear Evolution Eqautions and Their Applications, Springer Nature Switzerland AG, 2018. · Zbl 1414.34002 [16] J. Diestel, W.M. Ruess and W. Schachermayer, Weak compactness in \(l^1(\mu, x)\), Proc. Amer. Math. Soc. 118 (1993), 447-453. · Zbl 0785.46037 [17] M. Federson, J.G. Mesquita and A. Slavik, Measure functional differential equations and functional dynamic equations on time scales, J. Differerential Equations 252 (2012), 3816-3847. · Zbl 1239.34076 [18] W.E. Fitzgibbon, Semilinear functional equations in Banach space, J. Differential Equations 29 (1978), 1-14. · Zbl 0392.34041 [19] X. Fu, Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay, Evol. Equ. Control Theory 6 (2017), 517-534. · Zbl 1381.34097 [20] X. Fu and R. Huang, Approximate controllability of semilinear non-autonomous evolutionary systems with nonlocal conditions, Autom. Remote Control 77 (2016), 428-442. · Zbl 1348.93049 [21] X. Fu and Y. Zhang, Exact null controllability of non-autonomous functional evolution systems with nonlocal conditions, Acta Math. Sci. Ser. B (Engl. Ed.) 33 (2013), 747-757. · Zbl 1299.34255 [22] R.K. George, Approximate controllability of non-autonomous semilinear systems, Nonlinear Anal. 24 (1995), 1377-1393. · Zbl 0823.93008 [23] R. Goebel, R.G. Sanfelice and A. Teel, Hybrid dynamical systems, IEEE Control Syst. Mag. 29 (2009), 28-93. · Zbl 1395.93001 [24] D. Henry, Geometric Theory of Semilinear Parabolic Equations Lecture Notes in Mathematics, Springer, New York, 1981. · Zbl 0456.35001 [25] E. Hernandez and D.O. Regan, Controllability of Volterra-Fredholm type systems in Banach spaces, J. Franklin Inst. 346 (2009), 95-101. · Zbl 1160.93005 [26] C.S. Honig, Volterra-Stieltjes Integral Equations, North-Holland, 1975. · Zbl 0307.45002 [27] J.M. Jeong, E.Y. Ju and S.H. Cho, Control problems for semilinear second order equations with cosine families, Adv. Difference Equ. 2016 (2016), 1-13. · Zbl 1419.35175 [28] S. Ji, G. Li and M. Wang, Controllability of impulsive differential systems with nonlocal conditions, Appl. Math. Comput. 217 (2011), 6981-6989. · Zbl 1219.93013 [29] R.E. Kalman, Controllablity of linear dynamical systems, Contrib. Diff. Equ. 1 (1963), 190-213. [30] R. Kronig, and W. Penney, Quantum mechanics in crystal lattices, Proc. Royal Soc. London 130 (1931) 499-513. · Zbl 0001.10601 [31] S. Kumar, On approximate controllability of non-autonomous measure driven systems with non-instantaneous impulse, Appl. Math. Comput. 441 (2023), 127695. · Zbl 1511.93019 [32] S. Kumar and S.M. Abdal, Approximate controllability of nonautonomous second-order nonlocal measure driven systems with state-dependent delay, Internat. J. Control 96 (2023), no. 4, 1014-1025. [33] S. Kumar and S.M. Abdal, Approximate controllability of non-instantaneous impulsive semilinear measure driven control system with infinite delay via fundamental solution, IMA J. Math. Control Inform. 38 (2021), 552-575. · Zbl 1475.93012 [34] S. Kumar and N. Sukavanam, Approximate controllability of fractional order semilinear systems with bounded delay, J. Differential Equations 252 (2012), 6163-6174. · Zbl 1243.93018 [35] N.I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control. Optim. 42 (2003), 1604-1622. · Zbl 1084.93006 [36] N.I. Mahmudov, Approximate controllability of evolution systems with nonlocal conditions, Nonlinear Anal. 68 (2008), 536-546. · Zbl 1129.93004 [37] J.G Mesquita, Measure Functional Differential Equations and Impulsive Functional Dynamic Equations on Time Scales, Universidade de Sao Paulo, Brazil, 2012, Ph.D. thesis. [38] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Berlin, Springer, 1983. · Zbl 0516.47023 [39] L. Di Piazza, V. Marraffa and B. Satco, Measure differential inclusions: existence results and minimum problems, Set-Valued Var. Anal. 29 (2021), 361-382. · Zbl 1479.34008 [40] J. Prus, Evolutionary Integral Equations and Applications, Birkhauser, Basel, 1993. · Zbl 0784.45006 [41] R. Sakthivel and E.R. Anandhi, Approximate controllability of impulsive differential equations with state-dependent delay, Internat. J. Control. 83 (2003), no. 2, 387-393. · Zbl 1184.93021 [42] R. Sakthivel, and E. Anandhi, Approximate controllability of impulsive differential equations with state-dependent delay, Internat. J. Control. 83 (2010), 387-493. · Zbl 1184.93021 [43] B. Satco, Regulated solutions for nonlinear measure driven equations, Nonlinear Anal. Hybrid Syst. 13 (2014), 22-31. · Zbl 1295.45003 [44] W. Schmaedeke, Optimal control theory for nonlinear vector differential equations containing measures, SIAM J. Control. 3 (1965), 231-280. · Zbl 0161.29203 [45] A. Shukla, N. Sukavanam and D.N. Pandey, Approximate controllability of semilinear system with state delay using sequence method, J. Franklin Inst. 352 (2015), 5380-5392. · Zbl 1395.93119 [46] A. Slavik, Measure functional differential equations with infinite delay, Nonlinear Anal. 79 (2013), 140-155. · Zbl 1260.34123 [47] H.X. Zhou, Approximate controllability for a class of semilinear abstract equations, SIAM J. Control Optim. 21 (1983), no. 4, 551-565. · Zbl 0516.93009 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.