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doi: 10.3934/eect.2020076

## Approximate controllability of nonlocal problem for non-autonomous stochastic evolution equations

 Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

* Corresponding author: Pengyu Chen

Received  November 2019 Revised  April 2020 Published  June 2020

Fund Project: The first author is supported by NSF of China (No. 11661071), Project of NWNU-LKQN2019-3 and China Scholarship Council (No. 201908625016); The second author is supported by the Science Research Project for Colleges and Universities of Gansu Province (No. 2019B-047) and Project of NWNU-LKQN2019-13

In this paper, we are considered with approximate controllability for a class of non-autonomous stochastic evolution equations of parabolic type with discrete nonlocal initial conditions. Some new results about existence of mild solutions as well as approximate controllability are established under more natural conditions on nonlinear functions and control operator by introducing a new Green function and using the theory of evolution family, Schauder fixed point theorem and the resolvent operator condition. At last, as a sample of application, these results are applied to a class of non-autonomous stochastic partial differential equation of parabolic type with discrete nonlocal initial conditions. The results obtained in this paper is a supplement to the existing literature and essentially extends some existing results in this area.

Citation: Pengyu Chen, Xuping Zhang. Approximate controllability of nonlocal problem for non-autonomous stochastic evolution equations. Evolution Equations & Control Theory, doi: 10.3934/eect.2020076
##### References:
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Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar [24] K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl., 179 (1993), 630-637.  doi: 10.1006/jmaa.1993.1373.  Google Scholar [25] K. Ezzinbi, X. Fu and K. Hilal, Existence and regularity in the $\alpha$-norm for some neutral partial differential equations with nonlocal conditions, Nonlinear Anal., 67 (2007), 1613-1622.  doi: 10.1016/j.na.2006.08.003.  Google Scholar [26] Z. Fan and G. Li, Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Functional Anal., 258 (2010), 1709-1727.  doi: 10.1016/j.jfa.2009.10.023.  Google Scholar [27] S. Farahi and T. Guendouzi, Approximate controllability of fractional neutral stochastic evolution equations with nonlocal conditions, Results. Math., 65 (2014), 501-521.  doi: 10.1007/s00025-013-0362-2.  Google Scholar [28] W. E. Fitzgibbon, Semilinear functional equations in Banach space, J. Differential Equations, 29 (1978), 1-14.  doi: 10.1016/0022-0396(78)90037-2.  Google Scholar [29] X. Fu, Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay, Evol. Equ. Control Theory, 6 (2017), 517-534.  doi: 10.3934/eect.2017026.  Google Scholar [30] X. Fu and R. Huang, Approximate controllability of semilinear non-autonomous evolutionary systems with nonlocal conditions, Autom. Remote Control, 77 (2016), 428-442.  doi: 10.1134/s000511791603005x.  Google Scholar [31] 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.  doi: 10.1016/S0252-9602(13)60035-1.  Google Scholar [32] W. Grecksch and C. Tudor, Stochastic Evolution Equations: A Hilbert Space Approach, Akademic Verlag, Berlin, 1995.  Google Scholar [33] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981.  Google Scholar [34] R. E. Kalman, Controllablity of linear dynamical systems, Contrib. Diff. Equ., 1 (1963), 189-213.   Google Scholar [35] J. Liang, J. H. Liu and T. J. Xiao, Nonlocal Cauchy problems governed by compact operator families, Nonlnear Anal., 57 (2004), 183-189.  doi: 10.1016/j.na.2004.02.007.  Google Scholar [36] J. Liang, J. H. Liu and T. J. Xiao, Nonlocal Cauchy problems for nonautonomous evolution equations, Commun. Pure Appl. Anal., 5 (2006), 529-535.  doi: 10.3934/cpaa.2006.5.529.  Google Scholar [37] K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman and Hall, London, 2006.  Google Scholar [38] Z. Liu and X. Li, Approximate controllability of fractional evolution systems with Riemann-Liouville fractional derivatives, SIAM J. Control Optim., 53 (2015), 1920-1933.  doi: 10.1137/120903853.  Google Scholar [39] N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim., 42 (2003), 1604-1622.  doi: 10.1137/S0363012901391688.  Google Scholar [40] X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Ltd., Chichester, 1997.  Google Scholar [41] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [42] R. Sakthivela, Y. Ren, A. Debbouchec and N. I. Mahmudovd, Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions, Appl. Anal., 95 (2016), 2361-2382.  doi: 10.1080/00036811.2015.1090562.  Google Scholar [43] K. Sobczyk, Stochastic Differential Equations with Applications to Physics and Engineering, Kluwer Academic Publishers, London, 1991. doi: 10.1007/978-94-011-3712-6.  Google Scholar [44] H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Marcel Dekker, New York, USA, 1997.  Google Scholar [45] I. I. Vrabie, Delay evolution equations with mixed nonlocal plus local initial conditions, Commun. Contemp. Math., 17 (2015), 1350035, 22pp. doi: 10.1142/S0219199713500351.  Google Scholar [46] J. Wang, M. Fečkan and Y. Zhou, Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions, Evol. Equ. Control Theory, 6 (2017), 471-486.  doi: 10.3934/eect.2017024.  Google Scholar [47] R. N. Wang, K. Ezzinbi and P. X. Zhu, Non-autonomous impulsive Cauchy problems of parabolic type involving nonlocal initial conditions, J. Integral Equations Appl., 26 (2014), 275-299.  doi: 10.1216/JIE-2014-26-2-275.  Google Scholar [48] R. N. Wang and P. X. Zhu, Non-autonomous evolution inclusions with nonlocal history conditions: Global integral solutions, Nonlinear Anal., 85 (2013) 180–191. doi: 10.1016/j.na.2013.02.026.  Google Scholar [49] X. Zhang, P. Chen, A. Abdelmonem and Y. Li, Fractional stochastic evolution equations with nonlocal initial conditions and noncompact semigroups, Stochastics, 90 (2018), 1005-1022.  doi: 10.1080/17442508.2018.1466885.  Google Scholar [50] X. Zhang, P. Chen, A. Abdelmonem and Y. Li, Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroups, Math. Slovaca, 69 (2019), 111-124.  doi: 10.1515/ms-2017-0207.  Google Scholar [51] H. X. Zhou, Approximate controllability for a class of semilinear abstract equations, SIAM J. Control Optim., 21 (1983), 551-565.  doi: 10.1137/0321033.  Google Scholar

show all references

##### References:
 [1] P. Acquistapace, Evolution operators and strong solution of abstract parabolic equations, Differential Integral Equations, 1 (1988), 433-457.   Google Scholar [2] P. Acquistapace and B. Terreni, A unified approach to abstract linear parabolic equations, Rend. Semin. Mat. Univ. Padova, 78 (1987), 47-107.   Google Scholar [3] H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations, 72 (1988), 201-269.  doi: 10.1016/0022-0396(88)90156-8.  Google Scholar [4] P. Balasubramaniam and J. P. Dauer, Controllability of semilinear stochastic evolution equations with nonlocal conditions, Int. J. Pure Appl. Math., 19 (2005), 281-296.   Google Scholar [5] L. Byszewski, Application of preperties of the right hand sides of evolution equations to an investigation of nonlocal evolution problems, Nonlinear Anal., 33 (1998), 413-426.  doi: 10.1016/S0362-546X(97)00594-4.  Google Scholar [6] L. Byszewski, Existence and uniqueness of a classical solutions to a functional-differential abstract nonlocal Cauchy problem, J. Math. Appl. Stoch. Anal., 12 (1999), 91-97.  doi: 10.1155/S1048953399000088.  Google Scholar [7] P. Chen, A. Abdelmonem and Y. Li, Global existence and asymptotic stability of mild solutions for stochastic evolution equations with nonlocal initial conditions, J. Integral Equations Appl., 29 (2017), 325-348.  doi: 10.1216/JIE-2017-29-2-325.  Google Scholar [8] P. Chen and Y. Li, Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math., 63 (2013), 731-744.  doi: 10.1007/s00025-012-0230-5.  Google Scholar [9] P. Chen and Y. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys., 65 (2014), 711-728.  doi: 10.1007/s00033-013-0351-z.  Google Scholar [10] P. Chen, X. Zhang and Y. Li, Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators, Fract. Calcu. Appl. Anal., 23 (2020), 268-291.  doi: 10.1515/fca-2020-0011.  Google Scholar [11] P. Chen and Y. Li, Nonlocal Cauchy problem for fractional stochastic evolution equations in Hilbert spaces, Collect. Math., 66 (2015), 63-76.  doi: 10.1007/s13348-014-0106-y.  Google Scholar [12] P. Chen, Y. Li and X. Zhang, On the initial value problem of fractional stochastic evolution equations in Hilbert spaces, Commun. Pure Appl. Anal., 14 (2015), 1817-1840.  doi: 10.3934/cpaa.2015.14.1817.  Google Scholar [13] P. Chen, X. Zhang and Y. Li, Approximation technique for fractional evolution equations with nonlocal integral conditions, Mediterr. J. Math., 14 (2017), Art. 226, 16pp. doi: 10.1007/s00009-017-1029-0.  Google Scholar [14] P. Chen, X. Zhang and Y. Li, Nonlocal problem for fractional stochastic evolution equations with solution operators, Fract. Calcu. Appl. Anal., 19 (2016), 1507-1526.  doi: 10.1515/fca-2016-0078.  Google Scholar [15] 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.  doi: 10.1007/s10883-018-9423-x.  Google Scholar [16] P. Chen, X. Zhang and Y. Li, Study on fractional non-autonomous evolution equations with delay, Comput. Math. Appl., 73 (2017), 794-803.  doi: 10.1016/j.camwa.2017.01.009.  Google Scholar [17] P. Chen, X. Zhang and Y. Li, A blowup alternative result for fractional nonautonomous evolution equation of Volterra type, Commun. Pure Appl. Anal., 17 (2018), 1975-1992.  doi: 10.3934/cpaa.2018094.  Google Scholar [18] P. Chen, X. Zhang and Y. Li, Fractional non-autonomous evolution equation with nonlocal conditions, J. Pseudo-Differ. Oper. Appl., 10 (2019), 955-973.  doi: 10.1007/s11868-018-0257-9.  Google Scholar [19] P. Chen, X. Zhang and Y. Li, Non-autonomous parabolic evolution equations with non-instantaneous impulses governed by noncompact evolution families, J. Fixed Point Theory Appl., 21 (2119), Art. 84, 17pp. doi: 10.1007/s11784-019-0719-6.  Google Scholar [20] P. Chen, X. Zhang and Y. Li, Non-autonomous evolution equations of parabolic type with non-instantaneous impulses, Mediterr. J. Math., 16 (2019), Art. 118, 14pp. doi: 10.1007/s00009-019-1384-0.  Google Scholar [21] J. Cui, L. Yan and X. Wu, Nonlocal Cauchy problem for some stochastic integro-differential equations in Hilbert spaces, J. Korean Stat. Soci., 41 (2012), 279-290.  doi: 10.1016/j.jkss.2011.10.001.  Google Scholar [22] R. F. Curtain and P. L. Falb, Stochastic differential equations in Hilbert space, J. Differential Equations, 10 (1971), 412-430.  doi: 10.1016/0022-0396(71)90004-0.  Google Scholar [23] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar [24] K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl., 179 (1993), 630-637.  doi: 10.1006/jmaa.1993.1373.  Google Scholar [25] K. Ezzinbi, X. Fu and K. Hilal, Existence and regularity in the $\alpha$-norm for some neutral partial differential equations with nonlocal conditions, Nonlinear Anal., 67 (2007), 1613-1622.  doi: 10.1016/j.na.2006.08.003.  Google Scholar [26] Z. Fan and G. Li, Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Functional Anal., 258 (2010), 1709-1727.  doi: 10.1016/j.jfa.2009.10.023.  Google Scholar [27] S. Farahi and T. Guendouzi, Approximate controllability of fractional neutral stochastic evolution equations with nonlocal conditions, Results. Math., 65 (2014), 501-521.  doi: 10.1007/s00025-013-0362-2.  Google Scholar [28] W. E. Fitzgibbon, Semilinear functional equations in Banach space, J. Differential Equations, 29 (1978), 1-14.  doi: 10.1016/0022-0396(78)90037-2.  Google Scholar [29] X. Fu, Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay, Evol. Equ. Control Theory, 6 (2017), 517-534.  doi: 10.3934/eect.2017026.  Google Scholar [30] X. Fu and R. Huang, Approximate controllability of semilinear non-autonomous evolutionary systems with nonlocal conditions, Autom. Remote Control, 77 (2016), 428-442.  doi: 10.1134/s000511791603005x.  Google Scholar [31] 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.  doi: 10.1016/S0252-9602(13)60035-1.  Google Scholar [32] W. Grecksch and C. Tudor, Stochastic Evolution Equations: A Hilbert Space Approach, Akademic Verlag, Berlin, 1995.  Google Scholar [33] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981.  Google Scholar [34] R. E. Kalman, Controllablity of linear dynamical systems, Contrib. Diff. Equ., 1 (1963), 189-213.   Google Scholar [35] J. Liang, J. H. Liu and T. J. Xiao, Nonlocal Cauchy problems governed by compact operator families, Nonlnear Anal., 57 (2004), 183-189.  doi: 10.1016/j.na.2004.02.007.  Google Scholar [36] J. Liang, J. H. Liu and T. J. Xiao, Nonlocal Cauchy problems for nonautonomous evolution equations, Commun. Pure Appl. Anal., 5 (2006), 529-535.  doi: 10.3934/cpaa.2006.5.529.  Google Scholar [37] K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman and Hall, London, 2006.  Google Scholar [38] Z. Liu and X. Li, Approximate controllability of fractional evolution systems with Riemann-Liouville fractional derivatives, SIAM J. Control Optim., 53 (2015), 1920-1933.  doi: 10.1137/120903853.  Google Scholar [39] N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim., 42 (2003), 1604-1622.  doi: 10.1137/S0363012901391688.  Google Scholar [40] X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Ltd., Chichester, 1997.  Google Scholar [41] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [42] R. Sakthivela, Y. Ren, A. Debbouchec and N. I. Mahmudovd, Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions, Appl. Anal., 95 (2016), 2361-2382.  doi: 10.1080/00036811.2015.1090562.  Google Scholar [43] K. Sobczyk, Stochastic Differential Equations with Applications to Physics and Engineering, Kluwer Academic Publishers, London, 1991. doi: 10.1007/978-94-011-3712-6.  Google Scholar [44] H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Marcel Dekker, New York, USA, 1997.  Google Scholar [45] I. I. Vrabie, Delay evolution equations with mixed nonlocal plus local initial conditions, Commun. Contemp. Math., 17 (2015), 1350035, 22pp. doi: 10.1142/S0219199713500351.  Google Scholar [46] J. Wang, M. Fečkan and Y. Zhou, Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions, Evol. Equ. Control Theory, 6 (2017), 471-486.  doi: 10.3934/eect.2017024.  Google Scholar [47] R. N. Wang, K. Ezzinbi and P. X. Zhu, Non-autonomous impulsive Cauchy problems of parabolic type involving nonlocal initial conditions, J. Integral Equations Appl., 26 (2014), 275-299.  doi: 10.1216/JIE-2014-26-2-275.  Google Scholar [48] R. N. Wang and P. X. 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